Balanced mixed tournament plans: Introduction

7 = 3 + 4

Suppose you are running a tournament to play Skat, a three-player card game. Unfortunately seven players arrive. Seven is not divisible by three. How do you plan things?

You propose one standard three-table, and one four-player table where the players play Doppelkopf. The players agree to this, but they want everything to be "balanced". Each player should spend the same proportion of the day at the three-player table; and each player should play the same number of hands against each opponent.

The Fano plane

The Fano plane provides a solution. You assign each of the players to one of the seven vertices of the Fano plane. You divide the day into seven sessions: to each session you assign one of the seven "lines" of the Fano plane: in that session, the players whose points lie on that line will play at the three-player table, and the other four at the four-player table.

(Throughout these pages, we will use the word "line" to refer to the often curved arcs that link sets of points.)

This ensures that each player will spend 3/7 of the day at the three-player table; each pair of players will sit together at the three-player table exactly once; and each pair of players will sit together at the four-player table exactly twice.

This all works because the Fano plane is a "balanced block design". It has a set of vertices and a set of "blocks" (here represented by lines) arranged so that

It can be regarded as the Steiner system S(2,3,7), and as the block design 2-(7,7,3,3,1). Another page explains this notation.

7 = 3 + 3 + 1

Two interwoven Fano planes

The solid lines form one Fano plane, the broken lines form another. For each solid line, there is a broken line (shown in the same colour) that has no point in commmon with it.

Suppose the seven entrants in your Skat tournament aren't happy about playing Doppelkopf, they prefer to have two three-player tables and one player sitting out. Can we still arrange the tournament so that after seven sessions, each pair of players has played together once at each table?

It turns out that we can. The diagram to the right shows how to do it. For each session, we pick a solid line, and as before the three players whose points lie on that line play at one table. We then find the unique broken line having no point in common with it, and those three players play at the other table.

Block diagram

The columns represent the players, the rows represent the sessions. A red square shows that the player is sitting at the red table for the session, green likewise, light grey means he is sitting out.

This balanced block design is 2-(7,14,6,3,2). It can be represented by the incidence diagram shown to the left. This diagram is "live": you can rearrange it by using your mouse to drag-and-drop any square. Moving a square vertically moves its row, moving it horizontally moves its column, and moving it obliquely moves a row and a column. The result always still represents the same balanced block design.

Some terminology

These "balanced mixed tournament plans" are:
   balanced because each player is involved in an identical range of activities, and encounters each other player equally often
   mixed because more than one game is involved (in the second example above, the two Skat tables could be different, each player will spend half their time at each
   tournament plans because I am presenting these plans in the context of games tournaments; other applications are possible

Introduction to tournament plans
Explanation of list of tournament plans
List of tournament plans
Notation used for block designs
Some infinite sets of block designs
Further reading on block designs