Six-Handed Bridge

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A six suited deck was spotted at http://www.rightfast.com/empire/, as reported to me on November 2, 2002 by P. Genszler.

Back in the eighth decade of the twentieth century, while I was a student at the University of Waterloo in Ontario, we had a large number of people who played bridge. We played bridge during lunch breaks. Once or twice a week, we played duplicate bridge in the evening at the local clubs. About once a month, we travelled many miles to get to sectional, regional and national tournaments. (Around the year 2000, I started playing duplicate again, usually once a week, sometimes less, sometimes more. November 10, 2004, cycle will show rank change to LIfe Master. For part of October 2004, the listing on-line was at exactly 300.00 points.)

Since we had an ample supply of bridge players, and since students like to experiment, we also played other games. We played Diplomacy in my residence and Dungeons and Dragons in the mathematics student society offices.

We also played a six-handed version of bridge. Since I've not seen this described elsewhere on the web, I'll try to set down some rules and statistics.

Making a Six-Suit Deck: Take two normal packs of playing cards, with the same backing. Take the clubs and diamonds form the second pack and set the rest of that pack aside. Use a marker to replace the diamonds by red circles and the clubs by black circles. Now, with the first deck added, you have six suits. With our great imaginations, we called the new suits Red Rounders and Black Rounders. I don't remember who came up with the idea for this game, or if it was a group idea. I know I made myself a deck like that. (Was it the first? I would doubt it, but I have no proof.)

A few people have written to me recently and stated that they have seen six-handed decks, with the two extra suits called bells and stars, or called anchors and eagles. I do not have such a deck in my possession, nor do I know where one can purchase such a deck at this time (August 7, 2002). Of course if you do find out where to but one, I'd be happy to get one as a curiosity if it is not too expensive. It has been over twenty years (nearer thirty?) since I actually played the game. I don't recall that we had a scoring mechanism. If there was one, Red Rounders and Black Rounders would presumably have been worth 20 points each. We did order the suits Red Rounders, Black Rounders, Clubs, Diamonds, Hearts, Spades, No Trump. This preserved the perceived red-black-black-red-red-black symmetry.

Bidding would proceed as in normal bridge. Players bid for six tricks in addition to the number stated. Doubles and Redoubles are still permitted. There are two teams of three players, seated alternately. The contract is played by the first player (the declarer) who bid the trump suit on the team that wins the bidding. The player to the left of the declarer makes the opening lead. The other two members of the declaring side turn their hands face up (double-dummy play on every hand).

The two dummies make for some differences from ordinary bridge. For example, with hearts as trump:

Declarer	Dummy 1	Dummy 2
AK - xx xx - -	- xx xx xx - -	- xx xx xx - -

Declarer leads the Ace and King of Spades, discarding the diamonds from the first dummy and the clubs from the second. Now, declarer can cross-ruff the clubs and diamonds between the two dummies.

If the defenders had been on lead, they could have taken two clubs and two diamonds.

There are obvious other differences. In normal bridge, declarer can choose to lead from the dummy or from the closed hand (by taking the previous trick in the appropriate place). Now, declarer has three choices. It might be best to focus on which hand you'd like to play last. (A finesse towards the AQ in the second dummy would succeed 2/3 of the time, towards the first dummy only 1/3.)

One might argue that declarer has more power in this version, since he plays three hands, and each defender is at the mercy of two others. I don't recall that bothering us.

Distributions: (short version)

70% of all hands have a suit of 4 cards or longer.
21% of all hands have a suit of 5 cards or longer.
3.6% of all hands have a suit of 6 cards or longer.
0.4% of all hands have a suit of 7 cards or longer.

40% of all hands have a void.
97% of all hands have a singleton or void.

Distributions:

13-0-0-0-0-0	6
12-1-0-0-0-0	5070
11-2-0-0-0-0	182520
11-1-1-0-0-0	790920
10-3-0-0-0-0	2453880
10-2-1-0-0-0	34800480
10-1-1-1-0-0	37700520
9-4-0-0-0-0	15336750
9-3-1-0-0-0	319004400
9-2-2-0-0-0	261003600
9-2-1-1-0-0	1696523400	0.001%
9-1-1-1-1-0	612633450
8-5-0-0-0-0	49691070
8-4-1-0-0-0	1435519800	0.001%
8-3-2-0-0-0	3445247520	0.002%
8-3-1-1-0-0	11197054440	0.005%
8-2-2-1-0-0	18322452720	0.008%
8-2-1-1-1-0	26465765040	0.012%
8-1-1-1-1-1	2867124546	0.001%

7-6-0-0-0-0	88339680
7-5-1-0-0-0	3445247520	0.002%
7-4-2-0-0-0	11484158400	0.005%
7-4-1-1-0-0	37323514800	0.017%
7-3-3-0-0-0	8421716160	0.004%
7-3-2-1-0-0	179152871040	0.081%
7-3-1-1-1-0	129388184640	0.059%
7-2-2-2-0-0	48859873920	0.022%
7-2-2-1-1-0	317589180480	0.144%
7-2-1-1-1-1	114684981840	0.052%

6-6-1-0-0-0	2296831680	0.001%
6-5-2-0-0-0	20671485120	0.009%
6-5-1-1-0-0	67182326640	0.030%
6-4-3-0-0-0	42108580800	0.019%
6-4-2-1-0-0	447882177600	0.203%
6-4-1-1-1-0	323470461600	0.147%
6-3-3-1-0-0	328446930240	0.149%
6-3-2-2-0-0	537458613120	0.243%
6-3-2-1-1-0	2328987323520	1.056%
6-3-1-1-1-1	420511600080	0.191%
6-2-2-2-1-0	1270356721920	0.576%
6-2-2-1-1-1	1376219782080	0.624%

5-5-3-0-0-0	28423292040	0.013%
5-5-2-1-0-0	302320469880	0.137%
5-5-1-1-1-0	218342561580	0.099%
5-4-4-0-0-0	39476794500	0.018%
5-4-3-1-0-0	1231675988400	0.559%
5-4-2-2-0-0	1007734899600	0.457%
5-4-2-1-1-0	4366851231600	1.980%
5-4-1-1-1-1	788459250150	0.358%
5-3-3-2-0-0	1478011186080	0.670%
5-3-3-1-1-0	3202357569840	1.452%
5-3-2-2-1-0	10480442955840	4.753%
5-3-2-1-1-1	7569208801440	3.433%
5-2-2-2-2-0	1429151312160	0.648%
5-2-2-2-1-1	6192989019360	2.809%

4-4-4-1-0-0	285110182500	0.129%
4-4-3-2-0-0	2052793314000	0.931%
4-4-3-1-1-0	4447718847000	2.017%
4-4-2-2-1-0	7278085386000	3.301%
4-4-2-1-1-1	5256395001000	2.384%
4-3-3-3-0-0	1003587842400	0.455%
4-3-3-2-1-0	21349050465600	9.682%
4-3-3-1-1-1	7709379334800	3.496%
4-3-2-2-2-0	11644936617600	5.281%
4-3-2-2-1-1	37846044007200	17.164%
4-2-2-2-2-1	10321648365600	4.681%

3-3-3-3-1-0	2609328390240	1.183%
3-3-3-2-2-0	8539620186240	3.873%
3-3-3-2-1-1	18502510403520	8.391%
3-3-2-2-2-1	30276835205760	13.731%
3-2-2-2-2-2	4954391215488	2.247%
Total	220495674290430

Splits of one suit in the other three hands:

There are 84478098072866400 ways to distribute 39 cards in 3 hands.

As in regular bridge, the most likely splits are the ones that are nearest to even, without being even.

7 cards outstanding
3-2-2	28670669047238400	33.939%
4-2-1	23892224206032000	28.282%
3-3-1	17520964417756800	20.740%
4-3-0	6738832468368000	7.977%
5-1-1	3583833630904800	4.242%
5-2-0	3308154120835200	3.916%
6-1-0	75145360185600	0.870%
7-0-0	28274821545600	0.033%

6 cards outstanding
3-2-1	45053908502803200	53.332%
2-2-2	12287429591673600	14.545%
4-1-1	9386230938084000	11.111%
4-2-0	8664213173616000	10.256%
3-3-0	6353756327318400	7.521%
5-1-0	2599263952084800	3.077%
6-0-0	133295587286400	0.158%

5 cards outstanding
2-2-1	34814383843075200	42.211%
3-1-1	21275456792990400	25.185%
3-2-0	19638883193529600	23.247%
4-1-0	8182867997304000	9.686%
5-0-0	566506245967200	0.671%

4 cards outstanding
2-1-1	40616781150254400	48.080%
3-1-0	22912030392451200	27.122%
2-2-0	18746206684732800	22.191%
4-0-0	2203079845428000	2.608%

3 cards outstanding
2-1-0	56238620054198400	66.572%
1-1-1	20308390575127200	24.040%
3-0-0	7931087443540800	9.388%

2 cards outstanding
1-1-0	57800803944592800	68.421%
2-0-0	26677294128273600	31.579%

Department of Mathematical Sciences
Florida Atlantic University
777 Glades Road
Boca Raton, Florida 33431-0991
USA

Office: Room 286, Science & Engineering
Phone: (561) 297-3350
Fax: (561) 297-2436
URL: http://www.math.fau.edu/locke/Bridge6.htm

Last modified May 19, 2002, by S.C. Locke.