为什么Nim游戏靠二进制就能必胜？
Why does Nim have a binary winning strategy?
Nim取石子，把各堆数量写成二进制，若异或和为0则当前是必败态，否则可一步变成0。这就是尼姆和，是组合博弈论的基石。
In Nim, write each pile in binary; if the XOR sum is 0 the position is losing, otherwise you can move to make it 0. This Nim-sum is the cornerstone of combinatorial game theory.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#Nim #博弈
为什么国际象棋的马走日字能遍历全棋盘？
Why can a knight's move traverse the whole chessboard?
马的走法在8乘8上构成哈密顿路径问题，存在马巡游解（一笔画走遍64格各一次），是图论与组合优化的经典谜题。
The knight's move on 8x8 forms a Hamiltonian path problem; a knight's tour exists that visits all 64 squares once, a classic puzzle in graph theory and combinatorial optimization.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#马巡游 #图论
为什么囚徒困境里背叛是理性却导致双输？
Why is betrayal rational yet leads to mutual loss in the prisoner's dilemma?
对个人而言，不管对方怎么选，背叛都更安全；但两人都背叛的结局比都合作更差。这揭示个体理性与集体利益的冲突。
Individually, betraying is safer no matter what the other does; but both betraying yields a worse outcome than both cooperating. It reveals the conflict between individual rationality and collective interest.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#囚徒困境 #博弈
为什么七桥问题开启了图论？
Why did the seven bridges problem start graph theory?
欧拉证明柯尼斯堡七桥无法一笔画走完，条件是图中奇度顶点数为0或2。他用抽象的顶点与边取代地图，由此诞生了图论。
Euler proved the Konigsberg seven bridges cannot be traversed in one stroke; the condition is that a graph has 0 or 2 odd-degree vertices. Abstracting the map into vertices and edges birthed graph theory.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Euler《Solutio problematis ad geometriam situs》（1736）；Diestel《Graph Theory》
Euler, Solutio problematis ad geometriam situs (1736); Diestel, Graph Theory
独立核验依据 · Independent verification: 核验依据 1 ISBN 9783662575604 (2018)
#七桥 #图论
为什么猜帽子谜题要靠别人看不见的信息推理？
Why do hat puzzles rely on information others can't see?
这类题利用公共知识与我看到什么的推理链：每个人根据他人行为反推自己帽子颜色，常需约定策略，如前几人牺牲传递奇偶信息。
These puzzles use common knowledge and the chain of what each person sees; players infer their own hat from others' actions, often needing an agreed strategy like early players sacrificing to signal parity.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#猜帽子 #推理
为什么汉诺塔最少步数是2的n次方减1？
Why are the minimum Tower of Hanoi moves 2^n - 1?
递归思想：移n盘等于先把n-1盘移到中转柱(2^(n-1)-1步)，移最大盘(1步)，再把n-1盘移过去(2^(n-1)-1步)，合计2^n-1，指数增长极快。
Recursively: moving n disks means moving n-1 to a spare (2^(n-1)-1 steps), moving the largest (1), then n-1 again (2^(n-1)-1), totaling 2^n - 1, growing exponentially fast.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-03
来源 · Source
R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994
R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL3951639W：Concrete mathematics（首版
#汉诺塔 #递归
为什么井字棋先手不会输？
Why can the first player never lose at tic-tac-toe?
棋盘小、状态有限，先手用中心开局并对称防守即可保证不败。它是完全信息博弈中最优策略存在的最简例子。
The board is tiny with finite states; the first player opening in the center and defending symmetrically cannot lose. It is the simplest example of an optimal strategy in a perfect-information game.
难度 入门 · 适合 幼儿(3-6)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
Berlekamp, J. H., Conway, J. H. & Guy, R. K.《Winning Ways for Your Mathematical Plays》,MIT Press
Berlekamp, J. H., Conway, J. H. & Guy, R. K., Winning Ways for Your Mathematical Plays, MIT Press
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780120911011 (1985)
#井字棋 #博弈
为什么幻方阵每行每列和都相等？怎么造一个？
Why does a magic square have equal row and column sums, and how to make one?
幻方把1到n平方排成方阵，使行、列、对角线和相同（为n(n平方+1)/2），魔数即1到n平方总和除以n。如洛书：4 9 2 / 3 5 7 / 8 1 6；造奇数阶可用楼梯法：从顶行中位出发，依次右上斜移、越界回绕，遇已占位则下移一格。古代既作趣味也作神秘符号。
A magic square arranges 1 to n^2 so every row, column and diagonal sums equally (n(n^2+1)/2), the constant being the total 1..n^2 divided by n. The Luoshu: 4 9 2 / 3 5 7 / 8 1 6; for odd n the staircase method builds one: start top-middle, step up-right with wraparound, drop one row when a cell is taken. It was both a pastime and a mystical symbol.
难度 入门 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Martin Gardner,《My Best Mathematical and Logic Puzzles》Dover,1994
M. Gardner, My Best Mathematical and Logic Puzzles, Dover, 1994
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL460047W：My best mathematical and
#幻方 #趣味