1,000 Coins Puzzle

You have 1,000 coins, exactly 20 heads, and you cannot tell heads from tails. You may split them into two piles and flip any subset. The goal is to make the two piles contain the same number of heads.

Interactive idea

Pick any 20 coins to be Pile A. The other 980 coins are Pile B. Then flip all 20 coins in Pile A.

Suppose the arbitrary 20 chosen coins secretly contain 7 heads

0 heads in Pile A 20 heads in Pile A

Pile A · 20 coins

7 H

13 T

Pile B · 980 coins

13 H

967 T

Before flipping, the number of heads in the two piles is usually different.

Pile A visualized as 20 coins

A heads

B heads

Why it always works

Let the chosen 20-coin pile contain h heads. Then it must contain 20 − h tails.

Since there are exactly 20 heads total, the remaining 980-coin pile must contain 20 − h heads.

Flip every coin in the 20-coin pile. Its 20 − h tails become heads, and its h heads become tails.

After flipping Pile A:

Heads in Pile A = 20 − h
Heads in Pile B = 20 − h

Therefore, the two piles have the same number of heads.

Current example

If Pile A starts with 7 heads, then:
• Pile A starts with 7 heads and 13 tails
• Pile B has 13 heads
• After flipping all 20 coins in Pile A, it also has 13 heads