8 × 8 × 8 8\times8\times8 Nim

Logic Level 3

Dan and Sam play a game on a 8 × 8 × 8 8\times8\times8 cube, that consists of 512 objects; each one must take at least one object and at most a whole heap, in his turn.

As an explicit example: in the first turn, Dan can take just the bottom front left corner object, or can take a whole vertical central heap, or the whole bottom front horizontal heap. Thus, he can take any number of objects of one of the 512 initial heaps (heaps cannot be diagonals).

The winner is the one who takes the last object. If Dan begins, who will win? This means, who has a winning strategy?

This is the sixteenth problem of the set Winning Strategies .
Both Dan Neither Dimitri Sam

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2 solutions

Zubayr Ahmad
May 6, 2016

If Dan starts, he moves a heap of any possible amount (for simplicity sake he moves all except the farthest face of 64). Sam can then move the last heap. No matter the amount of steps, San will take the last heap

Another origin simmetry problem. As the length of the cube dimensions are even, Sam just has to mirror Dan's moves by rotating the space 180° from the cube center, so Sam wins.

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