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Dr Ian Plummer

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How to Arrange an All-Play-All Tournament

Does anyone have the format for 10 teams all play all in 9 rounds?

Phil Cordingley responds

Here is a generic method, suitable for programming, for generating n player all play all formats. I shall use the 10-player version as an example. Basically, it involves populating a matrix in this form:  

  J I H G F E D C B

where A - J are the ten players, and the numbers are the rounds. So, for example, A plays E in round 6, for that is the round number at the A/E intersection. Similarly C plays G in round 2, F plays J in round 9. And so on.

On inspection, there is a (slightly complicated) pattern to filling in the matrix for any even value of n. Basically, for each row, starting in the column after you started the previous row (start in column 1 for row 1). Fill in numbers consecutively from 1, until you reach the pivot column, which is the column representing the player playing him/herself. At this point, skip the number (the pivot number) you have reached, and continue numbering upwards. When you reach the top, continue across the row you started in from column 2, until you return to the starting point. Finally, put the pivot number in column 1. (There are one or two obvious special cases which I leave you to discover for yourself).

For example, to fill in the rounds for player D: start in column 4, fill in 1 2 3 across. Now skip the pivot (=4) and fill in 5 6 7 up. Now starting at column 2 for player D, fill in 8 9. Finally fill in column 1 with 4, the pivot number for player D.

For odd numbers of n, use the table for n+1, but consider the last player as BYE.

Actually, it's much easier to code than it is to describe. No idea why it works. Some smart arse could probably extend it to do lawn allocation as well.

Phil Cordingley

Kevin Carter adds:

This is the same as a common duplicate bridge movement, where 10 pairs, sitting at 5 tables, all play each other during the course of the session. A clever chap called Howell devised the 'Howell movement' for many numbers of pairs and printed them on cards (which he sold to bridge clubs).

Author: Phil Cordingley
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Updated 28.i.16
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