Competition problem — Combinatorics

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Combinatorics Difficulty 4.8 AIME Prove it United States

Problem:

Mr. Fat moves around on the lattice points according to the following rules: From point $(x, y)$ he may move to any of the points $(y, x)$ , $(3x, -2y)$ , $(-2x, 3y)$ , $(x+1, y+4)$ and $(x-1, y-4)$ . Show that if he starts at $(0,1)$ he can never get to $(0,0)$ .

Solution

Solution:

Observe that for each of Mr. Fat's moves, the value of $x + y \pmod{5}$ is invariant. Therefore, Mr. Fat can never reach $(0,0)$ from $(0,1)$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.