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Old 06-16-2010, 06:27 AM   #1
pdurrant
The Grand Mouse
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Location: Norfolk, England
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Escape from Prison Puzzle

There's a country that has a special prison. It's not a prison for people who've broken the usual criminal laws, it's a prison for people who've irritated the rest of the population beyond endurance by making up (and insisting on explaining) far too many logical puzzles.

So everyone in the prison is extremely good at logic. So much so, that they always make logically correct deductions from any information they have. And they all know that the other prisoners are equally logical, and that they all know this of each other, etc.

It quite a low-security prison. The prisoners are allowed to mingle freely, and they all get together for an evening meal in a large dining hall every day. So every prisoner gets to see the ink colour on the forehead of every other prisoner.

There's only one term of imprisonment — life. However, there is a (literal) get-out clause. On arrival every inmate is marked with indelible ink on their forehead — either green ink or blue ink, and the following is said to them:

Quote:
You have been marked with green ink or blue ink, If you can logically deduce the colour of the ink on your forehead, you can ask to be released. You may only apply to be released between 11am and 12 noon, since that's the only time the visiting logician is available to verify your conclusion. If you gain your freedom, it will be announced to the other prisoners at the evening meal. This same information is given to every prisoner on arrival.
Naturally, the prisoners are forbidden from discussing the ink colour on anyone's forehead, and there are no mirrors or reflecting surfaces in the prison. In addition, the prisoners are such keen logic problem setters and solvers that resorting to any such method of cheating is unthinkable to them. Finally, as to get their freedom they have to explain to the visiting logician how they deduced their ink colour, merely finding out by accident won't help.

Because the inks offer a hope of freedom, all the prisoners make careful mental notes of the ink colour on the foreheads of all the other prisoners, and they are able to do this because the can mingle freely and they all get together in one place at least once a day — for the evening meal.

The prison works well for several years. No-one gains their freedom. The governor randomly chooses the ink colour for each new prisoner. Eventually the 100th prisoner has been admitted, and, by chance, there are 45 prisoners with green ink and 55 with blue ink.

To mark the occasion, the governor gives a speech to the prisoners at that evening meal, being very careful, of course, not to mention any particular prisoner's ink colour. He says:

Quote:
On this day when the one hundredth prisoner has been admitted, I look at you all and consider how fortunate society is that you are in here where you can no longer torment us with your horrendous logic puzzles.

I thank you for your good behaviour in this prison, and I hope it will continue, as you spend your time and energy considering the chance of freedom offered by your foreheads with blue and green ink-spots that I see before me.

I am also sure that, since no-one has ever deduced their ink colour, that it is a false hope, and that society is safe from you forever.
As it happens, no more prisoners arrive at the prison until exactly 320 days after the governor's speech.

Do any prisoners ever gain their freedom from the prison? If so, who and why and when?

Please enclose your solution in spoiler tags, so others can keep trying.


(Quite a lot of the above text is only there to tighten constraints on the problem, so as to prevent quibbles with the solution I have in mind. But some bits of the text are red herrings.

It's a re-written form of the puzzle at I came across here: http://xkcd.com/blue_eyes.html )

Last edited by pdurrant; 06-16-2010 at 09:56 AM. Reason: minor typos — nothing significant.
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