perfect bridge hands


Forgive me if this is a repeat. I'm a recent member and I tried to search this topic.

Your challenge is to calculate the number of perfect bridge hands. There may be several ways to classify "perfect," but for the purpose of this challenge, perfect means any of the hands in this exercise will capture all 13 tricks of a legitimate declaration of 7 no-trump. The hand will stand alone with no contingincies as to distribution of the other 39 cards. Some hands are pure lay-downs and others might require an experienced bridge player to point out the proper sequence to play the cards. Accidental distribution in the dummy, or partial information gained by the initial lead can create other seemingly perfect hands, but a perfect hand stands on its own against all distribution contingencies, in other words, an experienced bridge player could demonstrate how the hand could take all 13 tricks by reference to the hand alone. To meet this standard, the hand would have to start with a stopper in all four suits - first clue.

Good luck to all who try,


I'll be back with my answer next Saturday (mine certainly subject to being wrong)


First up we must have all four aces otherwise a bad lead will hurt.
That leaves 9 cards to consider. The possible distributions of the remaining nine cards and their frequencies are:

        suit    longest                      shortest   number of
lengths suit suit possibilities
9 . . . AKQxxxxxxx A A A 4 * 10c7
8 1 . . AKQJxxxxx AK A A 4 * 3 * 9c5
7 2 . . AKQJTxxx AKQ A A 4 * 3 * 8c3
7 1 1 . AKQJTxxx AK AK A 4 * 3 * 8c3
6 3 . . AKQJT9x AKQJ A A 4 * 3 * 7c1
6 2 1 . AKQJT9x AKQ AK A 4 * 3 * 2 * 7c1
6 1 1 1 AKQJT9x AK AK AK 4 * 7c1
5 4 . . AKQJT9 AKQJT A A 4 * 3
5 3 1 . AKQJT9 AKQJ AK A 4 * 3 * 2
5 2 2 . AKQJT9 AKQ AKQ A 4 * 3
5 2 1 1 AKQJT9 AKQ AK AK 4 * 3
4 4 1 . AKQJT AKQJT AK A 4 * 3
4 3 2 . AKQJT AKQJ AKQ A 4 * 3 * 2
4 3 1 1 AKQJT AKQJ AK AK 4 * 3
4 2 2 1 AKQJT AKQ AKQ AK 4 * 3
3 3 3 . AKQJ AKQJ AKQJ A 4
3 3 2 1 AKQJ AKQJ AKQ AK 4 * 3
3 2 2 2 AKQJ AKQ AKQ AKQ 4

where the suite lengths only count the 9 remaining cards (so the suits are actually one longer than the figures above), NcM is the combination function and the x's are cards of that suit but we don't care about their exact value. We can do this because e.g. if we've got 10 cards in the suit, there are only three outstanding and after we've played A, K & Q they are all gone.

So unless I've missed something here or made an error summing the various counts (both of which are quite likely), the number of possibilities is: 3756.

- Pauli


Hi, Pauli, seems we're rhe only two interested, so I will jump ahead to Saturday to say that you and I agree on result and method of approach to solution. You are the first with the solution and I will leave it at that until Saturday just in case a late arrival wants to take the challenge. Of course, if your response doesn't fade by then,... well, it won't be a full challenge.

Several years ago, I responded to a columnist in the Dallas newspaper about this subject and they concluded that the awswer was 4 based on the trophy which was the distribution of your last hand ( TOP 4 3 3 3 ) adjusted for 4 suites. The Bridge columnist, I think it was Goren or someone like that, responded to my letter, and soothed me with, "You must know your way around a bridge table," but he never retracted his error.

Thanks for playing!



Well, Ron, I'd certainly like to be able to try. However, every time I read the bridge column in a newspaper, I realize that even if after reading a thousand of 'em, I still wouldn't have Clue One about how to play bridge... :o)

-- KS



Kinda like poker in the sense that you need to confront the social aspect and communicate, not really limited to technical skills. I managed a group of technical specialists at IBM several years ago and they played bridge for lunch any time there were four or more in town. Based on the theory that top specialists were adept in both technical and communications skills I gladly requisioned tables and chairs for their library. The ritual of Bridge offered a relief from the intensity of the technical support groups in those days of much lower product reliability. It gave our team a high level subject for discussion away from the minframes and operating systems for at least 60 minutes.



Also, bridge is pretty easy to get into. The rules are about as simple as they get in card games, and you can gradually work your way into the subtleties of bidding and playing over time. I can't think of any card game I have ever enjoyed more (except maybe poker, but that's different).

- Thomas


Absolutely right, Thomas!

Few games have the following that lasts generations. Like you said poker's good, but different. A good national common denominator, bridge, except within a large sample of marriages. I think it begins with a more rapid learning rate of one spouse or more time by one to study a new game. As I said earlier, the game demands good communication skills in all ways not just the coded signals.

One most certainly does not need to know how to calculate solutions to puzzles to play the game. That is artificial, like composed problems in chess. There are over 635 billion bridge hands and nearly 2.6 million 5 card stud hands in a deck of 52. The chance of being dealt either one is pretty slim, so why bother with the math except for amusement?



accoring to Pauli and I, the correct answer is 3756 perfect hands. The hands are contained within 18 formats with varying distributions among the models. Pauli sent the first response with the solution justified in a spreadsheet format. Why he (or someone) deleted the response I have no idea, but up to yesterday it still printed.

A little fun project for someone learning Excel might be to program a spreadsheet to solve the distribution of models after generating their profile description,i.e., AKQJ AKQ AKQ AKQ (or ( a little more difficult ), from the playing rules.



Why he (or someone) deleted the response I have no idea, but up to yesterday it still printed.

I don't see any deleted responses and I certainly haven't deleted by solution.

- Pauli

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