Ten Coins in the Dark
The puzzle
Ten coins lie on a table. Exactly five are heads up and five are tails up, but they are mixed together and you do not know which is which.
You are blindfolded, and the coins are identical to the touch - you cannot feel which way up any of them is. You may move coins and you may turn them over.
Divide them into two groups so that each group contains the same number of heads-up coins.
Stuck?
One at a time, smallest first. Open only as many as you need — the whole point of the puzzle is the part you work out yourself.
Hint 1
The two groups do not have to be the same size.
Hint 2
Take any five coins. Whatever is in that pile, what happens to its head count if you flip every coin in it?
Show the answer Hide the answer
Push any five coins to one side and turn all five of them over. The two piles now hold the same number of heads.
How to get there
Separate any five coins at random into a new pile. Say that pile happens to contain k heads - you have no idea what k is, and it does not matter.
Since there are five heads in total, the other pile contains 5 - k heads.
Now turn over every coin in your pile of five. Its k heads become tails, and its 5 - k tails become heads. The pile now holds 5 - k heads.
Both piles hold 5 - k heads, whatever k was.
The technique worth keeping
You never learn k and you never need to. The flip is chosen so that it converts the unknown quantity into exactly the quantity on the other side - the uncertainty cancels rather than being resolved. Any puzzle where you are denied information is asking for an operation whose result does not depend on it.