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  2. Next card is Red

  3. Today
  4. Pegity - white to win

    Yes. That’s a win for white
  5. Next card is Red

    Last card red might not be a good backup plan
  6. Next card is Red

    Or is it counterbalanced by the last card?
  7. Next card is Red

    better than 50%?
  8. Pegity - white to win

    Is O win if we get XOOOOOX?
  9. Next card is Red

    Thanks Thalia...
  10. Pegity - white to win

    OK so I was a kid once, back in the 40s and 50s, and we had this game called Pegity. It had a board with a 16x16 array of holes, and in turn each of 2-4 players inserted a peg of his own color into a hole. The object was to get 5 pegs of your color in a row: vertically, horizontally or diagonally. Not every game was won: similar to tic-tac-toe, you could run out of holes. But with only two players, there was usually a win. For simplicity let's shrink to a 9x9 board, mark the holes like in chess (A1, E7, etc.) and say that after O and X have each made three moves we have this position, with O to move: 1 2 3 4 5 6 7 8 9 A + + + + + + + + + B + + + + + + + + + C + + + + + + + + + D + + + + + + + + + E + + + + O + + + + F + + + O + O + + + G + + + + X + + + + H + + + X + X + + + I + + + + + + + + + With best play by both players, how soon can O win? Use chess notation (naming the holes, in two columns) to list the moves.
  11. Next card is Red

  12. Next card is Red

    @CaptainEd shuffles a standard deck of playing cards and begins dealing the cards face up on the table. At any time @plainglazed can say Stop and bet $1 that the next card will be red. If he does not interrupt, the bet will automatically be placed on the last card. What is the best strategy? How much better than 50% can @plainglazed do?
  13. Green and Yellow hats again, but harder

    That is amazing. And you've shown that 1 fewer than a power of 2 is the optimal value for n. Can you reduce it to only q memorized strings, and the success rate to (q-1)/q, where q is the highest power of 2 less than or equal to n? (Purposely leaving this un-spoilered, cuz this is a tough problem.)
  14. Baggage on a conveyor belt

    Yup. And a bit of possibly helpful thinking:
  15. A big, big, big hotel

    Yeah, I hate annoyances, too...
  16. Yesterday
  17. Obtuse triangles in a circle

    Not an answer, but a description of a potential approach and what would still be needed to make it work since this has gone for a while without being cracked.
  18. A big, big, big hotel

    Brilliant, surprisingly interesting and leading to a major annoyance. Well, now back to the original problem. Everybody leaves, the hotel is empty. The green and blue men arrive all together and go to the room they occupied previously. Is this problem equivalent to the first one? Precision: By equivalent, I mean that the problems are announced in a slightly different way, but the answers remain the same. Something this way: Aunt: If I give you 5 bananas and take back 2 bananas, how many bananas have you? John: I do not know. Aunt: I met your teacher and she told me you make this kind of problems at school. John: Yes, but we do it with apples. Just do not tell me that the men know now the room numbers. Besides being clairvoyant, they can move in time. Forwards, backwards and sideways.
  19. Last week
  20. Green and Yellow hats again, but harder

    I think I see the pattern here and am encouraged by the beauty of binary symmetry so here goes:
  21. When midnight strikes

    Sorry for 'discard', I meant these numbers build a {set} from which elements can be removed. And yes, we remain in rationals. What if l only paint those between 0.5 and 0.6, even leaving .53745 unpainted? Did I not paint an infinite number of numbers? We turn in round here. You claim that if you can remove an infinite amount of elements from a set, the set will be empty - I claim there may remain some elements (and even more then removed, depending on circumstances). I suppose you apply a 'law' or a 'theorem'. Can you give me the (Wikipedia) reference?
  22. Modified checkerboard destruction

    Can you think of a reason?
  23. Baggage on a conveyor belt

    I see, I didn’t pay attention. I really want to count only segments that are surrounded by shorter segments.
  24. Modified checkerboard destruction

    Thank you for the kind words, Bonanova. I fear my answer to this one is not so brilliant.
  25. A big, big, big hotel

    OK I think I just found your first blue man.
  26. Worth the Weight

    @aiemdao - Yes, nice solve indeed Aiem. Well done. @bonanova - thanks.
  27. Right Prime

    Any other prime numbers less than 1000.
  28. Worth the Weight

    @aiemdao - Nice solve. @plainglazed - Nice puzzle.
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