Main Article 7Z73 by Richard Pavlicek
Argh. Looks just like a credit-card number. Hopefully, I wont compromise somebodys credit card in this article; but just in case, Ill protect my ass with a disclaimer: The numbers in this file are not related to credit cards. [Knock at the door] Police officer: Are you Richard Pavlicek? Then, you have the right to remain silent
The ordering of a specific dealprint is arbitrary. My practice is to start with West, if only to keep West on the left where it belongs. The other three hands follow in clockwise order: North, East and South. For suits in each hand, almost everyone is accustomed to the order by rank: spades, hearts, diamonds and clubs.
How many specific dealprints exist? To determine this, I found all the ways the 560* hand patterns could be packed into a deal. For each specific West shape, I found which North shapes would fit; then with those paired, which East shapes would fit no further step was needed because, once three hands fit, whatever is left must represent the fourth hand. The total number of specific dealprints is exactly:
*Most people know that there are 39 generic hand patterns, but each of these can be permuted in 4, 12 or 24 ways to create 560 specific patterns. For example, the most common generic shape is 4-4-3-2, which comprises 12 specific shapes: 4=4=3=2, 4=4=2=3, 4=3=4=2, 4=3=2=4, 4=2=4=3, 4=2=3=4, 3=4=4=2, 3=4=2=4, 3=2=4=4, 2=4=4=3, 2=4=3=4 and 2=3=4=4.
The following table lists all the specific dealprints well, not really, as most of them are missing. I only listed the first and last, and some random selections in between, although all 37,478,624 were calculated. (To receive the full table, please send me a stamped, self-addressed envelope on second thought, better send a cargo ship to Port Everglades.)
The Combinations column shows the number of actual deals that can be created from each dealprint. To verify the accuracy of the table, I totaled the combinations and was pleased to see it agreed with the total number of bridge deals 52!/(13!)4.
|Total for 37478624 entries: 53644737765488792839237440000|
For convenience in writing dealprints, I use hexadecimal for the numbers above nine. A = 10, B = 11, C = 12, D = 13.
The last group (6412) represents the South hand, which is forced by the first three groups. Therefore, the South hand can be omitted, and the same dealprint is reduced to 3253-2452-2326. Similarly, the club suit in each hand is fixed by the other three suits, so it also can be eliminated. This reduces the dealprint to only nine significant numbers, 325-245-232, which can be thought of as a 3×3 matrix with the rows representing West, North and East, and the columns representing spades, hearts and diamonds.
Does every 3×3 matrix represent a specific dealprint? No. There are 14^9 = 20,661,046,784 possible 3×3 matrices where each digit can be from 0 to 13, inclusive. In order to represent a valid dealprint, the matrix must pass three tests:
If the above conditions are true, the matrix represents one, and only one, specific dealprint. To verify this, I tested each of the 20+ billion possible matrices, and exactly 37,478,624 passed. This is the same number that I found by my previous method of counting dealprints, so it must be right. Either that, or Im so far lost that Ive come full circle.
In order to implement this I needed to devise a scheme to transform a specific dealprint to its generic equivalent. I decided that each generic dealprint should start with the hand containing the longest suit (rather than always West) with its suits arranged in descending order of length, i.e., the first hand would always be one of the 39 generic hand patterns. I will call this the dominant hand.
Another consideration was whether to allow the four hands to be permuted (like the suits), but I decided that keeping the order was important. This is especially significant when you consider the play rotation of the cards. If two hands are swapped, say North and East, the deal becomes quite different and should not be of the same generic type. Therefore, I only allow the hands to be rotated, which essentially changes nothing. By this method, the total number of generic dealprints is relatively small:
In my article Patterns and Freakness I used a different scheme to define generic shapes and found 412,666. The difference arises when two suits have identical patterns the previous way did not account for this so they were considered unique.
To illustrate the transformation, consider this specific dealprint:
The dominant hand is East because it has the longest suit (seven hearts), so the hands are rotated to make East first. Note that the order of hands does not change as it becomes:
The next step is to arrange the suits of the first hand so its lengths are in order, longest to shortest. Since hearts is Easts longest suit, that suit will appear first, followed by clubs (next longest), diamonds and spades (shortest). Each change made in the first hand is replicated in the other three hands so the suit patterns around the table are unchanged. The final generic dealprint becomes:
The above example was easy, but what about ties? What if two or more hands have equally long suits? The dominant hand is then decided by the second longest suit, and if that still ties, the third longest suit. For example, 6421 wins against 6331, and 5431 wins against 5422. If two or more hands have identical dominant patterns, the tie is broken by the number of cards in its longest suit held by the next hand (clockwise), and if this ties, the next hand. If a tie still remains, it goes to the number of cards in its second longest suit held by the next hand, etc. If the tie cannot be broken, the dealprint is symmetrical, and it doesnt matter.
Once the dominant hand is selected, if it has two (or three) identical suit lengths (e.g. 5332 or 7222), these are ordered by the length of the suit in the next hand (clockwise), or the next hand, etc. Here also, if a tie cannot be broken, it doesnt matter. Below is a specific dealprint to illustrate tiebreaking:
Note that North and East tie for the dominant shape (7411). North wins because its seven hearts are followed by four hearts in East, while Easts seven clubs are followed by only two clubs in South. Rotating North to the front becomes temporarily:
Now there is another tie to break. Obviously, the suit order of the first hand will be 7411, but which of the 1s (spades or clubs) should be first? (This matters because the two patterns are different around the table.) Since the next hand has longer clubs than spades, the club suit wins the tie, and the final generic pattern becomes:
Generic dealprints also contain redundant information and can be fully represented in a 3×3 matrix. The only difference (compared to specific dealprints) is that the top row represents the dominant hand (rather than West) and the suits are ordered by length in the dominant hand (rather than by rank). The above generic dealprint becomes:
A file with all 393,197 generic dealprints in order of frequency is available on the Bridge Utilities page.
|Suit||Dealprints||Number of Deals||Percent|
The next table lists the most common generic dealprints (all that occur more than 0.1 percent of the time). The P column shows the number of distinct permutations that exist to form specific dealprints. This number is usually 96 (387,681 cases), occasionally 48 (5371 cases), rarely 24 (140 cases) and even more rarely 16 (5 cases). The total number of deals is shown, followed by the percent occurrence. Two patterns share the top spot at just over 0.2 percent.
|Generic Dealprint||P||Number of Deals||Percent|
© 2003 Richard Pavlicek