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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5698Twelve Coins, Unknown DirectionYou have 12 coins that look identical. Exactly one is counterfeit and has a different weight from the rest, but you do NOT know whether it is heavier or lighter. Using only a two-pan balance scale (each weighing reports left-heavy, right-heavy, or balanced), what is the minimum number of weighings that guarantees you both identify the counterfeit AND determine whether it is heavy or light? Weighings may be chosen adaptively.脑筋急转弯中等brainteaser未尝试免费5699Sorting Five With Fewest ComparisonsYou must sort 5 distinct numbers using only pairwise comparisons, each of which returns which of the two compared elements is larger. What is the information-theoretic lower bound on the worst-case number of comparisons any comparison-based sorting algorithm needs, AND is that bound actually achievable for 5 elements? Give the minimum worst-case number of comparisons that guarantees a full sort.脑筋急转弯中等brainteaser未尝试免费5700Ten Hats in a Line10 players stand in a line. Each wears a red or blue hat, assigned independently by a fair coin. Each player sees all hats in FRONT of them but not their own nor those behind. Starting from the back of the line, each player in turn announces a single guess of their own hat color, heard by everyone. They agree on a strategy beforehand (no communication after hats are placed except the public guesses). Using the optimal parity strategy, how many of the 10 players are GUARANTEED to guess correctly regardless of the hat assignment?脑筋急转弯中等brainteaser未尝试免费5701Guess the Number 1 to 1000An adversary picks a secret integer between 1 and 1000 inclusive. You may ask yes/no questions, each answered truthfully, and you may choose each question adaptively based on previous answers. What is the minimum number of questions that guarantees you can determine the secret number in the worst case?脑筋急转弯简单brainteaser未尝试免费5702One Poisoned Bottle, Binary TestersYou have 1000 bottles of wine, exactly one of which is poisoned. A tester who drinks any amount containing the poison dies after exactly the same fixed delay, and you can have each tester sip from any combination of bottles simultaneously in a single round (results observed after the delay, before the celebration). If you only get ONE round of testing, what is the minimum number of testers needed to guarantee identifying the poisoned bottle?脑筋急转弯中等brainteaser未尝试免费5703Eight Coins, One Known LighterYou have 8 visually identical coins; exactly one is counterfeit and is known to be LIGHTER than the rest. Using a two-pan balance scale (left-heavy / right-heavy / balanced per weighing), what is the minimum number of weighings that guarantees identifying the light coin in the worst case? Weighings may be adaptive.脑筋急转弯简单brainteaser未尝试免费