The scene seems doomed to trial and error: you know how many coins are face up, but you can't recognize any by touch. The surprising thing is that the solution does not depend on identifying anything, but on a gesture that transforms ignorance into symmetry.
The 100 coins blind
Riddle statement
There are 100 coins on a table. You know that exactly 20 are face up and 80 are face down, but you are completely in the dark and cannot tell one from the other by touch.
You can separate coins into two piles as you want and also turn over the coins as you need.
Can you make two piles so that both have the same number of face up coins?
Show solution
Solution
Answer: Separate any 20 coins into one pile and leave the remaining 80 in the other. Then turn over all the coins in the pile of 20.
Call \(x\) the number of face-up coins that landed in that first pile. Since the total number of heads is 20, the pile of 80 contains \(20 - x\) heads.
When you turn over the 20 coins in the first pile, the \(x\) heads become tails and the \(20 - x\) tails become heads. The first pile is left with exactly \(20 - x\) faces.
The second pile already had \(20 - x\) faces and was not touched. Both piles end up with the same number of face-up coins, regardless of the value of \(x\).