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5677Three-Pile Nim Winning MoveTwo players play standard Nim with three piles of sizes 3, 5, and 7. On a turn a player removes any positive number of stones from a single pile, and the player who takes the last stone wins. Does the first player win, and if so, give a winning first move (which pile and how many stones to remove).脑筋急转弯简单数值题未尝试免费5688Staircase NimCoins sit on a staircase numbered step 0 (the ground) up to step 4, with counts (step 0..4) = (2, 3, 1, 0, 5). A move takes any positive number of coins from a step s >= 1 and moves them down to step s-1. Coins on step 0 are off the board. The player who cannot move (all coins on the ground) loses. Under optimal play, does the first player win or lose?脑筋急转弯简单brainteaser未尝试免费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