# A reduced Howell for 12 pairs and 9 rounds

In newsgroup rec.games.bridge the question arose what would be the fairest form for a 6-table 3/4 Howell, with 9 rounds. In such a reduced Howell each pair does not meet some of the other pairs. To compensate for this we want such combinations of pairs to play more then average in the same compass direction. Criteria for the quality of the balance are the Quality factor, and the standard deviation of the score matrix. The balance may be optimized by switching directions at some tables ("arrow switches").

Starting from the reduced Howell for 6 tables, 9 rounds, given on page 60 of Hallén et. al. "Movements - a fair approach" (1994), a little bit of optimizing with program 'balans', and twiddling with parameters, I arrived at the following movement:

```Table 1       2         3         4         5         6
NS EW  b  NS EW  b  NS EW  b  NS EW  b  NS EW  b  NS EW  b
10- 1  1  12- 8  2  11- 5  3   7- 3  4   9- 6  5   2- 4  6
10- 2  2  12- 9  3  11- 6  4   8- 4  5   1- 7  6   3- 5  7
3-10  3  12- 1  4  11- 7  5   9- 5  6   2- 8  7   4- 6  8
10- 4  4  12- 2  5  11- 8  6   1- 6  7   3- 9  8   5- 7  9
10- 5  5  12- 3  6  11- 9  7   2- 7  8   4- 1  9   6- 8  1
6-10  6  12- 4  7  11- 1  8   3- 8  9   5- 2  1   7- 9  2
10- 7  7  12- 5  8  11- 2  9   4- 9  1   6- 3  2   8- 1  3
10- 8  8  12- 6  9  11- 3  1   5- 1  2   7- 4  3   9- 2  4
9-10  9  12- 7  1  11- 4  2   6- 2  3   8- 5  4   1- 3  5
```
A straightforward reduced Howell, with the only irregularity that pair 10 sits EW in rounds 3,6 & 9.
This movement has a quality factor of 91.3, whereas the original movement from Hallén et. al. has Qf=69.3.

The complete score matrix is:

```   *   5   5   3   3   5   3   7   3   5   3   3
5   *   3   7   3   5   3   3   5   5   3   3
5   3   *   3   5   3   5   5   3   3   5   5
3   7   3   *   5   5   3   3   5   5   3   3
3   3   5   5   *   3   7   3   5   5   3   3
5   5   3   5   3   *   3   5   3   3   5   5
3   3   5   3   7   3   *   5   5   5   3   3
7   3   5   3   3   5   5   *   3   5   3   3
3   5   3   5   5   3   5   3   *   3   5   5
5   5   3   5   5   3   5   5   3   *   3   3
3   3   5   3   3   5   3   3   5   3   *   9
3   3   5   3   3   5   3   3   5   3   9   *
sd= 1.311, Qc=90.69, Qf=91.13
```
The average amount of competition is 45/11, or slightly over 4. The largest deviation from this average occurs for pairs 11 and 12. The value 9 is about twice the average, which means that pairs 11 and 12 meet each other effectively twice. These two pairs compete at all boards in the same compass direction, which is an overcompensation of the absence of a direct meeting. The TD should make sure that these two pair numbers are occupied by two pairs of equal strength, and certainly not by one strong pair and one weak pair.

The original movement as given by Hallén has a score matrix with values running from -9 to +9, thus deviating from the average by values between -13 and +5