Nonillion-to-one Bridge Mystery

The actual odds against the British winning the world bridge championship from the U.S. were about 5 to 6. The theoretical odds against two identical hands occurring during the same evening's play were exactly 1,287,473,706,371,731,028,141,698,599,999 to 1.

Both happened in New York last week. England's crack team, already champions of Europe, defeated the top team of the American Contract Bridge League in eight sessions by a stunning 5,420 points. The identical hands came in the third session, within two hours of each other, from different decks, one green and one brown, the cards appearing exactly the same in all the positions around the table.

Although the "nonillion-to-one hand" (it actually was mediocre) did not affect the outcome of the match, the mystery almost obscured the tournament itself for the succeeding days. The British were playing brilliantly (see next page), but many onlookers were too busy asking very British but unbridgish questions to notice. Whodunit? Howdunit? Was there hanky-panky in the locked room? (Tournaments are played in both an open and a closed room.) Could it have been pure chance?

The actual circumstances were not very revealing. The third session began, as all tournament sessions do, with the players "preparing" the 10-inchlong numbered, aluminum tournament boards (center of table, below). These are delivered to the scene with fresh decks already separated into 13-card hands which are inserted in the boards' neat slots. Each competing player takes one or more of the boards that will be used in the next six or eight hands and prepares it in advance. This means removing the cards from the slots, putting them together, shuffling them, dealing them out in four new hands of 13 cards each, and returning them to the slots. When these chores are done, the players take the lowest numbered board, draw out their hands and begin to play. You must understand that in this kind of bridge the players never shuffle, cut and deal at the table the way you do when you have a foursome in your own home.

A little after midnight on the night of the third session, the U.S.'s Alvin Roth, playing Board 75 (the 75th hand of the 224 that comprised the match), suddenly said: "Wait a minute—I've played this hand before!" A quick check showed that Board 64, played some two hours earlier and by now safely stacked away for official recording, showed exactly the same hand. There was consternation, several official huddles and, finally, a ruling that Board 75 be reshuffled, redealt and replayed in both the closed and open rooms.

The next day The New York Times quoted the odds against the appearance of identical hands as 158,000,000,000 to 1. IBM put some of its best mathematicians on the problem, and a day later they raised the ante and offered "the right answer": 1 out of 5.3645 times 10 to the 28th power.

Then Editor Alfred Sheinwold of the American Contract Bridge League's monthly Bulletin proved that IBM, not knowing about bridge arrangements, had forgotten to multiply by 24, which is the number of possibilities for four bridge hands to be arranged on a board (NESW, NSEW, NWES, etc.). Sheinwold's calculation of "the exact odds" was the 31-figure number cited earlier—one million plus a couple hundred octillions. Bridge expert Oswald Jacoby concurred in this theoretical estimate.

A QUINTILLION YEARS

The Sheinwold figure meant that if all the people on earth shuffled and dealt cards all day, every day, it would take them quintillion years to deal nonillion hands. But it had not been quintillion years since the last identical deal—only 10, as a matter of fact. In 1945 some Scottish bridge players reported receiving the same cards twice during the same evening.

Something wrong with the shuffling? Yes, said the New York Times, something was amiss in the shuffling. But what? SI had assigned me to cover the championship matches; now I decided to play bridge detective as well. Boards 64 and 75 certainly showed some evidence of eccentric card distribution. Each of the four hands had three similar honors—three Jacks, three Queens, three Kings, three Aces. And several suits went around the table regularly—2, 3, 4, 5, etc. Perhaps the clue to the identical hands was not in the open and closed rooms at all, but farther back, in the offices of the ACBL, which provides both cards and boards.