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851Subtraction Game State 1Two players alternate removing stones from one pile of 27. A legal move is to remove any number in 1, 4, 6 . The player who takes the last stone wins. Under optimal play, does the first player win? If yes, what is an optimal first move?脑筋急转弯简单数值题未尝试免费861Locate the Losing Position 1In a subtraction game with legal moves 1, 4, 6 , positions are classified as winning or losing in the usual last-move-wins sense. What is the 8th positive losing position?脑筋急转弯简单brainteaser未尝试免费866End-Pick Coin Row 1Two players alternately take one number from either end of the row [4, 9, 2, 7, 5]. Each player wants to maximize the sum of the numbers they personally collect. Under optimal play, what should the first player take first, and what total can the first player guarantee?脑筋急转弯简单数值题未尝试免费867End-Pick Coin Row 2Two players alternately take one number from either end of the row [8, 1, 6, 3, 9, 2]. Each player wants to maximize the sum of the numbers they personally collect. Under optimal play, what should the first player take first, and what total can the first player guarantee?脑筋急转弯中等数值题未尝试免费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5678Six Pirates and 100 CoinsSix pirates ranked 1 (most senior) to 6 must divide 100 gold coins. Starting with the most senior, the proposer suggests an allocation; all living pirates (including the proposer) vote. If at least half vote yes the plan passes; otherwise the proposer is thrown overboard and the next most senior proposes. Pirates are perfectly rational and value, in order: survival, then maximizing their own coins, then (as a tiebreaker) seeing others thrown overboard. How many coins does the most senior pirate keep?脑筋急转弯中等数值题未尝试免费5679Race to the Last of 100A single pile holds 100 stones. Players alternate removing between 1 and 7 stones (inclusive). The player who takes the last stone wins. Does the first player win, and if so how many stones should they remove on the first move?脑筋急转弯简单数值题未尝试免费5680First to Say Twenty-OneTwo players build a running total starting at 0. On each turn a player adds 1, 2, or 3 to the total. The player who makes the total reach exactly 21 wins. Does the first player win, and if so what number should they add on the very first move?脑筋急转弯简单数值题未尝试免费5681Misere Nim of SingletonsFive piles each contain exactly 1 stone. Players alternate removing one or more stones from a single pile; here the player forced to take the LAST stone LOSES (misere play). Under optimal play, does the first player win or lose?脑筋急转弯简单brainteaser未尝试免费5682Queen on the Quarter-BoardTwo piles hold 3 and 5 tokens. On a turn a player either removes any positive number from one pile, or removes the SAME positive number from both piles. The player taking the last token (emptying both piles) wins. With the position at (3, 5), does the player to move win or lose under optimal play?脑筋急转弯中等brainteaser未尝试免费5683Sum of Two Take-1-or-2 GamesTwo independent piles are in play, of sizes 4 and 7. On a turn a player picks ONE pile and removes 1 or 2 stones from it. The player who takes the last stone overall (leaving both piles empty) wins. Under optimal play, does the first player win or lose?脑筋急转弯中等brainteaser未尝试免费5684Grundy Value at TenConsider the impartial subtraction game where from a single pile a player may remove 1, 3, or 4 stones, and the player taking the last stone wins. Compute the Sprague-Grundy value of a pile of size 10.脑筋急转弯中等数值题未尝试免费5685Three-Player Take-It-or-Leave-It SplitPlayers A, B, C divide 100 in integer dollars. A proposes a split (a, b, c) summing to 100. If B accepts, the split stands. If B rejects, 1 is burned (now 99 remain) and B proposes a split of the remaining money between B and C only; if C rejects that, another 1 is burned (98 remain) and C takes everything. All players are rational, maximize their own dollars, and accept when indifferent. How many dollars does A keep in A's optimal proposal?脑筋急转弯中等数值题未尝试免费5686Don't Take the Last OneA single pile of stones is played by removing 1 or 2 stones per turn, but the twist is misere: the player forced to take the LAST stone LOSES. Among pile sizes from 1 upward, list the rule for which starting sizes are losing for the player about to move, and state the 4th such losing size.脑筋急转弯简单brainteaser未尝试免费5687Moore's Nim with Two-Pile MovesIn Moore's Nim k a player may, on a single turn, remove any positive number of stones from AT MOST k distinct piles (different counts allowed). Play Nim 2 (k=2) with four piles of sizes 1, 2, 4, 7 and last-stone-overall wins. Under optimal play, does the first player win or lose?脑筋急转弯中等brainteaser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5689Euclid's GameStart with the pair (25, 7). A move replaces the larger number by the larger minus any positive multiple of the smaller, keeping both numbers nonnegative. The player who makes one of the numbers 0 wins (equivalently, the player unable to move loses). With (25, 7), does the player to move win or lose under optimal play?脑筋急转弯中等brainteaser未尝试免费5690Fibonacci NimA pile has 50 stones. On the FIRST move a player may remove any number of stones from 1 to 49 (not the whole pile). After that, a player may remove any number of stones up to TWICE the number the opponent just removed. The player taking the last stone wins. Does the first player win, and if so how many stones should they remove on the first move?脑筋急转弯中等数值题未尝试免费5691Chomp on a 2x3 BarChomp is played on a 2-by-3 grid of chocolate squares; the top-left square (row 1, column 1) is poisoned. A move picks any remaining square and eats it together with every square below and to the right of it. The player forced to eat the poisoned top-left square loses. Under optimal play, does the first player win or lose, and what is the standard argument?脑筋急转弯中等brainteaser未尝试免费5692Kayles Grundy ValueKayles is played on a single row of n adjacent bowling pins. A move knocks down either one pin or two ADJACENT pins, possibly splitting the row into two independent shorter rows. The player who knocks down the last pin wins. Compute the Sprague-Grundy value of a single row of 7 pins.脑筋急转弯中等数值题未尝试免费