In the classic three-door puzzle, switching your choice after one door is opened wins two times out of three.

The longer version

Your first choice is right one time in three, and nothing that happens later changes that. When a door is opened to reveal a losing option, the host has effectively offered you the other two doors as a bundle, and that bundle was always two thirds likely to contain the prize. Switching simply takes the bundle.

The fifty-fifty instinct comes from seeing two closed doors and forgetting that the opening was not random. Enlarge the setup to a hundred doors, let the host clear ninety-eight losers, and the advantage of switching becomes obvious. The whole argument depends on the host knowing where the prize sits and always avoiding it.

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.