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2020 AMC 12B Problem 19

Attempt Problem 19 of the 2020 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

19.

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

L,L, a rotation of 90∘90^\circ counterclockwise around the origin;

R,R, a rotation of 90∘90^\circ clockwise around the origin;

H,H, a reflection across the xx-axis; and

V,V, a reflection across the yy-axis.

Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying RR and then VV would send the vertex AA at (1,1)(1, 1) to (−1,−1)(-1, -1) and would send the vertex BB at (−1,1)(-1, 1) to itself. How many sequences of 2020 transformations chosen from {L,R,H,V}\{L, R, H, V\} will send all of the labeled vertices back to their original positions? (For example, R,R, R,R, V,V, HH is one sequence of 44 transformations that will send the vertices back to their original positions.)

2372^{37}

3⋅2363 \cdot 2^{36}

2382^{38}

3⋅2373 \cdot 2^{37}

2392^{39}

Answer: C
Concepts:transformationcasework
Difficulty rating: 2000
Small Hint:

The first 1919 transformations can be anything; the last one is then forced to be the unique inverse

Big Hint:

The forced last move must be one of the four allowed transformations; find the probability that it is

Solution:

Label the vertices 0,1,2,30,1,2,3 cyclically. Each allowed transformation has the form j↦εj+δ(mod4),j\mapsto \varepsilon j+\delta\pmod 4, where ε∈{1,−1}\varepsilon\in\{1,-1\} and δ∈{1,−1}.\delta\in\{1,-1\}. These four choices give exactly the two quarter-turns and the two stated reflections.

Under composition, the parity of δ\delta changes at every move. Thus a composition of 1919 allowed transformations again has odd δ,\delta, and so is one of the four allowed transformations. Its inverse is also allowed. Consequently every sequence of the first 1919 moves has exactly one choice for the final move, giving 419=2384^{19}=2^{38} successful sequences.

Thus, the correct answer is C.

Problem 18#18
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Problem 20#20

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