Know Without Trying: The Pigeonhole Principle

Methods in this video

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1. 1

   Every kind one short, plus one

   ▶ 0:37

   When to use it: The question asks how many you must take to be sure of getting some that match (two of the same colour, four of the same colour) or a pair (a left and a right of the same colour); or a total is shared among several groups (21 votes across 5 movies, 25 specimens in 4 species), and it asks at least how many the biggest group must have.

   • First decide what counts as one kind, and how many you need.
   • Worst luck: every kind is one short. If a kind runs out first, take all of it.
   • Add them up, then add one.
   • 21 votes across 5 movies: 4 each is only 20, so the extra vote must take one movie to 5.
2. 2

   Rival at their best, plus one

   ▶ 5:02

   When to use it: The contest isn't over, and the question asks the fewest points or votes that guarantee a win (guarantee, be sure to win, definitely). It often says a tie means nobody wins. The question may not name the person, so you find them in the table.

   • Look only at the top two. Skip the other rows.
   • Worst luck: second place gets everything left.
   • A tie isn't a win, so add one more point.
   • Take away what first place already has. That is the score needed.