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Threehanded Game of Chess

The III-COLOR-tournament mode


Now here is some tournament mathematics. A fair tournament or league for 3-player-games is shown by the example of six or nine players. If there are more than 9 players, this mode is not longer practicable, because the number of games to play grows up to high. So think about alternatives, if you like to. I don't know, if this tournament mode exists already in another threehanded game.

In math's language you may call this problem a design: every group of 2 (t=2) of n players (or teams) has to be put in groups of 3 (k=3), so they meet l times in the tournament. In our special case is l = n - t. Written in math:

t _ ( n , k , l )

The solution of this problem can be calculated with the binomial coefficients ( n over k).

For the 6-player-problem:

2_(6,3,4) =
6!/(3!*(6-3)!) =
4*5*6/(1*2*3) = 20 games

You have to plan 10 rounds, because 2 games can always be played simulteaneously.

For the 9-player-problem:

2_(9,3,7) =
9!/(3!*(9-3)!) =
7*8*9/(1*2*3) = 84 games

You have to plan 28 rounds, because 3 games can always be played simulteaneously. In a league with weekly contests you need about half a year.

Witch special matches have to happen? Some diagrams tell about. You have to divide a circle in n (here 6) same parts and mark them at the periphery. Now connect these marks to different triangle patterns. The triangels represent the three opponents of each game. By turning the patterns ahead you build the other matches. Here is the solution for 6 players:

6-player-tournement rounds 1-3 6-player-tournement rounds 4-9 6-player-tournement final round 10

You see 3 patterns. From the first one you get 3 rounds by turning it ahead, from the second 6 more rounds. The last diagram doesn't change by turning ahead. This one is the final round. If you write the encounters into a table, it looks like this:


round no. match no. I red II yellow III blue
1 1 1 2 3
1 2 4 5 6
2 3 2 3 4
2 4 5 6 1
3 5 3 4 5
3 6 6 1 2
4 7 1 2 4
4 8 3 5 6
5 9 2 3 5
5 10 4 6 1
6 11 3 4 6
6 12 5 1 2
7 13 4 5 1
7 14 6 2 3
8 15 5 6 2
8 16 1 3 4
9 17 6 1 3
9 18 2 4 5
10 19 1,3 or 5 1,3 or 5 1,3 or 5
10 20 2, 4 or 6 2, 4 or 6 2, 4 or 6

The last round is a real endmatch, because up to the 9th round every player has had three times the same colour and home games. The colours of the final round are found by drawing lots.

For a 9-player-tournament we do it the same way. Here are the 4 circle diagrams, you can take from all matches. The first 3 patterns generate 9 rounds in each case. The last pattern again represents the final round. Until the 27th round every player has had every colour and home games 9 times. In a league the last match has to be played on "neutral ground".

9-player-tournament rounds 1-9 9-player-tournement rounds 10-18 9-player-tournement rounds 19-27 9-player-tournement final round 28

From my opinion, there is no obstacle for playing STAR- and TRIANGLE-III-COLOR-Chess as new disciplines in the game of chess. Invite your friends to have a III-COLOR-CHESS-tournement! Dear chessmasters, please get busy with these new variants! It is a really fascinating stuff! Let my game idea grow on your critics! How we can make it better?

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