Let's approach it in a somewhat round-about way. The owner of a small club has a rather strange dream. He has exactly seven tables and decides to invite seven teams to come and play a match-pointed team game. It'll be a little bit like a board-a-match, but where each team's final score is the sum of all the match-points earned by the N/S pair added to those earned by the E/W pair. Another rule is that for every new round, each team's captain can decide which way to field its two pairs. There's just one snag. In each round, the two pairs of any one team will actually meet each other so the team can play the boards out if they like, but of course those scores won't make any difference to the result (they will balance out).
The plan is that at the end of each round, the boards will go down one table, the E/W pairs will go down two tables and the N/S pairs will go down three tables.
At the last minute, a pair of gate-crashers hobbles in – one on crutches, the other in a wheelchair. After a little thought, the director realizes that this is going to work out just fine. This pair will "bump" the N/S pair whenever an entire team is due to meet. As it happens, these meetings are always at the same table so this stationary pair will never have to move. This pair is assigned the number 8.
But the weird nature of the dream is only just beginning. As the game is finally about to start, the director notices that three of the tables seem to have had their chairs stolen and can't be used (so pairs arriving at these tables will be sitting – or standing – out the round). The team captains are quick to ensure that their best pairs (naturally, these are the pairs including themselves) are sitting at real tables. As the game gets started and all goes quiet, music from next door drifts in: Empty chairs at empty tables. Hearing this, one of the smarter players of the idle pairs realizes that as the movement progresses, whenever the pair makes it to a real table, the better half will be arriving at a chair-less table. The captain will no doubt switch them around so that the idle pair remains idle. In fact, this pair realizes that they are going to spend the entire evening standing around never getting to play a single card! They decide to go home. The other idlers follow them.
The remaining players finish the evening after seven rounds and the scores are totaled. Each of the original teams (1-7), gets the score achieved by their "good" pair, while the late (stationary) pair that so fortuitously arrived gets the score for pair 8. An enjoyable game of bridge between eight pairs with a single winner, and every pair playing every other pair, has just been completed. As it happens, the "weaker" pairs of each of the seven original teams never got to play a hand: they didn't really need to know how to play bridge (or whist). In fact, they don't need to exist at all – they're phantoms!
Perhaps Edwin Howell had a dream like this. Or maybe he came at the movement another way. But it was definitely a brilliant idea.
So let's now see how the tables are arranged and how it all works in real life. Logical table number corresponds to the table numbers in the dream. Physical table number is the number of the (real) table in a Howell movement. This example corresponds to the chart shown on the Bridge Guys web site. The table shows how everything is arranged for rounds one and two. Bumped pairs are shown in parentheses. Phantom pairs are marked in red italics.
| logical # | physical (play) # | Rnd 1: board set | Rnd 1: NS pair | Rnd 1: EW pair | Rnd 2: board set | Rnd 2: NS pair | Rnd 2: EW pair |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 8 (1) | 1 | 2 | 8 (2) | 2 |
| 2 | 2 | 2 | 6 | 5 | 3 | 7 | 6 |
| 3 | 3 | 3 | 4 | 2 | 4 | 5 | 3 |
| 4 | bye stand | 4 | 2 | 6 | 5 | 3 | 7 |
| 5 | 4 | 5 | 7 | 3 | 6 | 1 | 4 |
| 6 | bye stand | 6 | 5 | 7 | 7 | 6 | 1 |
| 7 | bye stand | 7 | 3 | 4 | 1 | 4 | 5 |
(see also Howell some more)
How do you determine which of the logical tables will be the bye-stand tables?
ReplyDeleteThat's a very good question - and I apologize for not noticing that you had asked it over a year ago. The movement specified in the article (E/W down 2, N/S down 3) is arbitrary. So of course is the movement of the boards but that's a bridge convention so we might as well stick with it. Once that movement has been decided upon, the bye stands can be determined by looking for the phantom pairs (in red in the table) that make the whole thing work. The one thing you know for sure is that table 1 (where the stationary pair resides) always has a real E/W pair throughout the movement.
DeleteAs it's now over three years since I published this article, I'm not sure that I can explain it any better - but this is how I recall it. It involves a little bit of trial and error.
It took me a while to figure out, but Howell derived his tables by superimposing orthogonal Latin squares. Latin squares were very popular puzzles in the 19th century and would have been well-known to him.
ReplyDeleteIt's not clear to me how much he knew (or needed to know) of group theory in order to create his method. I would be interested to see some of Howell's original table cards. Howell movements have many possible solutions and have been re-calculated many times since 1897, so many of the ones we use now are not likely to be the same as the ones he originally published.
Very interesting. Thanks.
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