How Many Auctions Are There? by Ted Muller

The number of possible 52-card bridge deals is:

53,644,737,765,488,792,839,237,440,000

The number of possible auctions is:

128,745,650,347,030,683,120,231,926,111,609,371,363,122,697,557

That is a great number indeed; it seems that auctions outnumber bridge deals by a factor of:

2.4 quintillion to one!  (that's 18 zeros)

Let us see how this monster figure is derived.

After a bid of 7 Notrump, there are seven possible continuations:

• P P P
• P P Dbl P P P
• P P Dbl Rdbl P P P
• P P Dbl P P Rdbl P P P
• Dbl P P P
• Dbl Rdbl P P P
• Dbl P P Rdbl P P P

These terminal patterns accrue after any bid is made and there is no subsequent overcall.  I refer to them collectively as the Basic Matrix.

After a bid of 7 Spades, any one of the basic follow-ups may occur.  Additionally, within the Basic Matrix, any of the three final passes in any of the seven sequences (shown in red) could be replaced by 7 Notrump, after which the same patterns would apply, with no further variations.  (Replacing earlier passes would be redundant.)  The possible sequences:

This total also can be represented as:    (7 × 1) + (7 × 21)  =  7 × 22

Now consider a bid of 7 Hearts.  Again the Basic Matrix applies, as well as any of the twenty-one final passes that could be replaced by either of two higher-ranking bids:

Substituting from the 7-spade calculation, this totals:   (7 × 22) + 21 × (7 × 22)  =  22 × (7 × 22)  =  7 × 22²

By continuing in this manner, a nice pattern emerges which can be extrapolated to all thirty-five opening bids:

The combined total for all opening bids can be summed as:   7 × (1 + 22 + 22². . . + 22³⁴)

There is a handy-dandy formula for summing the powers  0x  of integer n.  It is:  (n^x+1 − 1) ÷ (n − 1).

Application of this formula to our summation yields:  7 × (22³⁵ − 1) / 21, which factors to: (22³⁵ − 1) / 3.

We're not quite finished.  The above calculation must be multiplied by 4 to accommodate any number of initial passes.  Finally, the unique case of a passout is included.

Our grand total becomes:  4 × (22³⁵ − 1) / 3 + 1.  Rearranged for simplicity, we have:

(4 × 22³⁵ − 1) ÷ 3

Who would have guessed that the number of bridge auctions would be a function of the powers of 22?

Note:  In case you still lack a full appreciation of the size of that number, consider this hypothetical comparison:

• The number of grains of sand on the earth is estimated as  10²⁰  to  10²⁴.  Assuming the maximum, that's  100 million × 100 million × 100 million  grains.
   
• There are an estimated  10¹¹  stars in our galaxy, and an estimated  10¹¹  galaxies in the visible universe.
   
• Speculating that our own galaxy is of average size, that's  10 million × 10 million × 100 million  total stars.
   
• Envision an average of  12  earth-like planets orbiting every star in every galaxy.

Under those conditions, if you were to count all the grains of sand in the entire universe, the total still would fall short of the number of possible bridge auctions!

For the less astronomically minded, I offer this equivalent:

• If each of earth's 7 billion humans could simultaneously operate 7 billion computers, each of which could spit out 7 billion unique bridge auctions every second, it still would take nearly 12 billion years* to compile a complete listing!

Even simpler yet: That total is approximately equal to the odds against choosing 30 consecutive winners on a roulette wheel.  So the next time your partner suggests that your bidding system is ready, you might refer her/him to this article!

Yes, I appreciate that this mind-boggling concept can be difficult to reconcile; after all, it's just five different denominations on seven levels, right?  I submit that the doubter might glean a better understanding with this empirical test:  Just start at the level of, say, 7 Clubs, and begin listing the possible iterations on paper.  If you can manage to record an average of three auctions per minute amidst all your other activities, you will be nearly finished in a year's time.  Then go back and start the bidding at 6 Notrump, but plan on spending 22 years this time.

When you're finished with that, you might like to consider some truly large numbers.  Did you know that there are more than 288 billion different ways that the two players can make their first four moves in a chess game?  If you don't believe that, then try counting them; you won't even get through two moves.  And don't try to itemize the number of possible 40-move games, because you can forget about grains of sand.  That total far exceeds the number of atoms in the known universe!

Note: A Google search for something such as "count possible bridge auctions" probably will list at least one of the following pages, which feature differing methods of calculation:

• https://www2.stetson.edu/~efriedma/papers/bridge/bridge.html
• https://www.johnfaben.com/blog/counting-bridge-auctions
• http://www.kuliniewicz.org/blog/archives/2008/07/20/how-many-bridge-auctions-are-possible

Unfortunately, each of those pages displays an incorrect total of the possible auctions, because each author overlooked a key component of the formula; and one formula has a typo in it as well.

* Sorry, all.  When this article was featured in the ACBL Bulletin, January 2022, I was prompted to review it, and discovered an error of my own.  My original claim that the auction-count would take nearly a trillion years was off by a factor of 80!  Who is safe when even mathematicians falter?