bonanova
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Everything posted by bonanova
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Good logic!
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The answer is described by jcalonego in post #11, but it needed a slight tweak. See posts 15 and 16 for the tweak. If it's unclear, it can be explained in more detail. Hint: posts 11, 15 and 16 describe a way to tell everyone his hat color. And it's all done during the 15 minutes before any of them are asked their color. Does that make sense?
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Unist: Good approach, but wouldn't it be inappropriate for persons entering the store? I'm thinking the idea is that it would be usefully read from either side, but you could be right.
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On the right track ... but from the OP you should be able to conclude that you start the interval from a moment of a single chime. When does the interval end, that will make the longest time you'd have to wait until you're sure what time it is? Think about the time the interval starts and how long you might wait until you're sure what time it is. That's just a restatement of the OP, but it should show that some guesses are going to be wrong, so I hope that's a help.
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I tried the explicit formula for more than two terms and it got immediately complicated. I do think it can be shown that dP/da [not just the partial derivative] = dP/db = ... = dP/dz at the extremum. And then you can do a pair-wise proof for zero [as we both did] so they all are zero when all terms are k/n. I'm still working it out on paper - it's harder to do the formatting on the display Nice puzzle; and I used the Web as a refresher also in crafting my answer, mainly to reduce the steps, but also ... taking the log before differentiating was a step I did not see on my own. As a moderator of the site, I have to say I'm not sure this was so much a puzzle as a math exercise, but [1] I enjoyed working it out ... and seeing the surprise answer, and [2] the answer gave it a puzzle-like flavor.
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Great observation. It seems there is an adequate strategy - yours and PolishNorbi's - if you know [1] the split is 50-50 or if you know [2] the split is not 50-50. You need to advise the prisoners on the best strategy that will work whether it's [1] or [2]. When you find it, you'll slap your forehead. Maybe.
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Great thinking ... It looks at first like more than half could be certain of survival.
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I can think of words that work, but wouldn't make sense on a door.
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I hope I'm not exposing too much of a "sneaky" character trait here... I'm actually trying to be transparent about the constraints. But your observation about 50% expectation is absolutely correct! Now that the problem statement has been reduced to what looks like a coin flip, the strategy should be crafted to live with a 50% expectation, but nevertheless make the worst case scenario as good as possible. If it just remains a coin flip, the worst case is that they all could be executed. The idea is to see how many prisoners can be guaranteed a pardon.
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Good thinking, but no one has it yet. 1:45 [jcalonego] is close, but not quite right. 45 minutes [finance_it] is quite a bit off, and makes a mistake about when the interval could start. DST [GIJeff] doesn't figure in, because that change is made between 2:00 am and 3:00 am. 2:00 or how long it takes for the clock to strike more than once [itachi-san] is close, too, but not quite right. Looking at the clock [finance_it] I guess is allowed by the OP, but I meant how long, just by listening. My bad. Hint: read the OP carefullly:
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Close, but not quite. It's an even better one!
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Yes. Actually, only one more person -either of the end people - has to go to the middle, to remove the uncertainty of the 100th prisoner. Great job.
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A clock chimes the hour, every hour on the hour, and once each quarter hour in between. If you hear it chime once, what is the longest you may have to wait to be sure what time it is?
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Great Idea! And best of all, no communicating! But we're back to ensuring 99.5 prisoners are safe - actually guaranteeing 99. I claim we can get them all home safely, 100% guaranteed.
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Warden notices the back-turning and infers the prisoners are communicating. Everyone is executed.
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Interesting, and you're on the right track. But how does this work? Say I see two groups: one group has black hats and one group has white hats. Say I see an even number of black hats. How do I know which group to join? p.s. I can't walk over and ask someone, "Is this the group that sees an even number of black hats?"
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