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5693Dawson's Chess Grundy ValueDawson's chess is the octal game .137, whose Grundy values g(n) for n = 0,1,2,... are the well-known sequence 0,0,1,1,2,0,3,1,1,0,3,3,2,2,4,0,5,... (eventually periodic with period 34). In this game a position made of several independent strips is a win for the mover iff the XOR of the strips' Grundy values is nonzero. Using the published sequence, what is g(8)?脑筋急转弯困难数值题未尝试面试订阅5694Green Hackenbush StalksGreen Hackenbush is played on graphs of green edges rooted to the ground. A move deletes one edge; any edges no longer connected to the ground also disappear. The player unable to move loses. The position consists of three separate vertical stalks (paths) rising from the ground, of heights 4, 6, and 9 edges. Under optimal play, does the first player win or lose?脑筋急转弯简单brainteaser未尝试免费5695Turning Turtles (Coin-Turning Game)Turning Turtles is a coin-turning game on a row of coins numbered 1, 2, 3, ... A move chooses a coin showing HEADS at some position k, turns it to TAILS, and simultaneously turns over EXACTLY ONE other coin at a position j < k (to either face). The player who cannot move (all tails) loses. A standard result is that a single heads at position k is equivalent to a Nim heap of size k, and a position is the XOR (disjunctive sum) of its heads. If the only heads are at positions 3 and 6, does the player to move win or lose?脑筋急转弯中等brainteaser未尝试免费5696Grundy Values on a Game DAGA token sits on a vertex of a directed acyclic graph; a move slides it along one outgoing edge, and a player who cannot move (token on a sink) loses. The edges are: S to A, S to B, S to C; A to two distinct sinks; B to A and B to a sink; C to B. Sinks have Grundy value 0. Compute the Grundy values of A, B, C, and S.脑筋急转弯简单数值题未尝试免费5697Subtraction by Perfect SquaresFrom a single pile a player may remove any positive perfect-square number of stones (1, 4, 9, 16, ...). The player taking the last stone wins. Among pile sizes n = 0, 1, ..., 20, identify the losing (P-)positions for the player about to move, and state the 5th positive losing position.脑筋急转弯中等brainteaser未尝试免费