Combinations and permutations - more math than chess

Collapse
X
 
  • Filter
  • Time
  • Show
Clear All
new posts

  • Combinations and permutations - more math than chess

    Two chess players can be paired only 1 way: (1-2)

    Four players can be paired 3 ways: (1-2 and 3-4) (1-3 and 2-4) (1-4 and 2-3)

    Six players can be paired 15 ways: I'll spare my typing finger, and not list them; trust me.

    Eight players? How many ways can eight players be paired? Does anyone know the formula for the possible combinations for n players, where n is even? How about where n is uneven?

  • #2
    Re: Combinations and permutations - more math than chess

    Originally posted by John Coleman View Post
    Two chess players can be paired only 1 way: (1-2)

    Four players can be paired 3 ways: (1-2 and 3-4) (1-3 and 2-4) (1-4 and 2-3)

    Six players can be paired 15 ways: I'll spare my typing finger, and not list them; trust me.

    Eight players? How many ways can eight players be paired? Does anyone know the formula for the possible combinations for n players, where n is even? How about where n is uneven?
    Hi John. Since colour is an important part of the pairing process (2-1 and 1-2), why aren't you interested in in permutations? For combinations, I think the number you're looking for is 28 (but I forget the formula). :)

    https://en.wikipedia.org/wiki/Combination
    Last edited by Peter McKillop; Tuesday, 26th January, 2016, 08:30 PM.
    "We hang the petty thieves and appoint the great ones to public office." - Aesop
    "Only the dead have seen the end of war." - Plato
    "If once a man indulges himself in murder, very soon he comes to think little of robbing; and from robbing he comes next to drinking and Sabbath-breaking, and from that to incivility and procrastination." - Thomas De Quincey

    Comment


    • #3
      Re: Combinations and permutations - more math than chess

      Originally posted by John Coleman View Post
      Two chess players can be paired only 1 way: (1-2)

      Four players can be paired 3 ways: (1-2 and 3-4) (1-3 and 2-4) (1-4 and 2-3)

      Six players can be paired 15 ways: I'll spare my typing finger, and not list them; trust me.

      Eight players? How many ways can eight players be paired? Does anyone know the formula for the possible combinations for n players, where n is even? How about where n is uneven?
      The 15 setups for 6 players:

      12 34 56
      12 35 46
      12 36 45
      13 24 56
      13 25 46
      13 26 45
      14 23 56
      14 25 36
      14 26 35
      15 23 46
      15 24 36
      15 26 34
      16 23 45
      16 24 35
      16 25 34

      I think the answer for 8 players is 105.

      My reasoning:

      1 - 2 players
      1x3 - 4 players
      1x3x5 - 6 players
      1x3x5x7 - 8 players


      All 1 v 2 pairings, I think:

      12 34 56 78
      12 34 57 68
      12 34 58 67
      12 35 46 78
      12 35 47 68
      12 35 48 67
      12 36 45 78
      12 36 47 68
      12 36 48 57
      12 37 45 68
      12 37 46 58
      12 37 48 56
      12 38 45 67
      12 38 46 57
      12 38 47 56 = 15 different setups.

      So you would have the same sorts of pairings with 13, 14, etc. all the way to 18 (seven different groups); 7 x 15 = 105.
      Last edited by Tom O'Donnell; Tuesday, 26th January, 2016, 09:25 PM. Reason: clarity
      "Tom is a well known racist, and like most of them he won't admit it, possibly even to himself." - Ed Seedhouse, October 4, 2020.

      Comment


      • #4
        Re: Combinations and permutations - more math than chess

        I might add that I don't see any difference between say 7 or 8 players. Wouldn't it be the same number of possible pairings (i.e. 105)?
        "Tom is a well known racist, and like most of them he won't admit it, possibly even to himself." - Ed Seedhouse, October 4, 2020.

        Comment


        • #5
          Re: Combinations and permutations - more math than chess

          Originally posted by Tom O'Donnell View Post
          The 15 setups for 6 players:

          12 34 56
          12 35 46
          12 36 45
          13 24 56
          13 25 46
          13 26 45
          14 23 56
          14 25 36
          14 26 35
          15 23 46
          15 24 36
          15 26 34
          16 23 45
          16 24 35
          16 25 34

          I think the answer for 8 players is 105.

          My reasoning:

          1 - 2 players
          1x3 - 4 players
          1x3x5 - 6 players
          1x3x5x7 - 8 players


          All 1 v 2 pairings, I think:

          12 34 56 78
          12 34 57 68
          12 34 58 67
          12 35 46 78
          12 35 47 68
          12 35 48 67
          12 36 45 78
          12 36 47 68
          12 36 48 57
          12 37 45 68
          12 37 46 58
          12 37 48 56
          12 38 45 67
          12 38 46 57
          12 38 47 56 = 15 different setups.

          So you would have the same sorts of pairings with 13, 14, etc. all the way to 18 (seven different groups); 7 x 15 = 105.
          Since I had forgotten the formula for the number of combinations of eight things combined two at a time (it's been half a century since grade 11 math), here's how I came up with 28:

          Player 1 can be paired with each of 7 other players.
          Player 2 can be paired with each of 6 other players (the pairing of 1 and 2 was counted above).
          Player 3 .... 5 other players.
          .... and so on.

          i.e. 7 + 6 +5 +4 +3 + 2 + 1 = 28
          "We hang the petty thieves and appoint the great ones to public office." - Aesop
          "Only the dead have seen the end of war." - Plato
          "If once a man indulges himself in murder, very soon he comes to think little of robbing; and from robbing he comes next to drinking and Sabbath-breaking, and from that to incivility and procrastination." - Thomas De Quincey

          Comment


          • #6
            Re: Combinations and permutations - more math than chess

            John, n would always be even because you have in reserve, for uneven situations, the player called 'bye'.
            "We hang the petty thieves and appoint the great ones to public office." - Aesop
            "Only the dead have seen the end of war." - Plato
            "If once a man indulges himself in murder, very soon he comes to think little of robbing; and from robbing he comes next to drinking and Sabbath-breaking, and from that to incivility and procrastination." - Thomas De Quincey

            Comment


            • #7
              Re: Combinations and permutations - more math than chess

              Originally posted by Peter McKillop View Post
              Since I had forgotten the formula for the number of combinations of eight things combined two at a time (it's been half a century since grade 11 math), here's how I came up with 28:

              Player 1 can be paired with each of 7 other players.
              Player 2 can be paired with each of 6 other players (the pairing of 1 and 2 was counted above).
              Player 3 .... 5 other players.
              .... and so on.

              i.e. 7 + 6 +5 +4 +3 + 2 + 1 = 28
              No mathematician am I, but, I always thought the formula would be (# of players) x (# of players less 1) divided by 2.

              So, 8 players would be 8 x 7 \ 2 = 28

              A 15 player event would be 15 x 14 \ 2 = 105.

              This is assuming you meant a round robin of x number of players playing each participant once to obtain the number of games that would be played.

              Comment


              • #8
                Re: Combinations and permutations - more math than chess

                Originally posted by Tom O'Donnell View Post
                I might add that I don't see any difference between say 7 or 8 players. Wouldn't it be the same number of possible pairings (i.e. 105)?
                Thanks Tom, I came up with ths same number, but I couldn't prove it.

                Before I retired from chess organising, I developed a pairing system for kids which does not require that all round start at the same time. In fact, we don't even have rounds. We get 1400+ kids at our big event each year. We don't assign colours, the kids pick colours (chosing a pawn) themselves. I'm writing a pairing program to computerise this system, by way of keeping my brain active. It's an adventure.

                Comment


                • #9
                  Re: Combinations and permutations - more math than chess

                  Originally posted by J. Ken MacDonald View Post
                  No mathematician am I, but, I always thought the formula would be (# of players) x (# of players less 1) divided by 2.

                  So, 8 players would be 8 x 7 \ 2 = 28

                  A 15 player event would be 15 x 14 \ 2 = 105.

                  This is assuming you meant a round robin of x number of players playing each participant once to obtain the number of games that would be played.
                  Yes, you're right about the formula for n things combined 2 at a time.
                  "We hang the petty thieves and appoint the great ones to public office." - Aesop
                  "Only the dead have seen the end of war." - Plato
                  "If once a man indulges himself in murder, very soon he comes to think little of robbing; and from robbing he comes next to drinking and Sabbath-breaking, and from that to incivility and procrastination." - Thomas De Quincey

                  Comment


                  • #10
                    Re: Combinations and permutations - more math than chess

                    n is any even number of players (just add a bye for odd numbers to make them all even). Then P is the number of possible pairing arrangements for any individual round (assuming your players then randomize for white & black).

                    P(n) = 1 * 3 * 5 * ... * (n-1)

                    This gives us:

                    P(2) = 1
                    P(4) = 1*3 = 3
                    P(6) = 1*3*5 = 15
                    P(8) = 1*3*5*7 = 105
                    P(10) = 1*3*5*7*9 = 945
                    etc.
                    Tom O'Donnell's post does a good job of explaining the reasoning.

                    This is different than the number of ways to select a group of two people out of a group of n, which as mentioned already is n(n-1)/2, and gives you the total number of games that would be played in a single round robin tournament.

                    Comment


                    • #11
                      Re: Combinations and permutations - more math than chess

                      Originally posted by Andrew Peredun View Post
                      n is any even number of players (just add a bye for odd numbers to make them all even). Then P is the number of possible pairing arrangements for any individual round (assuming your players then randomize for white & black).

                      P(n) = 1 * 3 * 5 * ... * (n-1)

                      This gives us:

                      P(2) = 1
                      P(4) = 1*3 = 3
                      P(6) = 1*3*5 = 15
                      P(8) = 1*3*5*7 = 105
                      P(10) = 1*3*5*7*9 = 945
                      etc.
                      Tom O'Donnell's post does a good job of explaining the reasoning.

                      This is different than the number of ways to select a group of two people out of a group of n, which as mentioned already is n(n-1)/2, and gives you the total number of games that would be played in a single round robin tournament.
                      Thanks, Andrew. I misunderstood what John was looking for.
                      "We hang the petty thieves and appoint the great ones to public office." - Aesop
                      "Only the dead have seen the end of war." - Plato
                      "If once a man indulges himself in murder, very soon he comes to think little of robbing; and from robbing he comes next to drinking and Sabbath-breaking, and from that to incivility and procrastination." - Thomas De Quincey

                      Comment


                      • #12
                        Re: Combinations and permutations - more math than chess

                        Originally posted by John Coleman View Post
                        ...

                        Before I retired from chess organising, I developed a pairing system for kids which does not require that all round start at the same time. In fact, we don't even have rounds. We get 1400+ kids at our big event each year. We don't assign colours, the kids pick colours (chosing a pawn) themselves. I'm writing a pairing program to computerise this system, by way of keeping my brain active. It's an adventure.
                        Hi John, I would be interested in such a program. Thanks and regards, Aris.

                        Comment

                        Working...
                        X