Two-Thirds of the Room

How it works

A hundred people each write a number from 0 to 100. The winner is whoever comes closest to two-thirds of the average. Nobody sees anyone else's card and the rule is common knowledge. The story hands you the blank card and asks what you write. You lock in a guess, then step through the answer and breakdown.

How scoring works

No grading - your guess is displayed back beside the answer. The story teasers share one lifetime unlock for free users.

The key insight

Each level of reasoning multiplies the target by 2/3: random play averages 50, so level-1 writes 33, level-2 writes 22, and iterating forever lands on 0 - the equilibrium if everyone is perfectly rational and knows it. Real rooms stop at one to three levels, so winning numbers cluster around 20-30. The skill is estimating how deep this room goes and stopping one level past it, not racing to zero.

The full step-by-step solution lives at /brain-teasers/beauty-contest/solution.

A worked example

No winning guess can exceed 67, since two-thirds of the largest possible average is about 66.7. Delete those and nothing above 44 can win; delete again for 30, then 20, and the ladder only stops at 0. Against a normal crowd, best-responding to level-1 play means writing about 22.

Common mistakes

• Writing 33 and stopping - that best-responds to a random room but ignores that others reason too.

• Writing 0 against real people - correct only if the whole room iterates forever.

• Giving one number without conditioning on who is in the room; the honest answer is 0 against theorists, low twenties against a crowd.

Why interviews test this

Quant desks ask this because markets are this game: you are pricing what average opinion expects average opinion to be. The written solution at /brain-teasers/beauty-contest/solution covers level-k reasoning.