Bridge Auction Esoterica

I was surprised to find out that the total number of possible bridge auctions is greater than the total possible number of bridge hands by a factor that is an order of magnitude of 18 - metaphorically this blew my mind.

The total number of bridge auctions is 128,745,650,347,030,683,120,231,926,111,608,371363,122,697,557 - and I had to take a break whilst typing it.

The estimate for grains of sand on earth is 10^20-10^24. Therefore the total number of bridge auctions is likely greater than the square of the number of grains of sands on the planet.

The extra auctions got me thinking that it might be possible to show where every card in the deck is. My first idea was that the cheapest possible bid would deny possession of a card. Start with the two of clubs then the three .... then diamonds, hearts, and spades.* Note that only three actions are needed to determine the location of each card.

* Suit rank goes in aplhabetical order for those unfamiliar with bridge.

The auction would go like this:
p-p-p(2c)-1c
p-p-(3c)dbl-p
p(4c)-rdb-p-p(5c)
1d-p-p(6c)-dbl
p-p(7c)-rdb-p
p(8c)-p-1d-p(9c)
p-dbl-p(10c)-p
rdbl-p(Jc)-p-1h
p(Qc)-p-dbl-p(Kc)

Since we have got one suit out before the end of the first level we are clearly ahead of schedule. The problem with this is that dbl and rdbl are only available to one side at a time. An auction where the card was in the hand of the person who could not make one of those bids would look like this:


p-p-p(2c)-1c
-p-1d(3c)-p-1h(4c)
-p-1s

I gave up early because it is one card per bid. There are only 35 bids (and the three passes at the start) so this will not work.Instead we must leverage the dbl and rddbl by using them whenevr they are possible. This gives the following auction:


p-p-p(2c)-1c
-dbl-rdb(3c)-p-p
1d(4c)-dbl-rdbl-p(5c)
p-1h-dbl(6c)-rdbl
p-p(7c)-1s-dbl
rdbl(8c)-p-p-1N(9c)
dbl-rdbl-p(10c)p-
2c-dbl(Jc)-rdbl-p
p(Qc)-2d-dbl-rdbl(Kc)
2h-dbl-rdbl(2d)

We have used just over seven bids to show 13 cards so we will definitely be finished before the seven level,