2020 AMC 10B Problems/Problem 23

The following problem is from both the 2020 AMC 10B #23 and 2020 AMC 12B #19, so both problems redirect to this page.

Problem

Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:

$\quad\bullet\qquad$ $L,$ a rotation of $90^{\circ}$ counterclockwise around the origin;

$\quad\bullet\qquad$ $R,$ a rotation of $90^{\circ}$ clockwise around the origin;

$\quad\bullet\qquad$ $H,$ a reflection across the $x$ -axis; and

$\quad\bullet\qquad$ $V,$ a reflection across the $y$ -axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)

$\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$

Solution 1

For each transformation:

Each labeled vertex will move to an adjacent position.

The labeled vertices will maintain the consecutive order $ABCD$ in either direction (clockwise or counterclockwise).

$L$ and $R$ will retain the direction of the labeled vertices, but $H$ and $V$ will alter the direction of the labeled vertices.

After the $19$ th transformation, vertex $A$ will be at either $(1,-1)$ or $(-1,1).$ All possible configurations of the labeled vertices are shown below: $[asy] /* Made by MRENTHUSIASM */ unitsize(7mm); label("$A$",(1,0)); label("$B$",(1,1)); label("$C$",(0,1)); label("$D$",(0,0)); label("$C$",(5,0)); label("$D$",(5,1)); label("$A$",(4,1)); label("$B$",(4,0)); label("$A$",(9,0)); label("$D$",(9,1)); label("$C$",(8,1)); label("$B$",(8,0)); label("$C$",(13,0)); label("$B$",(13,1)); label("$A$",(12,1)); label("$D$",(12,0)); label("\textbf{Configuration}",(-5,0.5)); label("\textbf{The 20th}",(-5,-1.5)); label("\textbf{Transformation}",(-5,-2.25)); label("$L$",(0.5,-1.875)); label("$R$",(4.5,-1.875)); label("$H$",(8.5,-1.875)); label("$V$",(12.5,-1.875)); [/asy]$ Each sequence of $19$ transformations generates one valid sequence of $20$ transformations. Therefore, the answer is $4^{19}=\boxed{\textbf{(C)}\ 2^{38}}.$

~MRENTHUSIASM

Solution 2

Let $(+)$ denote counterclockwise/starting orientation and $(-)$ denote clockwise orientation. Let $1,2,3,$ and $4$ denote which quadrant $A$ is in.

Realize that from any odd quadrant and any orientation, the $4$ transformations result in some permutation of $(2+, 2-, 4+, 4-).$

The same goes that from any even quadrant and any orientation, the $4$ transformations result in some permutation of $(1+, 1-, 3+, 3-).$

We start our first $19$ moves by doing whatever we want, $4$ choices each time. Since $19$ is odd, we must end up on an even quadrant.

As said above, we know that exactly one of the four transformations will give us $(1+),$ and we must use that transformation.

Thus, the answer is $4^{19}=\boxed{\textbf{(C)}\ 2^{38}}.$

Solution 3

Notice that any pair of two of these transformations either swaps the $x$ and $y$ -coordinates, negates the $x$ and $y$ -coordinates, swaps and negates the $x$ and $y$ -coordinates, or leaves the original unchanged. Furthermore, notice that for each of these results, if we apply another pair of transformations, one of these results will happen again, and with equal probability. Therefore, no matter what state after we apply the first $19$ pairs of transformations, there is a $\frac14$ chance the last pair of transformations will return the figure to its original position. Therefore, the answer is $\frac{4^{20}}4 = 4^{19} = \boxed{\textbf{(C)}\ 2^{38}}.$

Solution 4

The total number of sequence is $4^{20}=2^{40}.$

Note that there can only be even number of reflections since they result in the same anti-clockwise orientation of the verices $A,B,C,D.$ Therefore, the probability of having the same anti-clockwise orientation with the original arrangement after the transformation is $\frac{1}{2}.$

Next, even number of reflections mean that there must be even number of rotations since their sum is even. Even rotations result only in the original position or $180^{\circ}$ rotation of it.

Since rotation $R$ and rotation $L$ cancels each other out, the difference between the numbers of them define the final position. The probability of the transformation returning the vertices to the original position given that there are even number of rotations is equivalent to the probability that

$|n(R)-n(L)|\equiv0\pmod{4}$ when $|n(H)-n(V)|\equiv0\pmod{4}$ or $|n(R)-n(L)|\equiv2\pmod{4}$ when $|n(H)-n(V)|\equiv2\pmod{4}$

which is again, $\frac{1}{2}.$

Therefore, $2^{40}\cdot\frac{1}{2}\cdot\frac{1}{2}=\boxed{\textbf{(C)}\ 2^{38}}.$

~joshuamh111

Video Solutions

Video Solution 1

https://www.youtube.com/watch?v=XZs9DHg6cx0

~MathEx

Video Solution 2 by The Beauty of Math

https://youtu.be/Bl2kn9oVxQ8?t=348

~Icematrix

2020 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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 AMC 10 Problems and Solutions

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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 AMC 12 Problems and Solutions

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