Top Tips For Chess Organisers/All-play-all Tournaments and the Berger Tables

The most reliable way to pick a winner from a group of competitors, whether individuals or teams, is for everyone to play everyone else. In the chess world this is called an all-play-all or APA. You might also hear "round-robin" or in Europe, "league".

When everyone plays everyone else, the issue of schedule strength disappears. The silver medalist can't complain that they had tougher opponents than the gold medalist, because everyone had the same opponents. Plus, you can publish every Player's complete tournament schedule in advance, which is tremendously useful! (We'll call them "Players" from here on, but this can refer to individuals or teams.)

The downside is that the length of the tournament has a linear relationship to the number of Players. If you are one of 10 Players, you have to play nine other people in nine games of chess. Recruit an 11th player, you now have 10 games each. So APAs are only really suitable for events with a limit on the number of Players, such as invitational norm tournaments, or divisions in a team league system.

In chess in particular, there is a further complication: having the White pieces more often than your rivals is an advantage. So in a single APA, where you play everyone else once only, you need some sort of method to keep the total number of Whites and Blacks for each Player approximately equal, and ideally prevent them having the same colour more than twice in a row. The Berger Tables provide a mathematically optimal fixture list satisfying these two requirements.

In high-level competitions where the exact balance of White vs. Black is important, a single APA needs to have an odd number of Players, so that the number of games played by each Player is even. (Every round will have one Player with a bye.) This means that one Player will have finished their schedule before all the others, which is a slight disadvantage - everyone else knows what they have to do to finish ahead of that Player.

The World Championship Candidates' Tournaments, with eight Players, solve the issue by using a double APA. Everyone plays everyone else twice, once with each colour. This avoids any complaints by the silver medallist that they had the Black pieces against the gold medallist in the decisive game. Early Candidates' Tournaments went further and used a quadruple APA. The ideal number of Players in a double APA is even, to avoid byes.

In team tournaments, it doesn't matter if a single APA has an even number of teams, because in a multiple-board match between Team A and Team B, Team A can have White on half the boards and Black on the other half. Or if there are an odd number of boards per match, and travelling to other teams isn't as simple as walking across a hall, the visiting team can have White on the odd-numbered boards (of which there are one more). In this kind of tournament, the balance of home and away matches is the important constraint rather than the balance of colours.

Berger Tables

So these just look like tables of numbers. What do they have to do with chess tournaments?

Let's say we have six players and we need them all to play each other within five rounds of games. Firstly, we need to be looking at the Berger Table of the correct size, the 5-6 Players table.

Berger Table For 5 Or 6 Players

Round 1		Round 2		Round 3		Round 4		Round 5
W	B	W	B	W	B	W	B	W	B
1	(6)	1	2	3	1	1	4	5	1
2	5	5	3	4	5	2	3	4	2
3	4	(6)	4	2	(6)	(6)	5	3	(6)

Now, assign every player to a number between 1 and 6. They will keep this number for the whole tournament, so don't pick one that they're scared of.

1. Alexander Alekhine
2. Barbara Botvinnik
3. Ciaran Carlsen
4. Daisy Dommaraju
5. Eustace Euwe
6. Fiona Fischer

Under Round 1, there are three pairs of numbers: 1-6, 2-5, 3-4. This means that in Round 1, Player 1 has the White pieces against Player 6, Player 2 has the White pieces against Player 5, and Player 3 has the White pieces against Player 4. The first number of each pair always has White (and/or is always the Home team if Players need to travel to each other's venues).

• AA vs. FF
• BB vs. EE
• CC vs. DD

The pairs of numbers (or "pairings") for Round 2 are 1-2, 5-3, 6-4, which generates the games

• AA vs. BB
• EE vs. CC
• FF vs. DD

And so on for Rounds 3, 4 and 5.

The Optional Player 6

The brackets around (6) indicate that there may not be a Player 6, if you only have five players in the APA. If you're using the 5-6 Player Berger Table with only five Players, you must assign them to numbers 1-5 and leave 6 vacant. The tables are designed so that the highest-numbered team, and only the highest-numbered team, can be removed without skewing any other Player's White-Black ratio.

If you ignored this advice and assigned "Bye" to number 1, it would cause Player 5, who already has three games with Black and only two with White, to lose their Round 5 game against 1 which is one of their two White games. Three Blacks and one White is unfair, especially when Player 2 would lose one of their two Black games and end up with three Whites and one Black. Assigning the bye correctly to number 6 means that every player loses one of their "excess" games - if they've got three Whites and two Blacks they lose a White, if they've got three Blacks and two Whites they lose a Black.

From the point of view of the other five Players, they have two games with White, two games with Black, and one game against Player 6 which may be either White or Black (or not exist). This fact is very useful in team competitions if you have one team with odd venue requirements. If you have a six-team APA, and one of the teams is a junior team who must play all of their matches at the junior club on Friday nights, that team should be Player 6. Every other team will get two regular home matches, two regular away matches, and one visit to the junior club (which may count as either home or away for the purposes of assigning colours). Likewise, a travelling team that doesn't have a venue of their own should be Player 6, so that every other team gets two home, two away, and one visit from the travelling team. (What to do when multiple Players have competing claims to be Player N, where N is the total number of Players, is a question we will return to.)

The existence of the Optional Player 6 should be your first indication that these Berger Tables are actually quite clever, but there's more.

Colour Alternation

The second mathematically optimal property of the Berger Tables is that no Player ever gets three of the same colour (or venue) in a row, and only gets two in a row on a maximum of one occasion.

It might seem as though the perfect set of fixtures should be able to achieve perfect alternation of colours for every Player. Alas, no two Players can have the exact same sequence of colours as each other, because they would never be able to play each other. (A game of chess where both players have the White pieces will not go well.)

So if two Players both have White in Round 1, one or both of those Players needs to have a break in the alternation to allow the fixture between the two to take place. A break in this case means two consecutive Whites, or two consecutive Blacks.

A quite amazing property of the Berger Tables is that, wherever a break occurs, one of the two consecutive games with the same colour will be against the Optional Team N. So if Team N doesn't exist, you have perfect alternation of colours after all!

	Round 1 Opponent	Round 2 Opponent	Round 3 Opponent	Round 4 Opponent	Round 5 Opponent
Player 1	6 (White)	2 (White)	3 (Black)	4 (White)	5 (Black)
Player 2	5 (White)	1 (Black)	6 (White)	3 (White)	4 (Black)
Player 3	4 (White)	5 (Black)	1 (White)	2 (Black)	6 (White)
Player 4	3 (Black)	6 (Black)	5 (White)	1 (Black)	2 (White)
Player 5	2 (Black)	3 (White)	4 (Black)	6 (Black)	1 (White)
Player 6	1 (Black)	4 (White)	2 (Black)	5 (White)	3 (Black)

In this table, which shows every Player's schedule round by round, you can see that replacing all the 6's with "Bye" (which is neither White nor Black) will result in the remaining Whites and Blacks for each Player alternating nicely.

3-4 Players

Round 1		Round 2		Round 3
W	B	W	B	W	B
1	(4)	1	2	3	1
2	3	(4)	3	2	(4)

7-8 teams

Round 1		Round 2		Round 3		Round 4		Round 5		Round 6		Round 7
H	A	H	A	H	A	H	A	H	A	H	A	H	A
1	(8)	1	2	3	1	1	4	5	1	1	6	7	1
2	7	7	3	4	7	2	3	4	2	2	5	6	2
3	6	6	4	5	6	7	5	6	7	3	4	5	3
4	5	(8)	5	2	(8)	(8)	6	3	(8)	(8)	7	4	(8)

Each team plays each other team in ascending numerical order, alternating colours/venues. When a team would be scheduled to play itself, it instead plays the highest-numbered team. This formula gives you two ways to generate each pairing - once from the point of view of each team.