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  2. Buried Proverb

    Both observations are on the right track. Will fix OP
  3. Today
  4. Green and Yellow hats

    The only problem is that it doesn't guarantee that anybody lives. Too much WiFoM
  5. Buried Proverb

    Not sure if these are supposed to go together or be 6 separate things.
  6. Yesterday
  7. "Flipping" Dimes and Pennies

    Agree, best I can do is 8.
  8. "Flipping" Dimes and Pennies

    I can do it in 8
  9. Place two dimes and two pennies in a line with a space between them like this: P P D D _ _ _ _ _ Using a sequence of moves, switch the two groups of coins to achieve this position: D D P P _ _ _ _ _ Moves are of two types: slide a coin to an (adjacent) empty space. jump a coin over another coin into an empty space. Type (1) move: P P D D _ _ _ _ _ Type (2) move: P D P D _ _ _ _ _ The underlines show the (only) five legal locations for the coins to occupy. What is the smallest number of moves needed?
  10. Buried Proverb

    The word buried here has but one letter. Did you find a jelly roll in Gearhardt's Bakery? It's the best one I've ever seen. The rug at her stairway was made in India. He's an old friend. Amos sold his bicycle to an old friend.
  11. Green and Yellow hats

    Yes and yes.
  12. Green and Yellow hats

    I figured it would be a "toughie" because hat puzzles tend to have counterintuitive answers. Now that we've become accustomed to that, we may be looking for an answer that isn't there given the new constraints.
  13. Green and Yellow hats

    @bonanova, is the warden perfectly logical? Are the prisoners perfectly logical?
  14. Green and Yellow hats

    This is basically my stumbling block. Any strategy you could come up with could be countered by the warden. If you could anticipate that the warden would do all in his power to flummox your strategy, you could (as a logical individual) decide to go completely against the strategy. That, however, doesn't guarantee any successes either.
  15. Building cars

    yes, exactly.
  16. You've just found a neat way to place points uniformly randomly inside a unit circle: simply place points at random inside a circumscribed square -- x and y uniformly chosen on [-1, 1] -- and ignore the points near the square's corners that are outside the circle. There are other ways, but this works, and it's simple to do. And why are you excited about this? The reason is that you've often wondered about the expected size of randomly drawn triangles inside a unit circle. And now you can find out. You sequentially place a million sets of three random points in the circle, calculating (and then averaging) the areas of the million triangles they define. And you find something pretty amazing: the million triangles had an average area that is only ~ 7.4% of the circle's area. You also note that the median area was ~ 5.4%. OK, so that's a fairly long set-up for a pretty short puzzle. Read on. You tell a friend about how amazingly small random triangles constrained by a circle are, and he replies with a question of his own: "That's cool," he says, "but I wonder what fraction of those triangles cover the circle's center?" You admit that was a piece of information that you did not take note of. "Oh, that's OK," your friend replies, "I think I can tell you." What answer did your friend (correctly) come up with?
  17. Green and Yellow hats

    Hmmm. OP does not rule out movement. But it does rule out communicating. So let's say that if the prisoners want to be at some preferred location in the room, that's permissible. But their chosen location can't be in any way influenced by hat color -- i.e., all movement must occur before the hats are placed.
  18. Green and Yellow hats

    Are they allowed to move at all, like to sort themselves without communication? Or do they start off in their positions in the room with hats placed on them, only allowed to say one word?
  19. In a long hallway, 100 prisoners are given red or blue hats, whose color only the other prisoners can see. At a signal given by the warden the prisoners must walk single file through a door and take their places inside a large room. The room is circular and its wall, ceiling and floor are featureless. Nothing is said, nor are any gestures made to prisoners as they enter the room and take their place. When the last prisoner has taken his place the warden inspects the configuration of their hat colors. If the colors form two monotonic groups separable, say, by some straight line, then all the prisoners are freed. If their hat colors instead are intermingled, they are all executed. Prisoners are allowed to discuss strategy before receiving their hats. What is their fate? Let's see, what else? Oh ya, they can't just pass their hats around. They're super-glued on their heads. Ouch. And no one has a magic marker to ... uh ... you know, make a line ... or anything like that.
  20. Green and Yellow hats

    Nope. They act on what they see. Sorry.
  21. Green and Yellow hats

    As always, we hope for some communication. The prisoners can see each other. It’s not clear they can hear each other (after all, if they shout simultaneously, they can’t benefit from hearing the others). Are they allowed to turn their bodies to face in a variety of directions, or some such thing? You said “no communication”, and I fear you mean it, but just askin’...
  22. Green and Yellow hats

    Here's a toughie. A room full of prisoners is given hats, whose color only the others can see. And just to be different, let's say they are yellow or green. No communication is permitted. At a signal, given by the warden, the prisoners must simultaneously shout out the color of their own hat. Those who guess wrong are subsequently executed. Beforehand, the prisoners meet to determine a strategy -- a set of rules, not necessarily the same for each prisoner -- that will guarantee the greatest number of survivors. As an added wrinkle, the warden may attend the meeting and then use his knowledge of their strategy when he chooses the colors of their hats. If there are 100 prisoners, how many can be assured of surviving?
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