# Balanced mixed tournament plans: explanation of list

The Mixed tournaments list is presented as a table, with each row detailing a set of tables (of the same or different sizes) and a block design that can be applied to them. The rows can be sorted by clicking on the ↑↓ symbols at the head and foot of each column.

The columns are listed below.

### Tables

The number of tables.

### Set

The set of tables. Some examples:

3,4 A 3-player table and a 4-player table. A 4-player table and another 4-player table at which a different game is played. Each pair of players must play together the same number of times at each table. You can envisage them as playing hearts at one table and Doppelkopf at the other: each must be balanced. Two interchangeable 4-player tables. Each pair of players must play together the same number of times at one of these 4-player tables, but it doesn't matter which. Two interchangeable 4-player tables and a 1-player table. The player at the 1-player table plays solitaire, or "sits out". Four 3-player tables forming two pairs, the members of a pair being interchangeable. Envisage two tables of Skat players and two of Tapp-Tarock.

### Total

The total number of players, as specified in the "Set" column. This is v in the block design formula t-(v,b,r,k,λ).

### λ

In the course of the tournament, each pair of players will play together λ times at each table or set of interchangeable tables. If the tables are "3,4", and λ is given as "1,2", each pair of players will meet once at the 3-player table and twice at the 4-player table. This is λ in the block design formula t-(v,b,r,k,λ).

### No. of rounds

The number of rounds in the tournament. This is b in the block design formula t-(v,b,r,k,λ).

### Combinations

The number of ways the players can be assigned to the tables for one round.

### Group order

The order of the symmetry group of the block design.

### Group

The symmetry group of the block design.

### Structure

The name of the Steiner system or other block design, or other algebraic or geometric structure corresponding to the design of the tournament.

### Listing

A list which specifies how the players will be arranged for each round.

For example, for 3,3, this row specifies "abc, abd, adf, cdf, bcf, bef, aef, ace, cde, bde". This shows for each round which three of players named {a,b,c,d,e,f} will sit at the first table.

For 2*4,1, it specifies "∀ a,b ∋ℤ3: (a±1,b±1)&(a±2,0,b±2)". Each player is identified by an ordered pair of members of ℤ3. For each of the nine rounds, we pick an ordered pair a,b from ℤ3, and seat players (a±1,b±1) at one table and (a±2,0,b±2) at the other, while player (a,b) sits out.

### Images

 Block diagram for 3,3.
 The 9-element affine plane, used for 3,3,3.

Two types of image are used to represent block designs: block diagrams as seen to the right and geometric diagrams as seen to the left.

A block diagram, or incidence matrix, has a column for each player and a row for each round. The colour of the square in that row and column indicates at which table that player should sit for that round. You can use your mouse to re-order the rows and the columns: this does not affect the underlying structure.

A geometric diagram has a vertex for each player and a line (which may be curved) for each round. It may not be obvious how such a diagram relates to the structure of the tournament. In the example to the left, for 3,3,3, three of the rounds will be structured according to the green lines, with the three green lines representing the three tables and trios of players moving from one of these to another over these three rounds; and the same for the other colours.

### Status

This is "confirmed" if I know the structure exists, "does not exist" if I know it not to exist, and "unknown" otherwise.