When I talk about piles they can be taken as piles of matches or stones or whatever, each with any number of objects. --------------------------------------------------------------------- --------------------------------------------------------------------- One pile, remove between at least one and at most half of the remaining matches. --------------------------------------------------------------------- Two piles, take either 1 or 2 mathces from either pile. --------------------------------------------------------------------- Given a decimal number, subtract any of its nonzero digits. Ends when you get 0. --------------------------------------------------------------------- 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. --------------------------------------------------------------------- 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. --------------------------------------------------------------------- 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. --------------------------------------------------------------------- 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. --------------------------------------------------------------------- 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.