Puzzles

Test your mettle with our staff-curated puzzles.

Missing Lights

There are 8 lights arranged in a circle. Each of them is randomly lit either red, green, or blue. What is the probability that there are six consecutive lights which are lit up in at most two colors?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Heads or Tails?

Winnie keeps flipping a coin until she gets either two heads in a row or three tails in a row, at which point she stops. What is the probability her final flip is heads?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Matching Cards

Two shuffled standard 52-card decks are placed side by side.

What is the reciprocal of the probability that at least one card is in the same position in both decks, to nine decimal places?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Ultimate End

Call a positive integer ultimate if it appears as the final digits of its square. For example, $25$ is ultimate because $25^2$ ends in $25.$ However, $125$ is not ultimate, since $125^2$ does not end in $125.$ How many positive integers less than $10$ million are ultimate?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Base Bedlam

For which positive integers n is the following equation true in base n?

$10^2 + 11^2 + 110^2 = 111^2$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Palindrome Pandemonium

A number is a palindrome in base n if its digits read the same forward and backward when written in that base.

How many numbers less than 1 million are palindromes in base 3? One such number is 23, written as 212 in base 3.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Happy New Year!

Using just the four operations of arithmetic and three $2021$ s, Grogg can make $2022$ as follows: $2021 + \frac{2021}{2021} = 2022.$
How can Grogg make $2022$ using the four operations of arithmetic, parentheses, and the fewest possible number of $17$ s?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Infinite Reciprocals

Winnie likes positive integers whose only prime factors are $7$ and $13.$ So, Winnie likes the numbers $1, 13, 49,$ and $91,$ but she does not like $14, 22,$ or $182.$

What is the sum of the reciprocals of all the numbers that Winnie likes?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Like Twenty-four

Using exactly one multiplication, one addition, one division, and one subtraction, what is the smallest positive number you can create using the numbers 4, 5, 6, 7, and 8 once each? For instance, we can form the number ((8÷5) x 7) - (4+6)= 1.2 in this way.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cubic Chess

Cubic chess is played on a 6 x 6 x 6 chessboard, where pieces occupy one of 216 individual cubes.

A queen attacks all cubes that are in a straight line parallel to an edge of the cube, in a diagonal line parallel to a face diagonal of the cube, or in a diagonal line parallel to a space diagonal of the cube (i.e. a diagonal connecting two opposite corners).

How many queens can you place on this cubic chessboard so that no two are attacking each other?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Unequal Equilateral Triangle

Given an equilateral triangle with ABC with side length 4, can you find points D, E, and F along its sides so that the areas x, y, and z are distinct positive integers.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Consecutive Christmas

Seven gifts are placed in a circle around the Christmas tree, and three of these gifts are chosen at random. What is the probability no two chosen gifts are adjacent with each other?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Age Conundrum

The sum of Dancer's, Dasher's, and Prancer's ages is 60. When Prancer was born, Dasher's age was double that of Dancer's. How old is Dancer today?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Diabolical Digits

Can you find a four-digit number $ABCD$ satisfying $ABCD= C \times D^{A}?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Polynomial Product

Suppose P(x) and Q(x) are two non-constant polynomials satisfying the equation P(Q(x)) = P(x) Q(x) for all x. What is Q(1)?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Composite Composition

For an integer n, the function τ(n) is the number of positive divisors of n. Among all positive integers n<1000, what is the greatest possible value of τ(τ(n))?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Geometric Sequence

Can you find a geometric sequence containing eight distinct 5-digit integers?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Simon's Favorite Rectangle Problem

How many rectangles with integer side lengths have the property that the numerical value of their area is 2021 times the numerical value of their perimeter?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Quasi-Fibonacci

Can you find integers $a$ and $b$ such that in the recursive sequence $F_n$ defined by $F_1 = a, F_2 = b,$ and $F_n = F_{n-1} + F_{n-2}$ for all integers $n>2$ , we have $F_{10}=2021?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Simple as a Square

What is the square root of the $4044$ -digit number $\underbrace{444\dots 44}_\text{2022 digits}\underbrace{888\dots88}_\text{2021 digits}9?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Easy as 1, 2, 3

A point $P$ is selected within square $ABCD$ so that $AP = 1, BP = 2,$ and $CP = 3,$ as shown in the diagram. What is the measure of $\angle APB?$

$[asy] import markers; size(250); pen s = fontsize(8); pair E = (1, 1); pair F = (1, -1); pair G = (-1, -1); pair H = (-1, 1); pair P = (.4, .2); draw(E--F--G--H--E); draw(P--E); draw(P--F); draw(P--G); pair X = (P+E)/2; pair Y = (P+F)/2; pair Z = (P+G)/2; dot(P); label("A", E, NE); label("B", F, SE); label("C", G, SW); label("D", H, NW); label("P", P, N); label("$1$", X, NW); label("$2$", Y, SW); label("$3$", Z, NW); markangle(n=1,radius=22, F, P, E, heavyred); [/asy]$

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Peculiar Polynomial

Suppose $P(x)$ is a polynomial with degree $2021$ so that $P(n) = \frac{1}{n}$ for all positive integers $n<2023.$ What is $P(2023)?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Infinite Roots

Which is greater: $2,$ or $\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \dots}}}}?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Squaring the Hexagon

Suppose the plane is tessellated by congruent hexagons as shown here. Can you find four vertices of this tessellation which create a square?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Listless Listing

Grogg lists five positive integers. Winnie then creates a new list by finding and recording the sum of each of the 10 pairs of Grogg's integers. If Winnie's list contains the integers 4, 40, 400, and 4000, then what is the largest integer that could possibly appear on Winnie's list?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Absolute Minimum

Can you find the minimum possible value of this expression as $x$ ranges over all real numbers?
$2|3x - 1| + 3|4x-1| + 4|5x-1| + ... + 99|100x-1|$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Egyptian Fractions

Can you find four positive integers $a,b,c,$ and $d$ so that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{20}{21}?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Angle-Bisector Bisection

In triangle $\triangle ABC,$ points $D, E,$ and $F$ are chosen along segment $BC$ so that $AD$ is an altitude, $AE$ is the interior angle bisector of angle $\angle BAC$ , and $F$ is the midpoint of $BC.$ Suppose that $AE$ also bisects the angle $\angle DAF.$ What is $\angle BAC?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Like Limbo

Let $\lceil x \rceil$ be the smallest integer greater than or equal to $x.$ Find $r$ given that $r \lceil r \lceil r \lceil r \lceil r \lceil r \rceil \rceil \rceil \rceil \rceil = 7.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tough Value

What is the value of the following expression? $\frac{1}{\sqrt{1+ \frac{1}{999^2} + \frac{1}{1000^2}}-1}$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Prime Time

Can you find all pairs of prime numbers $(p,q)$ such that $p^q + q^p$ is also a prime number? How do you know that you've found all of them?

Try your hand at solving this puzzle, and discuss your solution in this thread.

High Degree Difficulty

For which real numbers $N$ do there exist real numbers $x$ and $y$ such that $N = x^{2021} + y^{2021} = x^{2023} + y^{2023} = x^{2025} + y^{2025}?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Mathwalk Challenge: Mathscotch

Use the puzzle below for a Mathwalk challenge! After trying the puzzle yourself, chalk it out on the sidewalk, take a picture, and share it on social with #aopsmathwalk.

Mathwalk Challenge: Sum Shapes

Use the puzzle below for a Mathwalk challenge! After trying the puzzle yourself, chalk it out on the sidewalk, take a picture, and share it on social with #aopsmathwalk.

Count to N

Winnie and Alex play a game with a positive integer $N.$ Winnie counts all pairs of positive integers $(a,b)$ such that $a + b = N.$ Alex counts all pairs of positive integers $(c,d)$ such that $c^{-1} + d^{-1} = N^{-1}.$ Whoever has the higher count wins. For what numbers $N$ does Alex win?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Decimal Digits

How many $9$ 's are in the first $2021$ digits after the decimal point in the decimal expansion of $\frac{1}{9801}?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grazzling Grizzers

It takes Grogg 3 hours to fully grazzle a grizzer. It takes Winnie and Alex together 2 hours to grazzle a grizzer. One day, Alex grazzles a grizzer for 1 hour by himself, when Grogg and Winnie join him to finish grazzling the grizzer in 1 more hour. Who is the fastest grizzer-grazzler? You may assume beasts grazzle grizzers at fixed rates.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Picking Probability

Grogg picks a real number between 0 and 1. Alex picks a real number between 1 and 2. What is the probability that Alex's number is more than double Grogg's number?

Try your hand at solving this puzzle, and discuss your solution in this thread.

What's the Point?

Grogg is trying to draw an equilateral triangle $\triangle ABC$ with a special point $P$ inside the triangle, so that $AP = 1, BP=2,$ and $CP=3.$ Can he do it? If so, what's the side-length of his triangle? If not, why not?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Long Weekend!

Despising the short weekend, AoPS decided to invent Seconaturday: the Second Saturday, a day falling between Saturday and Sunday. The new calendar is completely the same with the typical 12 months spanning 365 days (except for leap years), although weeks now have 8 days total. If April 24th, 3000 is a Friday under the new calendar, how many Seconaturdays are in the year 3000?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Some Sums

If $x,y,$ and $z$ are positive real numbers that sum to $2021,$ what's the smallest possible value of $\frac{1}{x+16} + \frac{1}{y+32} + \frac{1}{z+64}?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Squares of Heptagons

A regular heptagon is divided as shown, where each variable or number denotes the length of the segment closest to it. What is $a^2+b^2+c^2?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Lovely Polygons

A polygon is lovely if all of its interior angles are equal and all of its side lengths are distinct. What is the smallest possible perimeter a lovely polygon could have, if all of its side lengths are integers?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Does It Average Out?

Can you find seven distinct integers that satisfy all of the following? Can you find six distinct integers?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Missing Digits

The six-digit number 4A8A2A is divisible by the two-digit number 8A for some digit A in base 10. What is A?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Hat Trick

In the line for a hat store, 10 people are wearing hats, each of which is either blue, green, or red. You notice every person standing behind someone wearing a green hat is themselves wearing a blue hat. Additionally, no two people wearing red hats are standing next to each other. How many possible orders are there for the colors of the hats in the line?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Find Four

What is the smallest n such that among any n integers, I can always find four integers whose sum is divisible by 4?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Numbering Numbers

Lizzie writes the following sequence on a paper. Her sequence has one 1, two 2's, three 3's, and so on, ending with one thousand 1000's. Lizzie crosses out every other term in this sequence, beginning with the first term. What is the sum of the numbers that Lizzie does not cross out?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Nice Triangles

A triangle is nice if the numerical values of all three of its side lengths and its area are distinct integers. If one side of a nice triangle has length 25, what is the smallest possible area it could have?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Quadratic Quiz

The quadratic equation $x^2 + mx + n = 1$ has roots $r$ and $s,$ while the quadratic equation $x^2 + rx + s = 1$ has roots $m$ and $n.$ Given that all of $m, n, r,$ and $s$ are non-zero real numbers, what is the greatest value any of them could be?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Perfectly Imperfect

Winnie likes numbers that are one more or one less than a perfect cube. Lizzie likes numbers that are perfect squares. How many positive integers less than 1,000,000 do either Winnie or Lizzie (or both Winnie and Lizzie) like?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Brain Breaker

This multiple choice question has at least one correct answer. If you guessed at random, what is the probability your answer would be incorrect?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Black Holes

How many paths are there from the rocket to the moon (located at the red vertices) if you can only move up or to the right along the blue edges, without falling into any black holes? (You may walk along the edges of the black holes.)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Multiples of the Year

What is the smallest positive multiple of 2021 whose digits are all distinct? What is the largest such number?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Inscribed Inscription

What has the larger area: A square inscribed in a square inscribed in a circle inscribed in a unit square, or a regular octagon inscribed in a square inscribed in a unit circle?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Divisibility Ability

What is the smallest positive integer with at least 25 positive divisors?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Circle Challenge

The figure pictured here is a quarter circle. The two smaller quarter circles within have the same area and are also tangent. Which area is greater, green or purple?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Cute Time

What is the probability that at a random time of day, the smaller angle formed between the hour and minute hands of an analog clock is less than 90°?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Rose Gold Ratio

Let the rose gold ratio be $\gamma = \frac{1 + \sqrt{13}}{2}$ . It turns out that $\frac{\gamma^9 - 651}{\gamma^3 - 3}$ is an integer. What number is it? How do you know?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cut Into Pieces

What is the largest number of enclosed regions you can create using any 3 lines and 3 circles? For example, this arrangement has 12 enclosed regions.

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Quadrilateral Quandary

Grogg draws a quadrilateral and its two diagonals. Lizzie connects the midpoints of opposite sides of the quadrilateral, and the midpoints of the two diagonals. Miraculously, all three of these lines intersect at the same point! Did Grogg and Lizzie get lucky, or will this work every time? Why?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Looking for a Match

Lizzie has three pairs of socks in different colors: one green, one blue, and one red. She grabs three socks randomly from her drawer. If they contain a pair of the same color, she wears it; if they are all different colors, she puts all three back. What is the probability she'll find a pair within her first three pulls?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tongue-Twister

If a trabble's nine trudds,
and a rabble's five rudds,
and a rudd and a trudd
are one half of a trabble,
then how many trudds
are in one hundred rabbles?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Two Much Fibonacci

What is the value of the infinite sum $\displaystyle \sum_{i=0}^{\infty} \frac{F_i}{2^i} = \frac{0}{1} + \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} +\frac{8}{64} + \cdots,$ where the numerators of each fraction form the famous Fibonacci sequence and the denominators of each fraction are successive powers of $2$ ?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Putting the Pentagon in Pentadecagon

A pentadecagon is a 15-sided polygon. How many pentagons can you construct by connecting 5 non-adjacent vertices in a regular pentadecagon?

Try your hand at solving this puzzle, and discuss your solution in this thread.

I Can't Believe It's Not Irrational!

The number $\sqrt[5]{7^5 + 43^5 + 57^5 + 80^5 + 100^5}$ is an integer. Without using a calculator, can you tell which integer it is? How do you know?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Counting Letters

Lizzie writes all the numbers from $1$ to $1{,}000$ on a chalkboard using words: $\text{one, two, three, ..., nine hundred ninety-nine, one thousand.}$
How many times does she write the letter $\textit{e}$ ?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cube Folding

You can cut the following shape out of a piece of paper, and fold it into a cube. Can you similarly cut and fold a $1 \times 7$ strip of paper (shown below) into a cube?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Data Difficulties

Grogg was collecting some data on social networks and noticed an interesting phenomenon. Among the $88$ beasts he collected data from, there were $2021$ pairs of beasts that were friends, and yet no three beasts were mutually friends with each other.

He explained his findings to Max, who thought about it for a minute before replying, "I think your data has an error!"

Is Max right? Why or why not?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Three Squares

In the figure below, two non-overlapping squares are placed inside a larger square. Why must the combined perimeters of the smaller squares be less than the perimeter of the largest square?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Loaded Dice

"Look!" Rosencrantz said. "I've got two dice that both have the numbers $1$ through $6$ on them. But they're loaded - on each die, some numbers have a higher probability of coming up than others. When I roll both of them, each of the sums from $2$ through $12$ has the same probability of coming up!"

"That's impossible!" Guildenstern exclaimed.

Who's right? Why?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tag

Winnie and Lizzie are playing tag in Beast Academy's circular playground, and Winnie is trying to catch Lizzie. Both beasts run at equal, constant speeds. Can Winnie always catch Lizzie in a finite amount of time?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Shaky Desk

"My new square teacher's desk is awful!" Ms. Q complained. "All four legs definitely have the same length, but the desk won't stop shaking!"

"Try rotating the desk about its center!" Professor Grok replied. "You'll definitely find a point where the desk stops shaking."

Is Professor Grok right? Why or why not?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tiling the AoPS Logo

The AoPS logo is a regular hexagon tiled with $27$ rhombi, each of which has two $60^{\circ}$ angles and two $120^{\circ}$ angles. There are three different rotational orientations for these tiles, and the $9$ tiles of each orientation are colored identically.

This is not the only way to tile the hexagon; you can see many others by looking at users' avatars on the AoPS Forums. However, any such tiling will always contain the same number of tiles of each orientation. Can you explain why?

Try your hand at solving this puzzle, and discuss your solution in this thread.

All But Two

Alex accidentally left his long division homework out in the rain. By the time he noticed, all but two of the numbers were unrecognizable! Can you help Alex figure out what this problem was supposed to read?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Perplexing Polynomial

Try your hand at solving this puzzle, and discuss your solution in this thread.

Squares in Squares

Can you fill in the 25 small squares below with the integers 1 to 25 once each, such that within each of the red and purple squares, the sum of the numbers in each row, column, and both main diagonals are the same?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Polygon Perimeters

Could the perimeter of Lizzie's new polygon be greater than the perimeter of her original polygon? Why or why not?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Doubling Function

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Scintillating Sequence

Prove that every number in the sequence below is composite.

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Gnarly Grid

Consider this grid of 25 squares.

Can you write one integer in each square so that the sum of the integers in each row is greater than 300, but the sum of the integers in each column is less than 200? Why or why not?

Can you write one integer in each square so that the product of the integers in each row is greater than 300, but the product of the integers in each column is less than 200? Why or why not?

Try your hand at solving this puzzle, and discuss your solution in this thread.

An Eccentric Equation

Find all real solutions to the following equation. Why aren't there any other solutions?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Five Quarter-Circles

Quarter-circles of the same color have equal areas. What's the ratio of the radius of the smallest quarter-circle to the radius of the largest quarter-circle?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grogg Tries to Skip School

Grogg said to his mom, "Why should I go to school when I don't have the time? I sleep $8$ hours a day, which adds up to about $122$ days per year. There's no school on weekends, which is $104$ days per year. We have $60$ days of summer vacation. I need $3$ hours a day to eat - that's $45$ days per year. And I need $2$ hours a day to have fun, or $30$ days per year. If I'm sick $4$ days per year, then there's no more days in the year, so I don't have time to go to school!"

Grogg's mom looked at Grogg's calculations and chuckled. "Your calculations aren't wrong," she said, "but that's no excuse not to go to school!"

What's wrong with Grogg's reasoning?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Dizzying Divisors

For every positive integer $n,$ let $\tau(n)$ be the number of divisors of $n.$ If $x$ and $y$ are positive integers, does the following equation have any solutions?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Tricky Triangle

The blue lines below divide an equilateral triangle with side length 7 into seven triangles with equal area. What is the combined length of the red segments?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Power Tower

What is the smallest positive integer $n$ such that

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Round Robin Paradox

In a round robin tournament, every player plays one game against every other player, and there are no ties. Is it possible that for every pair of people in a round robin tournament, there is some other player that wins against both of them? If so, what's the fewest number of people this could happen with?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Euclid's Game

Lizzie and Grogg are playing Euclid's Game. They start by writing two positive integers on a blackboard. Grogg goes first. On each player's turn, they must subtract some multiple of the smaller number from the larger number without making it negative, write that number on the board, and then erase the largest number from the board. The player that writes the integer $0$ wins.

For example, if it's Lizzie's turn and the numbers $81$ and $30$ are currently written on the blackboard, Lizzie could write down $81-30=51$ and erase $81,$ or write down $81-2 \cdot 30=21$ and erase $51.$

If Lizzie and Grogg start with the integers $123$ and $45,$ who has a winning strategy?

What if they start with the integers $m$ and $n$ ?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Prime Problem

Is this number a prime or composite? How do you know?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Odd Octagon

Which area is greater...purple or orange?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Drawing Doodles

Let's draw some doodles! Grab a blank piece of paper, and colored pencils or crayons in a few different colors. With a pencil, draw a doodle using the following rules:

Every stroke must be drawn using a continuous line or curve, without any retracing. A stroke may not intersect itself (though it may intersect other strokes).
Every stroke must either start and end at the same point, or start and end at different points along an edge or corner of the page.

Once you've drawn your doodle, color it in with as few colors as possible so that each region (bounded by pencil strokes) is colored in, and no two adjacent regions have the same color.

Try drawing a few doodles this way (or get someone else to contribute a doodle!). How are your doodles different? How are they similar? If you notice anything interesting, can you find a way to explain what's going on?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Dice Trick

Here's a math trick! Roll a pair of standard six-sided dice, and write down the following four products:

The product of the top numbers of the dice;
The product of the bottom numbers of the dice;
The product of the top number of the first die and the bottom number of the second die;
The product of the bottom number of the first die and the top number of the second die.

Add the four products. You should get $49$ !

Why does this trick work?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Dominono

Dominono is a two-player game played on a square grid, like the one below. Players take turns putting their mark on any vacant square, with one player using X's and the other using O's. The person who forms a domino - that is, who marks two squares that share an edge - loses.

Which player has a winning strategy on a $3\times 3$ board? How about a $6\times 6$ board? If you want a real challenge try thinking about a $7\times 7$ board.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Blanks

Fill in each blank below with a digit so that the equation is true and no digits are unnecessarily included. (For example, the rightmost digit cannot be a $3,$ as $12.3\overline{3}=12.\overline{3},$ so the digit in the blank would be unnecessarily included.)

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Word Store

At the word store, each vowel sells for a different price, but all consonants are free. The word "triangle" sells for $6, "square" sells for $9, "pentagon" sells for $7, "cube" sells for $7, and "tetrahedron" sells for $8. Try to find the longest word you can definitely buy for under $12!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cuckoo’s Egg

What number comes next in this famous sequence?

Try your hand at solving this puzzle, and discuss your solution in this thread.

WOW

How many whole numbers less than 10000 can be split into three numbers, each of which is a palindrome?

For example, you can split 113534 into 11 | 353 | 4 and each piece is a palindrome.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Data Divide

Grogg has a data set of 2n distinct points in the plane. Can he always draw a separating line so that there are n data points on either side of his line?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Queenpeace

Arrange n black queens and n white queens on a chessboard so that no black queen attacks a white queen. An example with n=4 is shown below.

Try to make n as big as you can!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Marker Problem 3

Five rectangles. The tilted black rectangle has area 16. What's the shaded area?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Polygon Symmetry

Start with a convex regular polygon $P.$ Make a new convex polygon $Q$ by using only some of the vertices of $P.$ Is it possible to make a polygon $Q$ in this way so that $Q$ has rotational symmetry, but no reflectional symmetry?

An example $P$ and $Q$ are shown below, with $Q$ drawn in red.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Marker Problem 2

Two copies of the same right triangle. What's the missing length?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Candyland

Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands.

Since Grogg is generous, he decides to give away his $n$ pieces of candy. He gives one piece of candy to each of the next $n$ beasts in line and then leaves the line.

The other beasts repeat this process: the beast in the front, who has $k$ pieces of candy, passes one piece each to the next $k$ beasts in line and then leaves the line.

For some values of $n,$ another beast (besides Grogg) temporarily holds all the candy. For which values of $n$ does this occur?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Marker Problem

What fraction of this rectangle is shaded?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Flip Side

Every card in this deck has a number on one side and a letter on the other. The same number can appear on more than one card.

Euna places four of these cards in a row, then flips over some (maybe all) of the cards and mixes them up. The before and after state is pictured below. What number is on the other side of the A?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Jewelry Design

10 silver beads and 10 gold beads are arranged randomly on a necklace. Is it always possible to make one straight line cut that divides the necklace into two pieces that each have 5 silver and 5 gold beads?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Weird Equations [Pt. 2]

Let weird(n) be the number of ways to write n as a + a + b + c, where n, a, b, and c are distinct positive integers. Is weird(n) an increasing function?

Surprisingly, weird(n) is not increasing... it sure seems like it should be! See if you can find a pattern for when weird(n) goes down. Can you explain what is going on?

Hint: the author wrote a script to print the first few values of n where weird(n) goes down and a few other things before the pattern became apparent! This problem is a good example of how computers can help us find patterns!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Last Year's Grid

Is it possible to fill in a 2020 x 2020 grid with the integers from 1 to 4,080,400 so that the sum of each row is 1 greater than the previous row? (USAMTS 2020)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Resigned Resolutions

Since I always fail my New Year's resolutions, this year, my New Year's resolutions are:

Make a new year's resolution
Fail my new year's resolution

Am I guaranteed to succeed? Guaranteed to fail? Something else?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Weird Equations

Let weird(n) be the number of ways to write n as a + a + b + c, where n, a, b, and c are distinct positive integers. Is weird(n) an increasing function?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Naughty or Nice

Christmas is over, and Santa needs to get to work preparing for next year. Remember, he needs to check his list twice after all. In order to check his list twice, he first writes the list, then puts an "x" next to anyone on the naughty list and a "p" next to anyone on the nice list. About how long can Santa spend per row on his list if he wants to be ready for next year? (This is an estimation problem — you don't have enough information to answer exactly!)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grogg's Snowmen

Grogg is building snowmen. He is very particular with his snowmen. He is only happy if each snowman he builds uses twice as much snow as either the snowman before it, or the snowman before that. The first two snowmen each used 1 cubic foot of snow. How many different possibilities are there for how many cubic feet of snow Grogg uses for the 10th snowman? How about the 100th snowman?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Happy New Year

Using the ice blocks below, construct a 3-digit number as follows: select a hundreds digit from the top block, a tens digit from the middle block, and a ones digit from the bottom block. Without reusing the same digit from a block, construct two more 3-digit numbers. For instance, you could create the numbers 967, 416, and 638. Find the sum of the three numbers. How do your choices affect the sum? Can you explain this?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Exaggerating Elf

Santa checks on his favorite three elves: Holly, Noel, and Kris. They have made 12 toys in total.

Santa says: "Good work. Who made the most toys? ''

Holly says: "I made twice as many as Noel.''

Noel says: "I made twice as many as Holly.''

Kris says: "I made as many toys as Noel and Holly combined.''

Santa deduces that exactly one elf is lying. Can you figure out who the liar is? Can you figure out who made the most toys?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Avoiding Fibonacci

Is it possible to split the positive integers into sets $A$ and $B,$ so that no two elements in $A$ add up to a Fibonacci number, and no two elements in $B$ add up to a Fibonacci number?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Santa's Flight Time

Santa needs to visit all of the houses below. The numbers between each pair of houses represent the time it takes for Santa to travel between those two houses (some houses have a lot of sky traffic between them!).

Try to find the shortest trip Santa can take that visits every house.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Favorite Number

Binary Bat, Ternary Toad, and Decimal Dog all have the same favorite number.

Decimal Dog says: "When I write down our favorite number, it has two digits!"

Ternary Toad says: "When I write down our favorite number, it has no 2s in it!"

Binary Bat says: "When I write down our favorite number, it has no 0s in it!"

What is their favorite number?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grogg's Secret Numbers

Grogg randomly picks two integers from 1 to 10. Lizzie writes down 3 numbers. She wins if she can guess at least one of Grogg's numbers.

What are the chances that Lizzie wins, and what is the average number of correct guesses Lizzie makes?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Student Puzzle: MaverickMan's Maze

(This problem was originally posted by one of our very own AoPS Community students, MaverickMan, and shared with their permission. Check out the message board for more of their puzzles.)

In the following maze, you start at the bottom green panel, and you have to get to one of the other 2 green panels. You move every one hour. However, if you touch a red panel, you move every 10 minutes, and if you touch a yellow panel, you move every 1 minute. What is the fastest path through the maze?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Boxes and Squares

Can you rearrange the boxes so that the sum of the numbers on any two neighboring boxes is a perfect square?

Try your hand at solving this puzzle, and discuss your solution in this thread.

R & G's Chord Game

Rosencrans and Gildenstern play a game. They have a circle with 30 points on it. On their turn, each beast (starting with Rosencrans) draws a chord between a pair of points in such a way that any two chords have a shared point (they either intersect or have a common endpoint). For example, two potential legal moves for the second player are drawn in with dotted lines.

The game ends when someone cannot draw a chord. The last person to make a move wins. Assuming they play perfectly, who do you think will win this game?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Baby Shark

The most-watched YouTube video of all time is now "Baby Shark", 2 minutes and 16 seconds of glorious singing. Doo doo doo doo doo doo. On November 2, 2020, the number of views of Baby Shark was 7.04 billion. On December 2, 2020, the number of views of Baby Shark was 7.37 billion. Given this information, how many people would you estimate are watching "Baby Shark" right now?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Impostor(s)

In a popular video game, 10 players compete, and 2 are randomly selected to be impostors.

As the game begins, the green player, who is not an impostor, says "red orange sus" with no knowledge of who is or isn't an impostor. What is the probability that

both red and orange are impostors?
at least one of red and orange is an impostor?

Try your hand at solving this puzzle, and discuss your solution in this thread.

One Ninth

The fraction $\frac 1 9$ has the infinite repeating decimal $0.\overline{111111}\dots$ . That means $\frac 1 9 = \frac 1 {10} + \frac 1 {100} + \frac 1 {1000} + \frac 1 {10000} + \cdots.$ Use this to compute $\left(\frac 19 \right)^2$ in two different ways. What does this tell you about the decimal representation of this number?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Pinball Wizard

A pinball shot is worth 1 million points. Every additional shot scores 500,000 more points than the one before. So, the second shot is 1.5 million points, the third shot is 2 million points, etc.

Grogg makes just enough shots to score a total of 100 million points, then walks away. How many points was Grogg's final shot worth?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Elusive Matsuura

The Fermat point is a point inside most triangles that forms three $120^\circ$ angles with segments to the three vertices.

A Matsuura triangle is a triangle whose side lengths are all integers, and whose three interior segment lengths from the $120^\circ$ point are also integers.

Find some Matsuura triangles, or prove they do not exist.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Leftovers

Grogg, Alex and Lizzie each bring a pie to their Thanksgiving dinner party. They each want to make one cut through every pie at the dinner table, which means the pies will have 3 cuts each.

Grogg suggested cutting his pie into 4 slices, like this. Alex said he would like to cut his pie into 6 slices. Lizzie found that there is a way to cut the pie into 7 slices!

It looks like there are many ways to cut these pies when they each made one cut!

Howard joined the party a bit later and bought another pie! So much pie! How many slices can they get if they each cut this pie?

Remember, this pie will be cut 4 times (since they will each be making one cut through the pie), but the cuts do not necessarily have to go through the center of the pie, and slices do not have to be the same size. The cuts do have to be straight and go all the way across the pie.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Four Corners

Four cups are placed on the four corners of a rotating table. Each glass is randomly face-up or upside-down. Your task is to turn them all in the same direction. Sounds too easy, so here's the catch:

you have to play blindfolded
you have to grab exactly two glasses at a time and optionally flip either of them over.
after you put the two glasses back, the table is spun randomly

Can you find a way to get all the glasses facing in the same direction?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Pepperoni Pizza Problem

The pizza below is covered with pepperoni. Can you make a straight cut to separate the pizza into two parts such that the ratio of the pepperoni on the two parts is 3:4? What other ratios can you get?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sense and Nonsense

Orangutoads tell the truth on Mondays, Wednesdays, Fridays, and Sundays. They lie on Tuesdays, Thursdays, and Saturdays.

Octapugs tell the truth on weekdays. They lie on weekends.

One day, an octapug and an orangutoad meet. The orangutoad says "I lied yesterday." The octapug replies, "so did I!" What day of the week is it?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Eight Hungry Broccolemurs

Six broccolemurs can eat six ounces of Thanksgiving leftovers in six minutes. How long will it take eight broccolemurs to eat eight ounces of Thanksgiving leftovers?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Pie Die Roll

Calamitous Clod and Winnie both want the last piece of pumpkin pie. Clod suggests a game to decide who gets it: they will each throw one of the dice shown below, and whoever gets the higher number gets the last piece.

"They all have different numbers on them," says Winnie. "Maybe you'll pick the best one."

"Okay," says Clod, "I'll let you pick first."

Should Winnie accept Clod's offer? If so, which die should she pick?

Here are the dice. They are drawn "unfolded" so you can see all the numbers.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Pie Logic

Six logicians have just finished their Thanksgiving repast, and are ready for dessert. Their host comes around and asks "Would all of you like some pumpkin pie?"

First logician: "I don't know."

Second logician: "I don't know."

Third logician: "I don't know."

Fourth logician: "I don't know."

Fifth logician: "I don't know."

Sixth logician: "No."

Who wants pumpkin pie?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Black Friday Strategy

Instead of going out to buy TVs on Black Friday, your friends choose to do a little geometry. You have the idea of the Shopper's Distance, where you measure how far away two aisle intersections are by counting the shortest number of aisles you must go through on a square grid.

For example, the two points below are a shopper's distance of 3 away from each other.

Can you place three friends in an equilateral triangle using shopper's distance? A square? Can you make a circle around an intersection point? What other shapes can you make, and how do they look different on Black Friday?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tryptophan

After eating too much turkey, Alex, Winnie, and Lizzie got very sleepy. They all fell asleep at exactly 8:00pm and have the following sleep patterns:

Lizzie sleeps for 5 hours at a time, wakes up, then immediately falls back asleep.
Winnie sleeps for 4 hours at a time, wakes up, then immediately falls back asleep.
Alex sleeps for 3 hours at a time, wakes up, then immediately falls back asleep.

The little monsters will all get up if they all happen to wake up at the same time. What time will it be when the three little monsters get up?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Picture Proof Jeopardy

This handy picture provides a "proof" of an interesting fact in arithmetic. Can you figure out what it is proving?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Clockwise

The numbers from $1$ to $2020$ are placed clockwise on a circle. We move around the circle clockwise erasing every other number until only one number remains. If the first number we erase is $1,$ what is the last remaining number on the circle?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Creepy Coincidence?

Grogg noticed some strange things last week. Help him determine if each observation is a coincidence or mathematically guaranteed to happen.

In Grogg's woodshop class, which has 27 people in it, 3 people were born in the same month of the year!
Grogg picked 11 numbers at random from 1 to 22. Two of the numbers added to 23!
Grogg and his friends (14 beasts in total) split 100 pieces of Halloween candy. Two of them ate the same number of pieces of candy!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Lo-Tech Baking

Alex and Lizzie are making brownies and need them to bake for exactly 15 minutes. Unfortunately, they only have an 11-minute hourglass and a 7-minute hourglass. Can you find a way for them to time exactly 15 minutes?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sideways Nim

Calamitous Clod has informed the little monsters that he will return Professor Grok if they can beat him in a game of Sideways Nim.

In a game of Sideways Nim, there are several piles of coins and each pile has some number of coins in it. On your turn, you can remove exactly one chip from any number of piles you want (with the caveat that you must remove at least one chip in total). The winner is the person who takes the final chip.

Since Calamitous Clod is very generous, he says he will let the little monsters choose how many piles of coins they want to play with and if they want to go first or second. Once they do this, a random number of coins will be placed in each pile and the game will begin.

Can you find a strategy to guarantee the little monsters have at least a $50\%$ chance of victory? How about $90\%?$ $99\%?$ $100\%?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Vexing Hexagon

In this regular hexagon, which is greater, purple or orange?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Two Groups of Three

Suppose that you're at a six-person party, where any two people are either acquaintances or strangers. Show that at this party, there must be two groups of three people who are either mutual acquaintances or mutual strangers. (These groups can overlap, and it's also possible to have one group of mutual acquaintances and one group of mutual strangers.) Must there be more than two such groups?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Dissecting the Heart

The following shape is formed by putting together three congruent squares. How many values of $n$ can you find such that this shape can be divided into $n$ identical shapes? (For example, $n=3$ works, since we can divide this figure into three identical squares.)

What if these shapes had to be smaller versions of the original shape?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tiling the Plane

There are three ways to tile the plane using only one regular polygon, by using regular hexagons, squares, or equilateral triangles.

How many different tilings of the plane can you find which use two or more regular polygons, all of which have the same side length?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Save Grogg

Oh no! Calamitous Clod has abducted Grogg and trapped him in the center of this maze.

Clod has given you the following rules for navigating the maze:
1. You must enter the maze from the arrow. The first tile you will step on is the tile in the bottom row labeled " $+1$ ".
2. Whenever you step on a tile, the next tile you step on must be that number of squares up, down, left, or right from the tile you're on. You cannot leave the maze. For example, the second tile you step on must either be one square up, left, or right from the bottom $+1$ tile.
3. You can only step on a tile once.
4. If you can reach Grogg and have the tile numbers you step on along the way sum up to $0,$ Calamitous Clod will set Grogg free!

Can you save Grogg?

Try your hand at solving this puzzle, and discuss your solution in this thread.

DIY Gerrymandering

In an upcoming election, each block votes for the yellow party or the green party, as shown below. The whole $5 \times 5$ grid is to be divided into five districts, so that each district consists of five connected blocks. For each district, the party with the most number of votes wins the district. Then the party who wins the most number of districts wins the election. Divide the grid into five districts, so that the yellow party wins the election.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sidewalk Options

How many ways are there to tile a $1\times 10$ rectangle with $1\times1$ squares and $1\times2$ dominos?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tug of War Tourney

Beast Academy is going to run a single elimination tournament to determine the best tug-of-war player. The tournament will consist of 1 vs 1 tug-of-war matches. Starting with 1000 competitors, how many matches will they need to play to determine the winner?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Knight Fight

How many knights can you place on an $8\times 8$ chess board so that no two attack each other? (It might be helpful to start with a $4\times 4$ board!)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Breaking the Bar

R&G are playing a game where you start with an $10\times 10$ chocolate bar, and players take turns choosing any piece and breaking it along a line into two pieces. You win if the other player can't make a move i.e. the bar is completely broken down into $1\times1$ pieces. Who has the winning strategy?

Try your hand at solving this puzzle, and discuss your solution in this thread.

3 out of 4

Wendell Willkie ran for President in 1940, but met only three of the four conditions for becoming President. Willkie was a natural-born citizen, at least 35 years old, and had been a resident of the US for at least 14 years. What was the fourth condition that Willkie was missing?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sum Equals Product

Howard bought three items at his local bodega, and when he added up the prices, it worked out to 9.96. Out of curiosity, Howard multiplied the prices, and to his surprise, the product was also 9.96! What were the prices of the three items that Howard bought?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Two Semicircles

A semicircle is constructed on line segment $\overline{AB}.$ Another semicircle is constructed on chord $\overline{CD},$ intersecting $\overline{AB}$ at $P$ and $Q.$ If $AP = 3,$ $PQ = 7,$ and $QB = 2,$ then find the length $CD.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Friends and Friends of Friends

There are ten people at a party. Any two people are either friends, or do not know each other. If there are three people A, B, and C such that A and B are friends, and A and C are friends, but B and C do not know each other, then we say that these three people form a Friend-of-Friend triple. What is the maximum number of Friend-of-Friend triples at the party?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Scheming

A rhyme scheme is A the first time,
Then is A again if it can rhyme,
But it switches to B,
If it changes, you see,
Then can go back to A (not A').

The rhyme scheme for the poem above is: AABBA.

Here are some other valid rhyme schemes for five-line poems:

AAAAA AABBB ABCDE ABBCA

Here are some things that are not valid rhyme schemes for five-line poems:

BABAB ABDAC ACBDE AACCA

How many possible rhyme schemes are there for five-line poems?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Kindred Polygons

Three regular polygons share and completely surround a vertex. What are some different options for the three regular polygons? How many can you find?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Area and Volume

A rectangular box has integer dimensions $a$ -by- $b$ -by- $c$ with $a \le b \le c$ . Its surface area, in square units, equals its volume, in cubic units.

What are some different options for the dimensions of the box? How many can you find?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Reciprocation

Three integers $a, b, c$ with $a \le b \le c$ have the property that their reciprocals add to exactly $\frac 1 2$ .

$\frac 1 a + \frac 1 b + \frac 1 c = \frac 1 2$ What are some different options for these integers? How many can you find?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Chords

On a unit circle, $n$ points are equally spaced. One point is selected, and chords are drawn from that point to the other $n-1$ points.

In terms of $n$ , what is the product of the lengths of these chords.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Algebeara's First Feast

Every bear attending the bear feast consumes exactly $6$ quarts of liquid, divided between porridge and honey. The newest bear, Algebeara, drank $\frac{2}{17}$ of the total amount of porridge and $\frac{1}{10}$ of the total amount of honey. At the end of the feast, all of the food has been consumed. How many bears are at the feast?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Reflecting on the Next Term

What's the next term in this sequence? (Yes, we know there are lots of possibilities, but what do you think should be next?)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Clod's Choice Claim

Calamitous Clod claims that for a certain choice of two numbers below, the product is nonnegative. Can you certify Clod's claim?

$\frac{65523246}{13580164} - 4.8249, \qquad 1.015 - \frac{3^{3^{\log \pi}}}{\pi^3}, \qquad \frac{2^{2^{2^{2^2}}}}{8^{50}} - 1.45$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Alex and His Primes

Alex found three numbers $a, b,$ and $c$ such that $a, b, c, a-b, b-c,$ and $a - c$ are all prime! What are $a, b,$ and $c?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Just Add the Reciprocal

I generate an infinite sequence as follows: The first term in my sequence is $1.$ Then, to get each following term in the sequence, I add the previous term in my sequence and its reciprocal. For example, the second term in my sequence is $1+\dfrac{1}{1}=2,$ and the third term in my sequence is $2+\dfrac{1}{2}=\dfrac{5}{2}.$

How many terms of my sequence will be integers?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sticky Squares

Eight sticks fill in this large red rectangle to create 5 squares total. How can you rearrange these sticks to create 7 squares total?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Lateral Quandary

The diagonals of this quadrilateral are perpendicular. The lengths of three of its sides, in some order, are 2, 3, and 4. What are the only two possible values for the fourth side of the quadrilateral?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sine

If A and B are both midpoints of this square, what is $\sin(\theta)$ ?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grid-addition

How can you place the numbers 1, 2, 3, 4, 5, and 6 in the six empty boxes so that each row and column of the square adds up to the same number?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sticky Triangle

A stick is randomly broken into 3 pieces. What is the probability that a triangle can be made from the three segments?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Orangutoad's Return

There are infinitely many lilypads in a single line. An orangutoad jumps 1 lilypad on its first jump, 2 lilypads on its second jump, 4 lilypads on its third jump, and so on, jumping a distance of $2^{n-1}$ lilypads on its $n$ th jump. Can you find a way for the orangutoad to return to its starting point? (The orangutoad can choose to jump in either direction on each jump.)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sum 25 Product

A set of positive integers has sum 25. What is the biggest you can make the product of the numbers?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Slow Motion

An object thrown into the air is called a projectile. The acceleration of a projectile on Earth's surface, ignoring air resistance, is
$a \approx -9.8 \;\mathrm{m/s^2}.$ Suppose you took a video of a projectile and played it at half speed. What would be the acceleration of the object in the video?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Triangles in a Pentagon

How many triangles can you find in the diagram below?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Deven's Eleven

Is there a positive integer that ends in 11 and has all of its prime factors less than 11?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Four Cards

Shawn has dealt four cards in front of you. He claims that if a card has an even number on one side of it, then the other side of the card is blue. Which cards do you need to turn over, in order to confirm if Shawn is telling you the truth?

Try your hand at solving this puzzle, and discuss your solution in this thread.

100 Toys

Bebe spent 100 dollars to buy 100 toys. A spinning top costs 50 cents, a pet rock costs 3 dollars, and a slinky costs 10 dollars. If Bebe bought at least one toy of each kind, then how many toys of each kind did she buy?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Honest Coin Flip

You and your friend have agreed to a large wager, which is to be settled by a coin flip. The problem is, you and your friend live in different cities. Your friend could flip a coin and tell you the result by phone, but since the stakes are so high, you don't trust your friend to give a truthful result. (Similarly, you could flip a coin, but your friend doesn't trust you!) Is there a way to flip a coin (or something equivalent) that both of you could trust?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Five Queens, Three Pawns

Place five queens and three pawns on a 5 x 5 chessboard, so that no queen attacks a pawn. Note: A queen attacks a pawn if they lie in the same row, column, or diagonal.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Venn Diagrams: Four or More

It is easy to draw a Venn Diagram for three sets. Is it possible to draw a Venn Diagram for four sets? How about five sets? How high can you go?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Every Day I'm Subtractin'

Pick a 3-digit number. Make the largest and smallest number you can from its digits. Then find the difference of these two numbers. Using this difference as the next 3-digit number, repeat this process.

For example, starting with 619, we first make 961 and 169, so we get the difference 961 - 169 = 792. Then, from 792, we make 972 and 279, recording the difference 972 - 279 = 693. We would then continue with 693, etc.

What happens in the end? Share your results!

Try your hand at solving this puzzle, and discuss your solution in this thread.

True or False?

Each statement is either true or false. How many of the following statements are true?

The answers to statements 2 and 3 are different.
The answers to statements 3 and 4 are different.
The answers to statements 4 and 5 are different.
The answers to statements 5 and 6 are different.
The answers to statements 6 and 7 are different.
The answers to statements 7 and 8 are different.
The answers to statements 8 and 1 are the same.
The answers to statements 1 and 2 are the same.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Pooch Problem

You're walking home, a distance of 1.5 km. You walk at 1.5 m/s. With you is your dog, who excitedly runs from you all the way home at a blazing speed of 9 m/s.

Since you aren't home yet, your dog turns around and runs back to you, again at 9 m/s. (There is no delay for your dog to change direction.)

Then your dog runs home again, then back to you, etc.

What is the total distance of all the back and forth trips your dog makes while waiting for you to finish walking home?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Shady Grid

Shade in some of the regions in the grid so that the shaded area is equal for each of the 11 rows and columns. Regions must be fully shaded or unshaded. At least one region must be shaded. The area of shaded regions must be at most half of the grid.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Rote's Representative

Sergeant Rote needs to pick a Beast Academy representative for the international tug-of-war championships. Sergeant Rote is willing to send anyone from the Beast Academy team, as long as he doesn't choose the weakest person on the team.

There are a total of 20 beasts, who each pull with a different constant strength. Sergeant Rote wants to design a tournament, with each round planned ahead of time, which will allow him to pick a representative for Beast Academy. Each round of the tournament is a 10-on-10 tug-of-war match. Each round may end in one side winning, or in a tie if the strengths on each side are equally matched.

How many games should Sergeant Rote schedule in the tournament? Try to schedule as few games as you can and still allow Sergeant Rote to pick a valid representative.

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Half Triangle Challenge

Calamitous Clod challenged Grogg to split the equilateral triangle below into two pieces of equal area using the shortest curve he could. Grogg's first idea was to draw a vertical line, like so.

Grogg's curve has length $\frac{\sqrt{3}}{2} \approx 0.866.$ Can you help Grogg find a shorter curve? What is the shortest possible curve that splits this equilateral triangle into two pieces of equal area?

Try your hand at solving this puzzle, and discuss your solution in this thread.

An Unlikely Pattern

Winnie got an answer of $\frac{1}{499}$ on her probability homework. She typed it into a calculator just to see what the decimal form was. She saw:

$\frac{1}{499} = 0.002004008016032064...$
What's going on there? Is that pattern a coincidence, or can you explain it?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Don't Flip Out

Alex has $25$ discs that are black on one side and white on the other. He arranges them on a $5 \times 5$ grid so that all the white sides are showing.

A move consists of taking any three consecutive discs in a row or column and flipping them over. You want the discs to make the checkerboard coloring shown below.

From the initial all-white position, what is the smallest number of moves needed to get to the checkerboard position?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Magic Star?

Can you place each number from 1 to 12 in the twelve circles below so that the sum along each of the 6 lines is equal?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Motley Crew

Snow White, Sirius Black, and Charlie Brown are wearing three solid-colored shirts, one each of white, black, and brown. "No one's shirt is the same color as their name," said the person in the brown shirt. "Wow, you're right!" replied Snow White.

Everyone is telling the truth. What color shirt is each person wearing?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Welcome to Kentucky

Fill each box below with a digit from 1 to 9 to make a true equation. Each digit must be used exactly once. For an extra challenge, try putting in your area code instead of 606 and see if you can still make a true equation!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Pace Yourself

In a 100-meter race, Amy, Brenda, and Calvin each run at a uniform pace throughout. If Amy beats Brenda by 10 meters and Brenda beats Calvin by 10 meters, by how many meters does Amy beat Calvin?

Try your hand at solving this puzzle, and discuss your solution in this thread.

On Brand

Can you divide the regular hexagon below into 12 congruent quadrilaterals? (A quadrilateral is a shape with 4 sides.)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Triangles in Triangles

A circle is inscribed in an equilateral triangle and an equilateral triangle is inscribed in that circle. What is the ratio of the areas of the two triangles?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Snail on a Cube

A snail starts in one corner of the unit cube and wishes to slide along the outside surface of the cube to the opposite corner. What is the shortest distance the snail would have to travel?

Bonus: What if instead the snail were traveling between opposite corners of a rectangular prism? Or an octahedron?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Decimal Dilemma

Fill in each of the blanks with a different non-zero digit. What digit is $x$ ?

If we replace the $3$ and $2$ with boxes as well, how many more solutions do we have?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Parentheses Puzzle

Can you add parentheses to the expression below to make a multiple of $50$ ? How about a perfect square?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Jars of Chocolate

Sophia has access to an unlimited number of jars of chocolate sauce. The jars each either contain 360 ounces of chocolate or 193 ounces of chocolate. Can you find a way for Sophia to measure exactly 1 ounce?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Infinite Fraction Fun

Can you write a simple expression to replace this infinite fraction?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Letter E

Each letter below is a variable. Compute the value of E.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sum Blob

Divide the square below into 4 different sections, each of which sums to the same value. For an added challenge, Calamitous Clod is covering three of the numbers that have the same value. What value is Calamitous Clod covering?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Parallelograms in a Grid

Pick four points on the grid below to form a parallelogram. Make sure that none of the sides intersect any other points in the grid (also, there should be no points in the interior of your parallelogram). What is the area? Can you get a different value for area if you add more rows/columns to the grid?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Birthday Seating Chart

To celebrate her birthday, Sophia throws a birthday party with Beast Academy characters. Grogg, Alex, Lizzie, Winnie, Calamitous Clod, and Professor Grok all attend. The cake is in the upper left corner of the table as pictured below. We know that:

Sophia sits closest to the cake;
Professor Grok and Calamitous Clod are as far away from each other as possible;
Grogg sits closer to the cake than Winnie does, and he sits further away from Clod;
Lizzie sits next to Sophia and across from Alex.

Where do each of the guests sit at the table?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Great Maze Escape

Begin at the shaded 7. Follow a skip-counting pattern* to escape by reaching the $75$. You can only move up, down, left, or right. Some of the numbers have been erased, but you can still use them if you figure out what they must be!

*A skip-counting pattern is a sequence of numbers with a constant difference between each of the terms. For example, if we started by going from the 7 to the 9, then the next square would need to contain an 11 = 9+2.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Transit Map of Beast Village

Below is a partial map of all the train stops in Beast Village. If the pattern continues, which train stop is directly west of train stop 100?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tip the Scale

A cup of water sits on a scale. You stick your finger into the cup of water, and the water does not overflow the cup. Your finger only touches the water, not the sides or bottom of the cup.

Does the reading on the scale go up when you put your finger in?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Three Circles and a Chord

A circle of radius $R$ has a chord of length $2t$ drawn. Two circles are inscribed above and below the chord as shown.

What is the area of the shaded region?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grid Town

Grid-town has a circular pond with radius $2$ and center $(0,4).$ There is also a horizontal river located on the $x$ -axis in Grid-town. Weven's tiny Grid-town home is located at $(7,20).$ Due to a recent drought, Weven needs to walk to the river and fill up a bucket, then walk to the pond and empty the bucket.

Of course, Weven is lazy, so he wants to walk as little as possible. Describe the shortest path that you can (draw a picture!) and find the length of your path.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Invisible Tetris

Fames is playing tetris with 1 x 3 blocks, on a grid that is 7 units wide.

Unfortunately, Fames is playing in invisible mode. He can't see the blocks he has already dropped or his total number of points.

Can you find a strategy that will eventually allow Fames to earn at least one more point?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Coin Flip Challenge

Two people each flip a coin. You have to guess the other person's result. If either person is right, you win. If both are wrong, you both lose. You can see the result of your own flip before guessing.

Before playing this game, you and your partner get to decide on a strategy you will use. Try to find a strategy that gives you as high of a probability of winning as you can!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Floating Ball

A uniform trough of water sits on a table with $51\%$ of the trough on the table and the other $49\%$ overhanging the edge.

A ball which is less dense than water is placed in the trough on the part over the top of the table, where it floats. The ball is pushed gently down to the far end of the trough. Will moving the ball down the trough in this manner run a risk of making it topple off the table?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Toothpick Puzzle

Grogg constructed the figure below out of toothpicks. Can you remove some toothpicks to leave exactly four rectangles in the picture? Try to do it by removing as few toothpicks as possible!

Try your hand at solving this puzzle, and discuss your solution in this thread.

Triangle in the Square

The square below has side length 1. There is a unique equilateral triangle that can be placed in this square with one vertex in the lower-left corner and the other two vertices on opposite sides, as shown. What is the area of this triangle?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cut the Net

A net for a polyhedron is cut along an edge to give two pieces. For example, we can cut a cube net along the red edge to form two pieces as shown.

Can you find two different polyhedra for which this process may result in the same two pairs of pieces?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Eight Numbers, Four Equations

Fill in the boxes with the numbers 1 through 8, using each number once, so that all the equations are true.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Half and Half

Kayla and Layla are twin sisters who have the same walking speed, and the same running speed. One day, they take a trip to the park. Kalya walks for half the distance, and runs for half the distance. Layla walks for half the time, and runs for half the time. Who arrives at the park first?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Folded Semicircle

A semicircle with a diameter of 6 is folded over a crease. The folded portion touches the diameter at a point that splits it into lengths of 4 and 2. Find the length of the crease.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Attack of the Knights

What is the smallest number of knights that must be placed on an $8 \times 8$ chessboard, so that all squares are under attack? Note: A knight also attacks the square it is on.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Three Piles of Coins

You are playing a game with three piles of coins. You are allowed to make the following move: You can move coins from one pile (say pile A) to another pile (say pile B), as long as you double the number of coins in pile B. Using these moves, is it always possible to make one of the piles empty?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Two Times Two

Solve the number puzzle below, where each letter stands for a different digit:

Try your hand at solving this puzzle, and discuss your solution in this thread.

Five Out of Six

Portia is thinking of four numbers. When she multiplies them in pairs, she gets six products. She tells you that five of these products are 2, 3, 4, 5, and 6. What is the sixth product?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Triangle in a Grid

Do there exist three lattice points in the coordinate plane, which form the vertices of an equilateral triangle? (A lattice point is a point where both coordinates are integers.)

Try your hand at solving this puzzle, and discuss your solution in this thread.

Magic Multiplication

You want to fill each square in the $4 \times 4$ grid below with a different positive integer, so that the product of the four numbers in each row, each column, and both diagonals is always the same constant. If $M$ is the largest number in the grid, how small can $M$ be?

Try your hand at solving this puzzle, and discuss your solution in this thread.

NEAR vs. EARN

In a dark corner at AoPS headquarters, a monkey is randomly pressing keys on a typewriter. If the monkey ever types the word NEAR, then it will be rewarded with a chocolate-covered banana. But if the monkey ever types the word EARN, then it will be immediately promoted and given a nice corner office. Which is more likely to occur first?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Magically Latin

Fill each square in the grid below with a number from 1 to 5, so that every row contains all the numbers from 1 to 5, every column contains all the numbers from 1 to 5, and both main diagonals contain all the numbers from 1 to 5. Some of the squares have already been filled in.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Points on a Pentagram

Ten points are marked in the pentagram below. Is it possible to color some of these ten points red, so that there are exactly three red points in each of the five lines of the pentagram?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Very Prime Product

A three-digit number and a two-digit number are multiplied. Each $p$ represents a prime digit and are not all necessarily the same. Determine the value of each digit.

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Fuse Problem

You have two fuses that look like ropes, and each fuse burns for exactly one minute. However, the fuses don't burn uniformly, so for example, 90% of a fuse might burn in the first 10 seconds, and then the remaining 10% burns for 50 seconds.

(a) Using the two fuses, measure an interval of 30 seconds.
(b) Using the two fuses, measure an interval of 15 seconds.

Try to generalize. What time intervals can you measure with $n$ fuses?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Two Rings in Space

Given a regular octahedron where all the edge lengths are 2, the incircle of one face is drawn (shown in red), and the circumcircle of an adjacent face is also drawn (shown in yellow). Find the shortest distance between a point on the red circle and a point on the yellow circle.

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Six-Digit Equation

(a) Enter the digits 1 through 6 into the boxes below, to get a true equation.
(b) Enter the digits 2 through 7 into the boxes below, to get a true equation.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cutting a Ring

What is the maximum number of pieces a ring can be cut into by 3 straight lines? In the example below, the ring is cut into 6 pieces.

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Egg Timers

I have two egg timers: One egg timer tells me when 7 minutes has passed, and the other egg timer tells me when 11 minutes has passed. Is it possible to use the egg timers to boil an egg for exactly 15 minutes?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Tetromino Tiles

Find the largest number of tetrominos below that can be placed on a $7 \times 7$ grid, without overlap. The tetromino can be rotated and/or reflected.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Divisibility Graph

For a positive integer $N \ge 3,$ we mark points labelled $3,$ $4,$ $5,$ $\dots,$ $N$ in the plane. We join two points with an edge if the label of one point divides the label of the other point. Find the largest $N$ for which it is possible to draw all the edges, so that no two edges intersect.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Twelve Circles

Place the numbers from 1 to 12 in the circles, so that the sum of the integers along each side of the square is 25.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Angle in a Square

In square $ABCD,$ $M$ is the midpoint of side $\overline{AB}$ and $N$ is the midpoint of side $\overline{BC}.$ Let $P$ be the intersection of $\overline{AN}$ and $\overline{DM}.$ Find $\angle BPM.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

What is my Polynomial?

I am thinking of a polynomial $p(x)$ with nonnegative integer coefficients. If you ask me, "What is the value of $p(n)?$ ", where $n$ is an integer, I will give you the answer. What is the minimum number of questions you need to ask me in order to determine $p(x)?$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Anyone's Game

Mr. X and Mrs. O are bored of playing regular tic-tac-toe, so they decide to come up with a variant: On any turn, a player can write either an X or an O. The first player to write three symbols in a row wins. In this variant, does either player have a winning strategy?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Harmonic Integers

Prove that
$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{n}$ is never an integer for $n > 1.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Twisted Square

Which is greater, teal or orange?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Everything Adds Up

Is it possible to enter the numbers from 1 to 9 into the boxes below, so that all the equations work?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Domino Problem

Can you cover this grid with $1 \times 2$ dominoes, without overlap?

Try your hand at solving this puzzle, and discuss your solution in this thread.

How to Tame Your Square Roots

Given positive real numbers a, b, and c, solve for x:

Try your hand at solving this puzzle, and discuss your solution in this thread.

Can You Trap the Monster?

A monster is moving in a straight line in the Cartesian plane at a constant speed so that every minute, it arrives at a lattice point (that is, a point with integer coordinates). Suppose that every minute you can throw a net at a lattice point that will trap the monster if it is there at that minute. You don't know where the monster started or where it is. Can you strategically throw nets in such a way that you are guaranteed to eventually trap the monster?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Star Stumper

In the star below, what is the sum of the angles marked in yellow?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Five Circles

The large circle has twice the diameter of each of the small circles, and none of the four orange areas overlap. Which area is greater, the total orange area or the total red area?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Cutting the AoPS Cube

Can you cut this $3 \times 3 \times 3$ cube into twenty-seven $1 \times 1 \times 1$ cubes using fewer than six straight cuts, each of which keeps individual $1 \times 1 \times 1$ cubes intact?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Magic Chessboard

I have a magic $8 \times 8$ chessboard: I can choose any row or column and swap the colors of every square in that row or column (so that every white square turns blue, and every blue square turns white). Can I turn the coloring in the left picture to that of the right picture?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Two Squares and a Semicircle

The diameter of the semicircle is 10. What's the total area of the two shaded squares?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Census Taker and the Children

A census taker walks up to a house, and knocks on the door. A woman answers.

Census taker: "Can you tell me about your children?"
Woman: "I have three children, and the product of their ages is 36. Also, the sum of their ages is the same as our house number."

The census taker observes the house number, and does some calculations.

Census taker: "I can't figure out the ages of your children."
Woman: "My oldest child plays the piano.
Census taker: "Now I know the ages of your children!"

How old are the children?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Not Quite a Venn Diagram

Which is greater: red or teal?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Revenge of the Decimals

Is the sum of all real numbers less than 1 with terminating decimal expansions finite or infinite?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Warped Tic-Tac-Toe

The game of Warped Tic-Tac-Toe is played just like regular Tic-Tac-Toe, but with two differences: Firstly, the game is played on an $n \times n$ grid instead of a $3 \times 3$ grid. Secondly, a player can win by making a warped diagonal - a diagonal that wraps around the sides of the grid. For example, when $n=5,$ X could win by placing five pieces into any of the following arrangements.

Can the player that goes second win Warped Tic-Tac-Toe?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Hexagonal Tiles

A triangle is composed of regular hexagons, with n hexagons on each side. The triangle for n = 6 is shown below.

For which values of n can the triangle be tiled by the tiles shown below?

Try your hand at solving this puzzle, and discuss your solution in this thread.

3 in a Line

Use this puzzle for a MathWalk! Chalk out the puzzle on the sidewalk, solve it, and take a photo to share in the comments! Then erase your work and leave the puzzle for the next walker to try.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Too Little Information?

The navy segment has length 100. Can you find the teal area?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Teal and Yellow Squares

Enter a number from 1 to 5 in each of the squares below, so that:

Each row contains the numbers 1, 2, 3, 4, 5.
Each column contains the numbers 1, 2, 3, 4, 5.
The sum of the numbers in the yellow squares is 38.
The product of the numbers in the teal squares is 86400.

Try your hand at solving this puzzle, and discuss your solution in this thread.

Counters in a Circle

We divide a circle into $n \ge 2$ sectors, and we place a counter in each sector. A move consists of choosing two counters, and moving each counter to an adjacent sector. (We can move the counters in the same direction, or in opposite directions.)

An example move is shown below. For which $n$ is it possible to move all the counters into the same sector?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Odd Coefficients

Show that for any positive integer $n,$ the number of odd coefficients in the expansion of $(x + y)^n$ is always a power of 2.

Can you determine which power of 2? Try to generalize your result, by finding the number of odd coefficients in the expansion of
$(x_1 + x_2 + \dots + x_k)^n.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

High-Low, High-Low, Round and Round We Go...

Pick your favorite three-digit number where not all the digits are the same. Put the digits in order from largest to smallest to form a number, and then in order from smallest to largest to form a second number (which might have leading zeros). Subtract the smaller number from the larger number. Then keep repeating this process with the number that you get.

For example, if my favorite three-digit number is 804, the number I'd get by putting the digits from largest to smallest is 840, and the number I'd get by putting the digits from smallest to largest is 048. Then, the number I'd get is 840-048=792, and I could repeat this process with the number 792.

What do you notice about the sequence of numbers you get? Can you explain your findings?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Four Points, Two Distances

Arrange four points in the plane, so that they determine only two different distances. For example, four vertices of a square will work, because there are only two different distances (teal and yellow).

How many other arrangements can you find?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Stymying Grid

In the grid below, what are a, b, and c?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Geometric Irrationality Proof

In the following sequence of pictures, all segments of the same color have the same length. How can you use these pictures to show that $\sqrt{2}$ is irrational?

Try your hand at solving this puzzle, and discuss your solution in this thread.

One Through Eight

Place the numbers 1-8 in the eight boxes below so that no two consecutive numbers are adjacent horizontally, vertically, or diagonally:

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Liars Club and the Very Large Number

At a meeting of the Liars Club, the president writes a large number N on the blackboard.

The first member says "N is divisible by 1."

The second member says "N is divisible by 2."

The third member says "N is divisible by 3," and so on, until the 31st member who says "N is divisible by 31".

Exactly two members were lying, and they spoke consecutively. Who were the two members who were not telling the truth?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Squares and Circles

Which is greater: teal or navy?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Bella’s Nine Cards

Bella has a set of nine cards, and each card has a number written on it. Note that there are $\frac{9 \cdot 8}{2} = 36$ ways to choose a pair of cards. Bella claims if you choose two of her cards and add the two numbers, the probability of getting that sum is the same as if you rolled two dice. (For example, the probability of getting a sum of 4 with Bella's cards would be the same as the probability of getting 4 with a pair of dice, which is $\frac{3}{36}.$ )

Can Bella be right? If so, what are the numbers on her cards?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Highly Divisible Cubic

Let $f(x) = ax^3 + bx^2 + cx + d,$ where $a,$ $b,$ $c,$ $d$ are integers and $a \neq 0,$ and let $n$ be a positive integer such that
$f(1) \mid f(2), \ f(2) \mid f(3), \ \dots, \ f(n - 1) \mid f(n).$ What's the highest value of $n$ that you can get?

Note: The notation $a \mid b$ means $b = ka$ for some integer $k.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

MATHCOUNTS Week: 2019 State Team #5

Mr. Schwin has a large jar containing M&Ms, each with the letter “m” stamped on it. He removes 1000 candies from the jar and removes the letter “m” from each one. He then returns all of the M&Ms to the jar. After thoroughly mixing up the candies in the jar, he randomly removes 1000 candies from the jar and finds that 245 of them do not contain the letter “m”.

What is the expected number of M&Ms in the jar?

Try your hand at solving this puzzle, and discuss your solution in this thread.

MATHCOUNTS Week: 2018 State Sprint #23

Bryan visits a carnival booth where Carl shows him 10 boxes. Exactly one of the boxes contains a gold coin; the other boxes are empty. Bryan randomly takes one of the boxes, but he doesn’t open it. Carl then opens five other boxes that he knows are empty and shows Bryan that they are empty. Carl then tells Bryan he can either keep his initially chosen box or return it and choose one of the remaining closed boxes instead.

If Bryan chooses to return his box and choose another one instead, what is the probability Bryan will choose the box with the gold coin?

Try your hand at solving this puzzle, and discuss your solution in this thread.

MATHCOUNTS Week: 2017 State Target #4

The addition table shown has rows and columns labeled with integers a, b, c and d, in that order. A few of the sums in the table are already filled in; for example, the table shows that a+d = n – 2.

When all sixteen sums are filled in, what is the sum of the sixteen entries in the table, in terms of n?

Try your hand at solving this puzzle, and discuss your solution in this thread.

MATHCOUNTS Week: 2016 State Sprint #29

In the list of numbers 1, 2, …, 9999, the digits 0 through 9 are replaced with the letters A through J, respectively. For example, the number 501 is replaced by the string “FAB” and 8243 is replaced by the string “ICED”. The resulting list of 9999 strings is sorted alphabetically.

How many strings appear before “CHAI” in this list?

Try your hand at solving this puzzle, and discuss your solution in this thread.

MATHCOUNTS Week: 2015 State Team #2

Using the figure of 15 circles shown, how many sets of three distinct circles A, B, and C are there such that circle A encloses circle B encloses circle C?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Party Problem

Leif and their partner invited 5 couples over for a party. Some of the people at the party met each other for the first time (but, of course, no one met themselves or their partner for the first time). At the end of the party, Leif asked everyone else how many new people they met at the party, and got 11 different answers. How many people did Leif meet at the party?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Sperry the Spinning Spider

Sperry the spider is practicing spinning a web, and wants to copy the spider's web below, which has 16 dots with 28 line segments connecting them. Sperry wants to start at one of the dots and spin a web from dot to dot along the given line segments, spinning across every line segment at least once. However, Sperry doesn't have a lot of thread, so they want to do this crossing the fewest number of line segments possible.

What's the shortest route you can find for Sperry? (Sperry can visit a dot or cross a line segment more than once.)

Try your hand at solving this puzzle, and discuss your solution in this thread.

#FlattenTheCurve

On the first day back at school, a math teacher gets their students to meet and greet each other by shaking hands. They specify that only one handshake is allowed between any two people, no one may shake hands with themselves, and all handshakes are between two people at a time. After a while, they say, "Stop - now write down on the board how many hands you shook. Then go wash your hands!" The students wrote down the following numbers on the board.

Can these students count?

$7, 4, 7, 2, 4, 10, 4, 4, 3, 5, 3, 4, 4, 2, 2.$

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Connected Network

Icarus Airlines is a new company vying for a place in the competitive airline market. Their CEO makes the following announcement: "We'll get you to your destination as fast as possible - we're only going to have direct flights that work in both directions. We're going to start flying from 25 cities, and fly to 12 cities from every city we serve. That may not sound like a lot, but we promise you can get between any of our 25 cities on Icarus Airlines flights."

Can the company uphold the CEO's promise?

Try your hand at solving this puzzle, and discuss your solution in this thread.

An Expanding Network

Icarus Airlines is expanding. Their CEO makes the following announcement: "Business has been good, and the time is ripe to expand the network of cities we serve. We will uphold our mission of only flying direct flight routes that work in both directions, but we're going to make sure we only run the flights that see the most demand. In particular, we won't run all three possible direct flight routes between any three cities we serve. We promise that by the end of our expansion, we'll be running a total of 650 different direct flight routes and serve 50 different cities."

Can the company uphold the CEO's promise?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Student Puzzle: Liars and Truth-tellers

(This problem was originally posted by one of our very own AoPS Community students, LivelyQuotient, and has been featured here with their permission.)

You come across a fork in the road, in an unknown country. The only thing you know is that some people tell the truth, and some people lie. You are trying to get to the capitol, but you have no map. There are two people there, and when you ask them how to get to the capitol, here's what they tell you:

The girl on the left says, "If you asked her, she'd tell you to take the right path."

The girl on the right glares at her and says, "She's lying!"

You thank them, and continue on your way. Which path do you take?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Three Semicircles

The area of the teal semicircle is 1. What's the navy area?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Star of David

Can you place the numbers 1 through 12 in the circles below so that the four numbers in each line add up to 26?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Ribbit! Ribbit!

Hey, remember these frogs from a few weeks ago? Well, some purple-eyed frogs came to join the party.

The red-eyed frogs and blue-eyed frogs want to trade places so that the group of red-eyed frogs is sitting on the right two lily pads, the group of blue-eyed frogs is sitting on the left two lily pads, and the two purple-eyed frogs are back where they started. All the frogs can move to the left or right, but they're pretty restricted in how they move: Each frog can either slide from one lily pad to an empty lily pad next to it, or jump over a frog of the other group if there is an empty lily pad behind that frog.

What's the smallest number of moves you can do this in?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Bouncing Ball

A bouncy ball is fired at a 45 degree angle from one corner of a (frictionless) metal box measuring 209 cm by 161 cm. How many times will it rebound off the sides of the metal box before it reaches a corner? Can you generalize?

Try your hand at solving this puzzle, and discuss your solution in this thread.

AoPS is Not Responsible for Your Broken Picture Frames

If you hang a picture with a string looped around two nails, as shown below, removing either nail will not cause the picture to fall.

Can you find a way to hang a picture with a string looped around two nails so that the picture would fall if either one of the nails were removed?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Triangle Sums

Place the numbers 1 through 9 in the circles below, so that the four numbers in each line add up to 20.

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Rhombus Riddle

What fraction of the rhombus is shaded?

Try your hand at solving this puzzle, and discuss your solution in this thread.

We're Not in Hanoi Anymore

Four blocks are situated on one of three towers, as shown below. Holes are drilled through each block, so that each block fits perfectly on the tower at its current position. A move consists of moving the top block of a tower to another tower.

A possible first move is shown below in the left image. You can move a block as long as it can fit, but a block cannot exceed the height of the tower. For example, in the right image, block 4 can be moved to the middle tower, but not to the right tower.

Your goal is to move all four blocks to another tower. What is the smallest number of moves you can do this in? Can you generalize this to n blocks?

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Confounding Clock

I have a weird clock: It tells me whether the current time is in A.M. or P.M., but the hour and minute hands are identical. Are there times during the day when I can't tell the time by looking at my clock? How many times does this occur in a 24-hour day?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Not Quite Fermat's Last Theorem

If $x,y,z$ are positive integers, do there exist any solutions to the equation $x^5+y^6=z^{11}$ ?

Can you generalize?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Winding Road

Which color is greater: teal or navy?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Ribbit!

There are three red-eyed frogs and three blue-eyed frogs sitting in a row of lily pads with a single empty lily pad separating the two groups. The two groups of frogs want to trade places so that the group of red-eyed frogs is sitting on the right three lily pads, and the group of blue-eyed frogs is sitting on the left three lily pads. Unfortunately, though, the frogs are pretty restricted in how they move:

Red-eyed frogs can only move to the right, and blue-eyed frogs can only move to the left.
Each frog can either slide from one lily pad to an empty lily pad next to it, or jump over a frog of the other group if there is an empty lily pad behind that frog.

What's the minimum number of moves required for the frogs to trade places? Why? What if more frogs come along?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Increasing Distances

Fill the teal squares with the numbers 1 through 15, so that the distance between squares 1 and 2 is less than the distance between squares 2 and 3, which is less than the distance between squares 3 and 4, and so on, up to the distance between squares 14 and 15.

Try your hand at solving this puzzle, and discuss your solution in this thread.

The Points Don't Matter

1000 points are placed on the plane. Can you always draw a line l such that the two regions on either side of l (excluding l) contain exactly 500 points each?

Try your hand at solving this puzzle, and discuss your solution in this thread.

A Peculiar Product

The expression $\dfrac{p}{p-1}$ gets closer and closer to $1$ as $p$ gets larger. But is the infinite product
$\displaystyle\prod_{p \text{ prime}} \dfrac{p}{p-1} = \dfrac{2}{1} \cdot \dfrac{3}{2} \cdot \dfrac{5}{4} \cdot \dfrac{7}{6} \cdot \dfrac{11}{10} \cdots$ finite or infinite?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Mushroom Puzzle

Which color is greater: teal or navy?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Grid Problem

Each symbol below (circle, triangle, square) has a weight. The number to the right of a row or below a column represents the total weight of the symbols in that row or column. What total weight should replace the question mark (?) in the third row?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Digit Sorting

Can you sort the digits 1 through 9 into the top or bottom row, using each digit exactly once, so that the product of the digits in the top row equals the sum of the digits in the bottom row?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Numbercross

Every box contains a digit such that when the numbers in each row or column are read from left-to-right or top-to-bottom, they fit the clues above. Can you solve the puzzle? What's 5-Across?

Across
1 $5^3$
4 A multiple of $9$
5 The smallest three-digit number in the grid

Down
2 A multiple of $7$
3 A power of $2$

Try your hand at solving this puzzle, and discuss your solution in this thread.

Bot Golf

In the Bot Golf Challenge, we give you a starting number of golf balls, and a target number of golf balls. You need to make a combination of Bots that turns the starting number into the target number.

Below are three different Bots we can use. How few Bots can you use to get from 100 —> 99?

Try your hand at solving this puzzle, and discuss your solution in this thread.

Keep Learning With Our AoPS Puzzle Library