1)
Okay, draws will be allowed at Shogi.
2)
The problem which Thomas noticed is not merely a consequence that 9 is a
prime number, but more specifically that 3 is a divisor of 9. It is
avoided if there is no game between player1 and player4, player2 and
player5 and so on. So it works, because there are (exactly) six offsets
(1, 2, 4, 5, 7, 8) which are prime with 9
1 2 3 4 5 6 7 8 9
1 C X S s x c
2 c C X S s x
3 x c C X S s
4 x c C X S s
5 s x c C X S
6 S s x c C X
7 S s x c C X
8 X S s x c C
9 C X S s x c
(True, one could say that there are three groups of players which don't
meet each other, that is, player1, player4 and player7 don't meet, and so
on.)
However, with 8, 10 or 12 players, it won't be possible to find six
convenient offsets so I guess we'll have to make do with it.
3)
I think I have overlooked Fergus' idea of having a champion at each game.
Yes, it is possible to have a Chess champion, a Xiangqi champion and a
Shogi champion between the players who have scored two wins at each game.
If there are two players with two wins at one game, they play two games,
playing once as White and once as Black. In case of equality, they play
one game with the higher-ranked player playing as White (and then as Black
in case of a further draw, and then as White, and so on).
(If both players so choose, they can play only one game with the
higher-ranked player playing as White, and then as Black, and so on.)
If there are three players with two wins at one game, they meet each
other, playing once as White and once as Black. In case of equality, the
two better-ranked players play one game with the higher-ranked player
playing as White (and then as Black in case of a further draw, and then as
White, and so on).
If there are four players with two wins at one game, they meet each other,
the two higher ranked players playing twice as White and once as Black. In
case of equality, the two better-ranked players play one game with the
higher-ranked player playing as White (and then as Black in case of a
further draw, and then as White, and so on).
If there are five or more players with two wins at one game, they play
exactly two games, and the survivors will fight a subsequent round.
True, a player may get eliminated of a Chess, Xiangqi or Shogi playoff
because of a poor overall ranking in the other games, but only in
combination with one loss or two draws in that playoff.
Let's say there are nine participants:
1 2 3 4 5 6 7 8 9
1 C X S s x c
= 1 0 1 1 1 4.5, two wins at Xiangqi
2 c C X S s x
= 0 1 0 0 1 2.5, two wins at Xiangqi
3 x c C X S s
0 1 0 = 1 1 3.5, two wins at Shogi
4 x c C X S s
0 1 1 1 1 0 4 (BS=11), two wins at Chess
5 s x c C X S
1 = 0 = 0 1 3, two wins at Shogi
6 S s x c C X
0 1 0 = 0 0 1.5
7 S s x c C X
1 0 1 1 0 1 4 (BS=10.5), two wins at
Xiangqi
8 X S s x c C
0 0 0 1 1 1 3, two wins at Chess
9 C X S s x c
0 0 1 0 0 0 1
player1, player3, player4 and player7 enter the general play-offs (1 is
always White, 3 is always Black).
The Chess title is played between player4 (White) and player8 (Black). If
they have one win (or two draws) each, player8 will become White. Let's
say Player8 wins.
The Shogi title is played between player3 (White) and player5 (Black). If
they have one win each, player5 will become White. Let's say player3
wins.
The Xiangqi title is played between player1, player2 and player7. Let's
say they score one win and one loss each, player1 is White and wins,
player7 is Black and loses, player2 only has some reason for complaining.
4)
It is possible to qualify a Chess champion, a Xiangqi champion, a Shogi
champion and an overall round-robin champion (plus replacement players in
case of overlap, which means the second-ranked player will probably
qualify) for the final round of four, like Thomas suggested, but then, the
general play-off will have to follow the other play-offs.
Maybe it is just as well.
If we proceed this way, player8, player3, player1 and the higher-ranked
remaining player (player4, thanks to a better Buchholz-Sokoloff index) vie
for the combined title, with player1 is always playing as White and player3
always playing as Black.
(Of course player7 won't be happier than player2 before, but he also blew
several chances.)
5)
So there should be three rounds. Do you agree, and if so, how much time
should take each round?
6)
That formula requires at least seven players. Are there enough volunteers?
1) Okay, draws will be allowed at Shogi. 2) The problem which Thomas noticed is not merely a consequence that 9 is a prime number, but more specifically that 3 is a divisor of 9. It is avoided if there is no game between player1 and player4, player2 and player5 and so on. So it works, because there are (exactly) six offsets (1, 2, 4, 5, 7, 8) which are prime with 9 1 2 3 4 5 6 7 8 9 1 C X S s x c 2 c C X S s x 3 x c C X S s 4 x c C X S s 5 s x c C X S 6 S s x c C X 7 S s x c C X 8 X S s x c C 9 C X S s x c (True, one could say that there are three groups of players which don't meet each other, that is, player1, player4 and player7 don't meet, and so on.) However, with 8, 10 or 12 players, it won't be possible to find six convenient offsets so I guess we'll have to make do with it. 3) I think I have overlooked Fergus' idea of having a champion at each game. Yes, it is possible to have a Chess champion, a Xiangqi champion and a Shogi champion between the players who have scored two wins at each game. If there are two players with two wins at one game, they play two games, playing once as White and once as Black. In case of equality, they play one game with the higher-ranked player playing as White (and then as Black in case of a further draw, and then as White, and so on). (If both players so choose, they can play only one game with the higher-ranked player playing as White, and then as Black, and so on.) If there are three players with two wins at one game, they meet each other, playing once as White and once as Black. In case of equality, the two better-ranked players play one game with the higher-ranked player playing as White (and then as Black in case of a further draw, and then as White, and so on). If there are four players with two wins at one game, they meet each other, the two higher ranked players playing twice as White and once as Black. In case of equality, the two better-ranked players play one game with the higher-ranked player playing as White (and then as Black in case of a further draw, and then as White, and so on). If there are five or more players with two wins at one game, they play exactly two games, and the survivors will fight a subsequent round. True, a player may get eliminated of a Chess, Xiangqi or Shogi playoff because of a poor overall ranking in the other games, but only in combination with one loss or two draws in that playoff. Let's say there are nine participants: 1 2 3 4 5 6 7 8 9 1 C X S s x c = 1 0 1 1 1 4.5, two wins at Xiangqi 2 c C X S s x = 0 1 0 0 1 2.5, two wins at Xiangqi 3 x c C X S s 0 1 0 = 1 1 3.5, two wins at Shogi 4 x c C X S s 0 1 1 1 1 0 4 (BS=11), two wins at Chess 5 s x c C X S 1 = 0 = 0 1 3, two wins at Shogi 6 S s x c C X 0 1 0 = 0 0 1.5 7 S s x c C X 1 0 1 1 0 1 4 (BS=10.5), two wins at Xiangqi 8 X S s x c C 0 0 0 1 1 1 3, two wins at Chess 9 C X S s x c 0 0 1 0 0 0 1 player1, player3, player4 and player7 enter the general play-offs (1 is always White, 3 is always Black). The Chess title is played between player4 (White) and player8 (Black). If they have one win (or two draws) each, player8 will become White. Let's say Player8 wins. The Shogi title is played between player3 (White) and player5 (Black). If they have one win each, player5 will become White. Let's say player3 wins. The Xiangqi title is played between player1, player2 and player7. Let's say they score one win and one loss each, player1 is White and wins, player7 is Black and loses, player2 only has some reason for complaining. 4) It is possible to qualify a Chess champion, a Xiangqi champion, a Shogi champion and an overall round-robin champion (plus replacement players in case of overlap, which means the second-ranked player will probably qualify) for the final round of four, like Thomas suggested, but then, the general play-off will have to follow the other play-offs. Maybe it is just as well. If we proceed this way, player8, player3, player1 and the higher-ranked remaining player (player4, thanks to a better Buchholz-Sokoloff index) vie for the combined title, with player1 is always playing as White and player3 always playing as Black. (Of course player7 won't be happier than player2 before, but he also blew several chances.) 5) So there should be three rounds. Do you agree, and if so, how much time should take each round? 6) That formula requires at least seven players. Are there enough volunteers?