Difference between revisions of "2023 AMC 8 Problems/Problem 4"
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The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime? | The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime? | ||
| − | |||
<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math> | <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4</math> | ||
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==Solution== | ==Solution== | ||
| − | + | We fill out the grid, as shown below: | |
| + | <asy> | ||
| + | /* Made by MRENTHUSIASM */ | ||
| + | size(175); | ||
| + | |||
| + | void ds(pair p) { | ||
| + | filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); | ||
| + | } | ||
| + | |||
| + | ds((0.5,4.5)); | ||
| + | ds((1.5,3.5)); | ||
| + | ds((3.5,1.5)); | ||
| + | ds((4.5,0.5)); | ||
| + | |||
| + | add(grid(7,7,grey+linewidth(1.25))); | ||
| + | |||
| + | int adj = 1; | ||
| + | int curUp = 2; | ||
| + | int curLeft = 4; | ||
| + | int curDown = 6; | ||
| + | int curRight = 8; | ||
| + | |||
| + | label("$1$",(3.5,3.5)); | ||
| + | |||
| + | for (int len = 3; len<=7; len+=2) | ||
| + | { | ||
| + | for (int i=1; i<=len-1; ++i) | ||
| + | { | ||
| + | label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i)); | ||
| + | label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj)); | ||
| + | label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i)); | ||
| + | label("$"+string(curRight)+"$",(3.5-adj+i,3.5-adj)); | ||
| + | ++curDown; | ||
| + | ++curLeft; | ||
| + | ++curUp; | ||
| + | ++curRight; | ||
| + | } | ||
| + | ++adj; | ||
| + | curUp = len^2 + 1; | ||
| + | curLeft = len^2 + len + 2; | ||
| + | curDown = len^2 + 2*len + 3; | ||
| + | curRight = len^2 + 3*len + 4; | ||
| + | } | ||
| + | |||
| + | draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); | ||
| + | </asy> | ||
| + | From the numbers that appear in the shaded squares, <math>\boxed{\textbf{(D)}\ 3}</math> of them are prime: <math>19,23,</math> and <math>47.</math> | ||
| − | ~MathFun1000 | + | ~MathFun1000, MRENTHUSIASM |
==Video Solution by Magic Square== | ==Video Solution by Magic Square== | ||
Revision as of 03:14, 26 January 2023
Problem
The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number 7. How many of these four numbers are prime?
Solution
We fill out the grid, as shown below:
![[asy] /* Made by MRENTHUSIASM */ size(175); void ds(pair p) { filldraw((0.5,0.5)+p--(-0.5,0.5)+p--(-0.5,-0.5)+p--(0.5,-0.5)+p--cycle,mediumgrey); } ds((0.5,4.5)); ds((1.5,3.5)); ds((3.5,1.5)); ds((4.5,0.5)); add(grid(7,7,grey+linewidth(1.25))); int adj = 1; int curUp = 2; int curLeft = 4; int curDown = 6; int curRight = 8; label("$1$",(3.5,3.5)); for (int len = 3; len<=7; len+=2) { for (int i=1; i<=len-1; ++i) { label("$"+string(curUp)+"$",(3.5+adj,3.5-adj+i)); label("$"+string(curLeft)+"$",(3.5+adj-i,3.5+adj)); label("$"+string(curDown)+"$",(3.5-adj,3.5+adj-i)); label("$"+string(curRight)+"$",(3.5-adj+i,3.5-adj)); ++curDown; ++curLeft; ++curUp; ++curRight; } ++adj; curUp = len^2 + 1; curLeft = len^2 + len + 2; curDown = len^2 + 2*len + 3; curRight = len^2 + 3*len + 4; } draw((4,4)--(3,4)--(3,3)--(5,3)--(5,5)--(2,5)--(2,2)--(6,2)--(6,6)--(1,6)--(1,1)--(7,1)--(7,7)--(0,7)--(0,0)--(7,0),linewidth(2)); [/asy]](http://latex.artofproblemsolving.com/6/e/c/6ec6820e4734959c2e1ea32455e6e37e6ee38967.png)
From the numbers that appear in the shaded squares, 
of them are prime: 
and 
~MathFun1000, MRENTHUSIASM
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=5392
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
