Quote:
Originally Posted by LazyScot
Bang goes my first idea.....  
Spoiler:
When I pick 21 coins, I'll end up with x heads and 21 - x arms. Since there are 21 heads in total, the gaoler will have 21 - x heads.
I want to maximise the chance that I don't get executed, i.e. I want to have the same number, or more, heads than the gaoler. Mathematically I want to optimise the chances of "my heads" >= "gaoler's heads".
From a simple selection, this is the chances of "x" >= "21 - x".
Observation
The coins are identical, and so can be turned over with the result that a head becomes an arms and an arms becomes a head.
So I have a choice of either x heads or 21 - x heads, if I turn them all over.
Turning then changes my chances of freedom to the cases where
21 - x >= 21 - x
This is clearly always true, so I'll always go free.
Unfortunately for, and unbeknownst to, me my greatest enemy was also sitting blindfolded in the room. He reached over to the gaoler's coins, and turned them all over, meaning that the goaler now had 59 + x heads, and so I was executed regardless of what I did with my coins.
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Spot on. Well done. I'm not sure I like your extension to the game.