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#41 There are three prisoners

Set by Christopher Dearlove

Three mathematicians are held prisoner and made to play a game in which they each have a blob painted on their forehead in green or blue, with an independent 50 per cent chance of either colour for each prisoner. They must all simultaneously guess their own blob colour or say nothing. They lose the game if nobody guesses or if someone guesses wrong.

How can they plan to play so they guarantee a 50 per cent chance of winning?

How can they increase their chances of winning to 75 per cent?

The prison guards overhear the mathematicians plotting. To thwart the prisoners’ strategy, the guards change the rules so they now choose the blob colours. If each prisoner has a fair coin they can flip once, can they use them to come up with a modified strategy with a 75 per cent success rate? The prisoners can see all three coins, but the guards can’t see any of them.

Solution next week

#40 There are four lights

Solution

There are four ways to get from light D to light A: blue-blue-red, blue-red-blue, red-blue-blue and red-red-blue.

Light B cannot be lit directly after blue is pressed, as no blue arrow points to it.

Pressing blue-red-blue guarantees that either A or D is lit. Pressing blue first ensures that A, C or D is lit. Pressing red then causes D, A or B to be lit, and the final blue doesn’t change D or A, but will turn B to A.

Pressing blue-red-blue-red-blue guarantees that A is lit. We know after blue-red-blue that A or D is lit. Pressing red makes that A or B, then the final blue turns B to A or leaves A unchanged.