the pigeonhole game
N+1 pigeons. N holes. good luck.
level 1
🐦
there was never a way.
if you have more pigeons than holes, two share. that one-line observation is the pigeonhole principle, and it proves real theorems:
n+1 > n
the hair theorem
london has about 9,000,000 people and fewer than 1,000,000 possible hair counts. two people match exactly. guaranteed.
the crowded square
any 5 points in a unit square: two land in the same quadrant, so they sit within √2/2 ≈ 0.71 of each other.
the repeating decimal
dividing by q leaves only q possible remainders. infinitely many division steps, so one repeats, and the digits cycle. every rational repeats.
the party trick
365 holes this time. how many random pigeons until two share? place your bet.
50
0
pigeons placed
2
23
70+
median pigeons to a collision ·
half of all parties with 23 people share a birthday. the pigeonhole principle's drunk cousin.