When I talk about piles they can be taken as piles of matches or stones or whatever, each with any number of objects.

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One pile, remove between at least one and at most half of the remaining matches.

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Two piles, take either 1 or 2 mathces from either pile.

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Given a decimal number, subtract any of its nonzero digits.  Ends when you get 0. 

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Any number of piles:  Move any number of matches from a pile to a strictly larger pile.  Reasonably easy for 3 piles, probably unknown in general.

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Two pieces on squares on an n-by-n board.  In a turn, move one piece any distance down or left, but not on top of or beyond the other one.  Known for n up to 4 or 5, beyond that only some cases of losing/winning positions are known.

Easier if a piece is allowed to move on top of or beyond the other, known.

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Two piles, remove either the size of the smaller from the bigger or their difference from the smaller, if the difference is at most that big.  Probably unknown.

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Given a decimal number n, subtract 2*k where k is any of its nonzero digits (or n if 2k>n).
Probably unknown. Seems to have a surprising, nice property.

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Wythoff's game:  Two piles, take any number from one pile, or any equal number from both.  Known, losing positions form a nice (non?)-pattern when plotted in 2D.