Escape from Prison Puzzle

[pdurrant]

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:

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:

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 )

[omk3]

and, by chance, there are 45 prisoners with green ink and 55 with blue ink.

Do the prisoners have this information? I would guess they only know about the other 99 people, not all 100

[pdurrant]

Originally Posted by omk3

Do the prisoners have this information? I would guess they only know about the other 99 people, not all 100

You have this information The prisoners do not.

Prisoners with green ink only know that they can see 55 prisoners with blue ink and 44 with green ink.

Prisoners with blue ink only know that they can see 54 prisoners with blue ink and 45 with green ink.

They are not allowed discuss the number of prisoners they see with various ink colours with each other. And they are all such puzzle nerds they won't even try to cheat.

[pomposity]

Nice puzzle! So here's my attempt (I've tried to hide it so everyone else can keep trying independently):

Spoiler:

I think the governor gives away a key bit of information, 'your foreheads with blue and green ink-spots that I see before me' means that both colours are present in the audience.

After this, anyone who saw 0 others with a particular colour could deduce that they must carry that same colour themselves, and could apply to leave.

The next day, if no-one has applied to leave, that means that everyone must see at least 1 other prisoner with each colour; consequently, anyone who sees exactly 1 other with a particular colour could deduce that they must carry that same colour themselves, and could apply to leave.

The next day, if no-one has applied to leave, that means that everyone must see at least 2 other prisoners with each colour; consequently, anyone who sees exactly 2 others with a particular colour could deduce that they must carry that same colour themselves, and could apply to leave.

The nest day, if no-one has applied to leave, that means that everyone must see at least 3 other prisoners with each colour; and so on....

it's kind of like a proof by induction.

Each day the minimum number of individuals who must carry each colour ratchets up by one, until it gets to 44, at which point all of the green ink prisoners will be able to deduce that they carry green and will be able to leave, after which those with blue will too.

I'm a bit worried by that 320 days thing, but I'm guessing it's a red herring...

[poohbear_nc]

After reading through the entire set-up for the puzzle I'm ready to put you into this prison! With red ink on your forehead! Talk your way out now puzzle fan!!

[pdurrant]

Originally Posted by pomposity

Nice puzzle! So here's my attempt (I've tried to hide it so everyone else can keep trying independently):

Completely right in your analysis (including the red herring). I'm not sure you've actually given a completely correct answer in the specifics.

Spoiler:

Remember that the Governor's speech was given in the evening and prisoners can only apply to be released between 11am and midday.

To further appreciate the problem, you might like to consider exactly what information the governor gave away — after all, I've told you that there were 45 green and 55 blue, so all the prisoners already know that there were both green and blue ink prisoners...

[pdurrant]

Originally Posted by poohbear_nc

After reading through the entire set-up for the puzzle I'm ready to put you into this prison! With red ink on your forehead! Talk your way out now puzzle fan!!

I hoped people would appreciate the sentiment in the set-up.

[zelda_pinwheel]

GAAAAA !!!!!

must.... resist.... clicking.... spoiler tag !!!!!!!

[omk3]

Originally Posted by zelda_pinwheel

GAAAAA !!!!!

must.... resist.... clicking.... spoiler tag !!!!!!!

Did you succeed in resisting? I'm still trying...

I've encountered a puzzle like this before, and the only thing I remember is that I didn't solve it.

My head's not clear enough at the moment...Will it ever be? Who knows...

Can we maybe post our thoughts (not solutions) without spoilers, so that each can build on the others' reasoning, like in the first puzzle with the probabilities? I liked the team work then

[zelda_pinwheel]

Originally Posted by omk3

Did you succeed in resisting? I'm still trying...

yes, i still haven't clicked : i guess work is good for *something* after all.

I've encountered a puzzle like this before, and the only thing I remember is that I didn't solve it.

My head's not clear enough at the moment...Will it ever be? Who knows...

i think i'm feeling the same way. also, i followed the link to the xkcd version, and why is there apparently no solution posted for that one ???? that is the sort of thing that will drive me INSANE. and i mean that. unless the solution for that one is the same as the solution for this one, and i can figure it out by clicking that spoiler tag if necessary... (pdurrant ? confirmation ? or at least tell me you do know the solution to the other one. you must. maybe you could post it here, in a spoiler tag.)

Can we maybe post our thoughts (not solutions) without spoilers, so that each can build on the others' reasoning, like in the first puzzle with the probabilities? I liked the team work then

i like that idea, personally. if other people prefer to work it out alone, maybe you could use spoiler tags but write something like "not the solution" or "thoughts" or something so we know which ones are safe to click. what do you think about that pdurrant ?

[pdurrant]

Originally Posted by zelda_pinwheel

(pdurrant ? confirmation ? or at least tell me you do know the solution to the other one. you must. maybe you could post it here, in a spoiler tag.)


i like that idea, personally. if other people prefer to work it out alone, maybe you could use spoiler tags but write something like "not the solution" or "thoughts" or something so we know which ones are safe to click. what do you think about that pdurrant ?

The solution to the xkcd puzzle is very similar to the solution for this puzzle. Really, they're the same puzzle, just stated differently, and with different numbers.

There is a solution to the xkcd puzzle on the xkcd site, it's just the solution page isn't linked to from the puzzle page. You really have to want to find the solution to find it — via an appropriate google search. I urge you to resist doing so.

If people want to collaborate on a solution, I'm happy for such messages to be posted here, although I think it might be best to post them

Thoughts towards a solution:

Spoiler:

like this

so that those who want to battle on solo can do so.

[zelda_pinwheel]

Originally Posted by pdurrant

The solution to the xkcd puzzle is very similar to the solution for this puzzle. Really, they're the same puzzle, just stated differently, and with different numbers.

thank god for that !

There is a solution to the xkcd puzzle on the xkcd site, it's just the solution page isn't linked to from the puzzle page. You really have to want to find the solution to find it — via an appropriate google search. I urge you to resist doing so.

GAAAAAAAAA !!! now i have to resist clicking the spoiler button AND resist doing a google search ???? you're trying to kill me, aren't you ?

If people want to collaborate on a solution, I'm happy for such messages to be posted here, although I think it might be best to post them

Thoughts towards a solution:

Spoiler:

like this

so that those who want to battle on solo can do so.

yep, seems reasonable to me. so, omk, have you got any thoughts ? i'll tell you right now i haven't got a single one yet. not worth / appropriate for posting, anyway.

[pdurrant]

[QUOTE=zelda_pinwheel;963166GAAAAAAAAA !!! now i have to resist clicking the spoiler button AND resist doing a google search ???? you're trying to kill me, aren't you ?
[/QUOTE]

Only two temptations? Let me add a third. I wonder if we could keep this up for forty days and forty nights?

Clue:

Spoiler:

Consider what would happen if there were just two prisoners, one with green ink and one with blue ink...

[poohbear_nc]

I vote we solve this puzzle with a prison riot that kills the governor and frees all the inmates. Conducted logically of course.

Does this give any indication how out of ideas I am on this one?

[zelda_pinwheel]

Originally Posted by pdurrant

Only two temptations? Let me add a third. I wonder if we could keep this up for forty days and forty nights?

Clue:

Spoiler:

Consider what would happen if there were just two prisoners, one with green ink and one with blue ink...

oh yes, you are the devil ! come here you, i want to put a little spot of ink on your forehead. then just go right through that door, please. no not the one you came in through. that one over there...

Originally Posted by poohbear_nc

I vote we solve this puzzle with a prison riot that kills the governor and frees all the inmates. Conducted logically of course.

Does this give any indication how out of ideas I am on this one?

actually that is starting to sound good to me too. but i'm going to try to figure out how that clue is supposed to be of any help at all to me first.