Skip to content
  • Categories
  • Recent
  • Tags
  • Popular
  • Users
  • Groups
Skins
  • Light
  • Cerulean
  • Cosmo
  • Flatly
  • Journal
  • Litera
  • Lumen
  • Lux
  • Materia
  • Minty
  • Morph
  • Pulse
  • Sandstone
  • Simplex
  • Sketchy
  • Spacelab
  • United
  • Yeti
  • Zephyr
  • Dark
  • Cyborg
  • Darkly
  • Quartz
  • Slate
  • Solar
  • Superhero
  • Vapor

  • Default (No Skin)
  • No Skin
Collapse

The New Coffee Room

  1. TNCR
  2. General Discussion
  3. Puzzle time- Numbers on foreheads

Puzzle time- Numbers on foreheads

Scheduled Pinned Locked Moved General Discussion
19 Posts 5 Posters 127 Views
  • Oldest to Newest
  • Newest to Oldest
  • Most Votes
Reply
  • Reply as topic
Log in to reply
This topic has been deleted. Only users with topic management privileges can see it.
  • jon-nycJ jon-nyc

    My intuition is that choosing the number closest to the mean of those you see is the best solution but I can't back it up. Maybe tomorrow's hint will help.

    KlausK Offline
    KlausK Offline
    Klaus
    wrote on last edited by
    #8

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

    My intuition is that choosing the number closest to the mean of those you see is the best solution but I can't back it up. Maybe tomorrow's hint will help.

    I don't quite get the logic behind that.

    Why should the median of the other numbers influence the distribution of your number?

    1 Reply Last reply
    • jon-nycJ Online
      jon-nycJ Online
      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.

      Only non-witches get due process.

      • Cotton Mather, Salem Massachusetts, 1692
      KlausK 1 Reply Last reply
      • jon-nycJ Online
        jon-nycJ Online
        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.

        Only non-witches get due process.

        • Cotton Mather, Salem Massachusetts, 1692
        1 Reply Last reply
        • 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 Offline
            KlausK Offline
            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 Online
              jon-nycJ Online
              jon-nyc
              wrote on last edited by
              #13

              That's clever.

              Only non-witches get due process.

              • Cotton Mather, Salem Massachusetts, 1692
              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 Offline
                KlausK Offline
                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 Online
                  jon-nycJ Online
                  jon-nyc
                  wrote on last edited by
                  #15

                  You choose the visible number closest to the median.

                  Only non-witches get due process.

                  • Cotton Mather, Salem Massachusetts, 1692
                  1 Reply Last reply
                  • jon-nycJ Online
                    jon-nycJ Online
                    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.

                    Only non-witches get due process.

                    • Cotton Mather, Salem Massachusetts, 1692
                    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 Offline
                      KlausK Offline
                      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 Offline
                        AxtremusA Offline
                        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?

                          1 Reply Last reply
                          Reply
                          • Reply as topic
                          Log in to reply
                          • Oldest to Newest
                          • Newest to Oldest
                          • Most Votes


                          • Login

                          • Don't have an account? Register

                          • Login or register to search.
                          • First post
                            Last post
                          0
                          • Categories
                          • Recent
                          • Tags
                          • Popular
                          • Users
                          • Groups