You’re blindfolded, faced with a deceptively simple task:

You have 1,000 coins scattered on the floor.
Exactly 20 are heads-up, the rest are tails.
You can’t see or distinguish the coins by touch.
You may divide them into two piles, and flip any subset of coins.
Your goal: Create two piles with the same number of heads.

Sounds impossible without sight? It’s not—just clever.

The Key Trick: Use Randomness Against Itself

You can force symmetry using a neat, invariant-based move.

Step-by-Step Solution

  1. Randomly select any 20 coins from the 1,000 coins. (You know there are 20 heads total, but not where they are.)

  2. Place those 20 coins into a separate pile (Pile A).
    The remaining 980 coins go into Pile B.

  3. Flip every coin in Pile A.

Done!

Why It Works

Let’s say in the 20 coins you picked for Pile A, ( h ) were originally heads-up.
Then there are ( 20 - h ) tails in Pile A.

That means in Pile B (the remaining 980 coins), there must be ( 20 - h ) heads—because there are 20 total heads.

When you flip the ( h ) heads and ( 20 - h ) tails in Pile A, it becomes:

  • ( h ) → tails
  • ( 20 - h ) → heads

So now Pile A has exactly ( 20 - h ) heads.
Pile B already had ( 20 - h ) heads.

Result: Both piles now contain the same number of heads.

Final Answer

Pick any 20 coins to make one pile. Flip all of them.

Now both piles have an equal number of heads.

Reference