[LazyScot]

Originally Posted by pdurrant

Here are the answers to the questions:

Spoiler:

1. Pick 21 coins at random, but as you place them flat on the table in front of you, be sure to turn them over.
2. You are certain to be freed.

But I suspect that those who are still puzzled by the puzzle won't be happy with just bare answers, and would like to know why.

Spoiler:

In the 21 coins you pick at random (before you turn them over), there will be X heads, where X is any number from 0 to 21 inclusive.
In the 80 coins the Gaoler has, he will have the remaining (21-X) heads.

After you turn your 21 coins over, you will also have (21-X) heads.

So no matter how many heads are in the 21 coins that you pick, after you turn them all over you will have exactly the same number of heads in your 21 coins as the gaoler has in his 80 coins. You go free.

Spoiler:

My understanding is that this works because the number of coins you draw exactly matches the number of coins that were originally laid out with heads.

How would the answers be altered if the number of coins you drew where not the same as the number of heads? Could you still influence the probability of your being freed?