The birthday problem shows that a room of roughly twenty-three people has better than even odds of a shared birthday.

The longer version

Work from the opposite end and ask how likely it is that everyone differs. The second person must avoid one date out of three hundred sixty-five, the third must avoid two, and so on, and multiplying those near misses gives a chance of no match just under half once twenty-three people are in the room.

What tricks intuition is the number of chances involved. Twenty-three people form two hundred fifty-three distinct pairs, and any one of those pairs can collide, so the odds accumulate far faster than a single comparison suggests. By about seventy people a shared birthday is nearly certain. Leap days and seasonal birth patterns shift the totals only slightly.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.