2020 AMC 10B Problem 23

Problem 23 of 25HarderGeometry Number Theory

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: •

L,

a rotation of

90^{\circ}

counterclockwise around the origin; •

R,

a rotation of

90^{\circ}

clockwise around the origin; •

H,

a reflection across the

x

-axis; and •

V,

a reflection across the

y

-axis. Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying

R

and then

V

would send the vertex

A

(1,1)

(-1,-1)

and would send the vertex

B

(-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.)

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Solution

Each of

L,R,H,V

moves every vertex to an adjacent corner of the square. Therefore after an odd number of transformations the labeling is in one of the four odd-parity states, and after an even number it is in one of the four even-parity states. After any first

19

transformations, the square is in an odd-parity state. From each odd-parity state, exactly one of

L,R,H,V

sends the labeled vertices back to their original positions. Thus every sequence of the first

19

transformations has exactly one valid final transformation. There are

4^{19}=2^{38}

choices for the first

19

transformations, so there are

2^{38}

valid sequences. Thus, C is the correct answer.

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Solution

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