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# bushindo

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1. ## A simple (*not* bonanova 'simple') hat problem

Ah, I see what you are getting at then. Your claim is absolutely right. This is a superbly designed puzzle, by the way.
2. ## A simple (*not* bonanova 'simple') hat problem

My impression of the claim you intend the BD to prove is the following: Claim: Given that the fishermen figure out a way to alert all the other players when they've figured out their hat, then the fishermen should be able to solve what color hat they are wearing through just logic; that is, they do not need to confer among themselves for a common strategy. In other words, let's say that we remove the fishlines from the problem, and just assume that there is a royal announcer. Every time participant X raises his hand, the royal announcer would then intone within all participant's hearing
3. ## A simple (*not* bonanova 'simple') hat problem

This is puzzling, since a previously-agreed-upon strategy is the hallmark of these everyone-must-be-correct-to-win problems. I would like some clarification. Do you mean to say "If there was no time to devise and agree on a strategy, hence allowing the fishermen only to raise their hand when they knew their own color of hat, what is the algorithm for the first fisherman such that the 5 participants are guaranteed to win?" Or perhaps you mean to ask for an algorithm for the first fisherman such that the chance of fishermen winning is as high as possible?
4. ## A simple (*not* bonanova 'simple') hat problem

Well, here's are some strategies Minor details about the fish lines
5. ## A simple (*not* bonanova 'simple') hat problem

I would like some clarification. My impression is that whenever a participant raises his hand, the other 4 participant would be aware of this fact. For instance, if participant 1 raises his hand after the first gong sounding but before the second gong sounding, the other 4 participants would be aware that number 1 raised his hand, even though they would not know the color of 1's hat. Is this true?
6. ## A really simple connect-the-dots problem

Here's an attempt,
7. This is indeed the correct probability. Congrats! (14.swapnil.14 made the same conjecture but made some mistake in the calculations). However, the solution to this puzzle is slightly incomplete (for me) because the probability computation relies on the unproven fact that There is a neat proof for this fact, and I think the fine denizens of the den would appreciate the challenge of working why the above statement is true. Instead of posting that fact as a new puzzle, I think I'll declare this puzzle to be half-solved, and revise the OP to include this new challenge.
8. Suppose that there is a thin, circular ring of wire with circumference of 10 meters, and that we have some ants with the following properties, 1) All the ants when placed on the wire will travel at a fixed, constant speed of 1 meter/minute in the direction that they are heading. 2) When any two ants collide on the wire, each ant will instantaneously turn around and travel in the opposite direction at the same speed. Suppose that we simultaneously place 14 ants at random locations on the ring. The orientation in which each ant is headed (clockwise or counter-clockwise) is determined by a
9. ## Cut To The Chase

First of all, let me say that this is an excellent problem. Thanks for sharing it.
10. ## Cut To The Chase

The full expression for the chance of beating Vern at this game is
11. Here's a non-calculus guess
12. ## The best game to win

You didn't read the choices incorrectly. I was just being faceticious :-). Please carry on.
13. ## The best game to win

This is an easy one
14. Thanks for the problem. By the way, congrats on the 1000th post.
15. Thanks for sharing these little gems with us, superprimastic. I enjoy the journey greatly, and feel enriched by the experience. Also, thanks for you recommendation. I picked up the Stanford Mathematics Problem Book for my ereader, and look forward to spending some time on it.
16. Let me just chime in and say that I like your puzzles as well. I'm always happy to see new topics that you post. Can you recommend one or two of these expository texts? There are some time slots in my life that I'd like to put to better use.
17. I would like some clarification. What do you mean by 'thinking of the numbers in their binary representations'? My interpretation is that nCk is odd if and only if n = k and k > 0, but that interpretation is clearly false. Clarification would be appreciated. Thanks.
18. This is such an elegant problem. I'm amazed at how the algebra works out nicely. Thanks
19. Clear and elegant as well. Good job!
20. Thanks superprismatic and witzar. That is really a silly oversight in the wording of the puzzle. The intent is to show that under certain starting lamp patterns, there is a strategy which will guarantee that the butler will never be able to turn all the lights on. I'll fix it in the OP. Thanks for pointing that out.
21. Let's say that you are setting the dining table for a party of 60 guests. The 60 guests will be seated at a giant circular table. Each seat has a lamp, and currently some of the 60 lamps are on and some are off. The butler offers to play a game with you. Let's say that the chairs are labelled with the guests names, which for convenience we label starting from the top and going clockwise as C1, C2, ..., C60, and the chairs are stationary. The giant circular dining table along with the lamps, however, can rotate. Each lamp has a switch, and flipping the switch when the lamp is on turns it off, a
22. That's brilliant, witzar. Good work!
23. Some thoughts, mostly 90% hand waving and 10% proof by exhaustion
24. Adding the proof to complete the puzzle. Please excuse the formatting; typing a proof in the brainden editor is quite complicated.
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