revised May 2017

# A nice 4 table 24 board Howell

This page describes a movement I received from John Probst, and some improvements of it. The result is a very nice movement for 7 or 8 pairs, playing 8 rounds.

## Original movement.

If you want to play 24 boards, choices are a movement of 8 rounds, of 3 boards each, and a movement of 6 rounds of 4 boards each. If 8 pairs participate, with 8 rounds each pair meets one single pair twice, while with 6 rounds each pair meets one single pair not at all. This skewness applies to only 3 boards out of the 24 in one case and to 4 boards in the other. The 8-round movement is therefore preferrable.

The solution of John Probst was to apply the Worger principle to a standard Howell movement for 8 pairs and 7 rounds. This means that an extra set of boards is introduced at a few strategically chosen places in the movement, to substitute the original sets of the Howell movement. The boards thus released are played in an extra 8th round. This sounds complicated, but is really very simple as shown below:

```VERSION 1

1       2       3       4
3- 6 A  7- 2 C  5- 4 H  8- 1 E
4- 7 B  1- 3 D  6- 5 E  8- 2 F
5- 1 C  2- 4 E  7- 6 H  8- 3 G
6- 2 D  3- 5 F  1- 7 G  8- 4 A
7- 3 E  4- 6 G  2- 1 A  8- 5 B
1- 4 F  5- 7 A  3- 2 H  8- 6 C
2- 5 G  6- 1 B  4- 3 C  8- 7 D
2- 3 B  6- 7 F  4- 5 D  8- 1 H
```
Compare this to the original Howell:
```  1       2       3       4
3- 6 A  7- 2 C  5- 4 D  8- 1 E
4- 7 B  1- 3 D  6- 5 E  8- 2 F
5- 1 C  2- 4 E  7- 6 F  8- 3 G
6- 2 D  3- 5 F  1- 7 G  8- 4 A
7- 3 E  4- 6 G  2- 1 A  8- 5 B
1- 4 F  5- 7 A  3- 2 B  8- 6 C
2- 5 G  6- 1 B  4- 3 C  8- 7 D
```
As we see, the only difference is the extra 8th round, and, the occurence of board set H in rounds 1, 3 and 6, at table 3.

An elegant movement.

## Optimalization.

Version 1 shown above turns out to have s.d.= 2.56, Qf=65.63. This may be easily improved by program "balans" to s.d.= 1.40, Qf=87.50. In fact a multitude of possibilities exist that have this same quality factor. After some experimenting it was found that keeping the first 7 rounds fixed already yields such a movement:
```VERSION 2

1       2       3       4
3- 6 A  7- 2 C  5- 4 H  8- 1 E
4- 7 B  1- 3 D  6- 5 E  8- 2 F
5- 1 C  2- 4 E  7- 6 H  8- 3 G
6- 2 D  3- 5 F  1- 7 G  8- 4 A
7- 3 E  4- 6 G  2- 1 A  8- 5 B
1- 4 F  5- 7 A  3- 2 H  8- 6 C
2- 5 G  6- 1 B  4- 3 C  8- 7 D
3- 2 B  7- 6 F  5- 4 D  8- 1 H
```
Compared to version 1 all that had to happen was to undo the arrow switches in the last round!

## Similar movement derived from Superperfect Howell

As discussed elsewhere on this site, 4-table Howells with perfect balance do not all have the same properties. Some of them are more perfect then others, as shown by the behaviour in the "2 strong pairs" model. It was tempting to see if an 8-round movement derived from such a "superperfect" Howell behaves differently. Only a small twitch (i.e. an arrow switch at table 2) is needed to convert our Howell to a superperfect one, and, following the same recipe as above, and 8-round movement may be derived.
```VERSION 3.

1       2       3       4
3- 6 A  2- 7 C  5- 4 H  8- 1 E
4- 7 B  3- 1 D  6- 5 E  8- 2 F
5- 1 C  4- 2 E  7- 6 H  8- 3 G
6- 2 D  5- 3 F  1- 7 G  8- 4 A
7- 3 E  6- 4 G  2- 1 A  8- 5 B
1- 4 F  7- 5 A  3- 2 H  8- 6 C
2- 5 G  1- 6 B  4- 3 C  8- 7 D
3- 2 B  7- 6 F  5- 4 D  8- 1 H
```
Movement version 3 has again s.d.= 1.40, Qf=87.50.
We will show this version is the best. Tablecards for this movement are available.

For those who prefer the Universal Begin Position we also give a renumbered version:

```VERSION 3b.

1       2       3       4
1- 2 A  3- 4 B  5- 6 C  7- 8 D
1- 5 E  8- 6 F  3- 2 G  4- 7 A
1- 3 H  7- 2 C  8- 5 A  6- 4 D
1- 8 B  4- 5 G  7- 3 E  2- 6 H
1- 7 F  6- 3 A  4- 8 H  5- 2 B
1- 4 C  2- 8 E  6- 7 B  3- 5 D
1- 6 G  5- 7 H  2- 4 F  8- 3 C
1- 2 D  3- 5 F  6- 4 E  7- 8 G
```
This is essentially the same movement as version 3.

## Tests on the balance

As noticed already the standard deviation and quality factor are the same for versions 2 and 3.

We will now test them with the "two strong pairs" model. The following graphs show the distribution of the scores for all possible choices of the two strong pairs.

It is clear that the original version 2, is somewhat less fair to the average players then version 3, derived from the superperfect Howell. The scores with version 2 have a larger spread, and the extreme values are further away from the ideal value 38.1.

The difference in quality is larger in the Bussemaker model. With movement version 2 the following percentage probability table is obtained:

```Pair  1  2  3  4  5  6  7  8
Pos
1   80 19  1  .
2   18 64 17  2  .
3    2 15 59 16  8  .
4    .  2 21 56 19  2  .
5       .  2 19 56 21  2  .
6          .  8 16 59 15  2
7             .  2 17 64 18
8                .  1 19 80
Qd 64.7
```
We recall that in this model all pairs have different strengths and are ordered by their strength, pair 1 being the strongest pair. The table shows that within the framework of the model, pair 1 has an 80% probability of ending at the first place.

For movement version 3 the similar table looks as follows:

```Pair  1  2  3  4  5  6  7  8
Pos
1   90 10  .
2   10 78 11  .  .
3    . 11 69 13  7  .
4       1 19 62 18  1
5          1 18 62 19  1
6          .  7 13 69 11  .
7             .  . 11 78 10
8                   . 10 90
Qd 74.7
```
A considerable improvement. The numbers on the diagonal are now a lot closer to the ideal value 100.

## A movement for 7 pairs?

It would be nice if the movement would also be useful for 7 pairs. Sometimes a pair does not show up, and a subsitute pair is not available. Then we would like to use the same movement where now one pair is absent, and the opponents of that pair have a sit-out. (By the way, this is another reason to choose the 8-round movement: the sit-out is now only for 3 boards rather then 4).

In movement version 2 we obtain Qf ranging from 51.82 to 73.08, depending on which of the pairs is absent. The best values are for leaving out pair 1, s.d. = 1.49, Qf = 73.08.

In movement version 3 the s.d. = 1.03, and Qf = 86.36 when one of the pairs is absent. This is independent of which pair is absent!

In other words movement version 2 has Qf1av=61.62 and Qf1max=73.08, while movement version 3 has Qf1av = Qf1max=86.36

It turns out that in this respect movement version 3 is really superior to version 2.

In conclusion we may say that version 3 (or the equivalent version 3b) is the winner, and is the best movement for 4 tables and 8 rounds.

Note. With a few extra arrow switches it is possible to further improve the 7-pair movement to s.d. = 0.70, Qf=95, but that implies separate movements for 7 and 8 pairs.