鸽巢原理与策略

考什么 · What it tests

需要先会 · Prerequisites

• 除法与取整 / Division and rounding
• 最不利原则（最坏情况） / The worst-case principle
• 对称的概念 / The idea of symmetry

常见套路 · Common moves

1. 鸽巢先找「抽屉」：想保证有两个同类，先想清有几种「类别」（抽屉），东西比抽屉多 1 个就必有重复——通用第一步。

   For pigeonhole, first find the boxes: to force two of a kind, work out how many categories (boxes) there are — one more item than boxes guarantees a repeat, the universal first step.
2. 求「至少拿几个保证……」用最不利：先把不想要的情况拿到极限，再加 1 个逼出目标，别按运气最好算。

   For "how many to guarantee...," use the worst case: push the unwanted outcomes to their limit, then add one more to force the goal — never assume the luckiest draw.
3. 博弈用对称策略应对：后手常可以「模仿/凑成定值」维持对称，把必输的局面丢还给对方。

   Meet a game with a symmetry strategy: the second player can often mirror or top-up to a fixed total, keeping symmetry and handing the losing position back.

易错点 · Common pitfalls

• 「保证」题按最好情况算：问「至少摸几个保证两只同色」要算最倒霉的摸法，不是摸 2 个就够。

  Solving a "guarantee" problem with the best case: "how many to guarantee two of one color" needs the unluckiest draw, not just 2.
• 鸽巢把「抽屉」数搞错：抽屉是类别数，不是物品数，先定清分成几类。

  Miscounting the pigeonholes: the boxes are the number of categories, not of items — settle how many categories first.