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The New Coffee Room

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  3. Puzzle time- Numbers on foreheads

Puzzle time- Numbers on foreheads

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  • jon-nycJ Offline
    jon-nycJ Offline
    jon-nyc
    wrote on last edited by jon-nyc
    #9

    The idea is that the median is only slightly affected by your number, less so if your number is close to it.

    Imagine if there were 1,000 people doing that. Clearly the differences in median that each would calculate would be a tiny rounding error, thus they would all converge on the same number, including at least one person who had that number.

    "You never know what worse luck your bad luck has saved you from."
    -Cormac McCarthy

    KlausK 1 Reply Last reply
    • jon-nycJ Offline
      jon-nycJ Offline
      jon-nyc
      wrote on last edited by
      #10

      You can also improve over the random probability by doing something with modes like Horace suggested. Even with just 10.

      I still haven't been bothered to work out probabilities. I'm hoping the hints reveal a more clever intuitive approach.

      "You never know what worse luck your bad luck has saved you from."
      -Cormac McCarthy

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      • HoraceH Offline
        HoraceH Offline
        Horace
        wrote on last edited by
        #11

        A lot would hinge on whether the median was a single-occupancy number or multi-occupancy. In the case of a single-occupancy median, that strategy is a guaranteed loss. It's a guaranteed win for multi-occupancy median when it's still the median after removing one of them. It begins to be similar to the mode idea from that perspective.

        Education is extremely important.

        1 Reply Last reply
        • KlausK Online
          KlausK Online
          Klaus
          wrote on last edited by Klaus
          #12

          I think I got a solution that works with probability 1. Good old pigeonhole principle.

          click to show

          The prisoners number themselves 0 to 9.

          Let s the the sum of all numbers on all heads and let d be its rightmost digit.

          Now consider prisoner number p. He computes the sum t of all numbers he sees on the other's heads.
          He guesses that d = p. Hence he guesses (p - t) modulo 10 as his number.

          This will yield a correct guess for prisoner d.

          Proof. Consider prisoner d.

          His guess will be (d-t) modulo 10, which is the same as s - t, which is the number on his forehead.

          1 Reply Last reply
          • jon-nycJ Offline
            jon-nycJ Offline
            jon-nyc
            wrote on last edited by
            #13

            That's clever.

            "You never know what worse luck your bad luck has saved you from."
            -Cormac McCarthy

            1 Reply Last reply
            • jon-nycJ jon-nyc

              The idea is that the median is only slightly affected by your number, less so if your number is close to it.

              Imagine if there were 1,000 people doing that. Clearly the differences in median that each would calculate would be a tiny rounding error, thus they would all converge on the same number, including at least one person who had that number.

              KlausK Online
              KlausK Online
              Klaus
              wrote on last edited by
              #14

              @jon-nyc said in Puzzle time- Numbers on foreheads:

              The idea is that the median is only slightly affected by your number, less so if your number is close to it.

              Imagine if there were 1,000 people doing that. Clearly the differences in median that each would calculate would be a tiny rounding error, thus they would all converge on the same number, including at least one person who had that number.

              I don't think so.

              Let's consider a uniform distribution of the numbers. Then the median of the 1000 would very likely be 4 or 5. But that number is not more likely to show up than any other number.

              1 Reply Last reply
              • jon-nycJ Offline
                jon-nycJ Offline
                jon-nyc
                wrote on last edited by
                #15

                You choose the visible number closest to the median.

                "You never know what worse luck your bad luck has saved you from."
                -Cormac McCarthy

                1 Reply Last reply
                • jon-nycJ Offline
                  jon-nycJ Offline
                  jon-nyc
                  wrote on last edited by jon-nyc
                  #16

                  I think your solution isn't right. They each (potentially) have a different value of t.

                  "You never know what worse luck your bad luck has saved you from."
                  -Cormac McCarthy

                  KlausK 1 Reply Last reply
                  • jon-nycJ jon-nyc

                    I think your solution isn't right. They each (potentially) have a different value of t.

                    KlausK Online
                    KlausK Online
                    Klaus
                    wrote on last edited by
                    #17

                    @jon-nyc said in Puzzle time- Numbers on foreheads:

                    I think your solution isn't right. They each (potentially) have a different value of t.

                    I made a small correction and elaboration on why this is correct.

                    1 Reply Last reply
                    • jon-nycJ jon-nyc

                      Each of 10 prisoners will have a digit between 0 and 9 painted on his forehead (they could be all 2's and 3's, for example).  At the appointed time, each prisoner will be exposed to all the others, then taken aside and asked to guess his own digit.

                      All the prisoners will be freed, provided at least one prisoner guesses correctly.

                      The prisoners have an opportunity to conspire beforehand; how can they maximize their probability of success?

                      AxtremusA Away
                      AxtremusA Away
                      Axtremus
                      wrote on last edited by
                      #18

                      @jon-nyc said in Puzzle time- Numbers on foreheads:

                      The prisoners have an opportunity to conspire beforehand; how can they maximize their probability of success?

                      If prisoners get to choose their own number, then in the “conspire” phase all prisoners agree to one number (e.g., every prisoner chooses to paint himself with the number 0), then every one knows that his own number is 0.

                      1 Reply Last reply
                      • taiwan_girlT Offline
                        taiwan_girlT Offline
                        taiwan_girl
                        wrote on last edited by
                        #19

                        Not sure if I understand, but to me, there is no way to make a guess any more accurate. If it is a true random number on the forhead, then there is a 10% chance of guessing correctly.

                        It doesnt matter if every other number is "8", your number could still be anything from 0-9. etc

                        What am i missing?

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