The Book of Bids Vol2 was inspired by a web based complete book of bridge deals. http://bridge.thomasoandrews.com/impossible/

To be a book the possible outcomes must map( or encode)  to a unique page number and there should be no gaps or blank pages. The mapping scheme should be deterministic both ways. You might get away with the odd ‘page intentionally left blank’ but the real magic is having a clean one-to-one mapping.

This web page is about how BOBV2 was constructed using unfamiliar number systems with bases other than 10. The encoding chosen for BOB is a 35 digit  sequence that becomes a number expressed in base 22.

Base 22 numbers need 22 symbols to cover the range of column numbers 0 to 21. The symbol used are  0 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L. Here A=10, B=11, L=21 etc.

The number CC0DIJ = 12x22⁵ +12x22⁴ +0x22³ +13x22² +18x22¹ + 19x22⁰ =64661363 decimal. In any base other than 10, arithmetic can seem strange but 1J+B=28 is good in base22.

All the base 22 numbers  00000000000000000000000000000000000 _base22      thru to

11111111111111111111111111111111110 _base22 are used in this scheme so a continuous and unique numbering system is obtained.  

The latter is the number of legal bidding sequence where at least one  bid has been made in base 22. 

Remember the encoding for the sequence1h,2d,3s,4d,5c,5s,6h which enumerates to the binary number   [00100,01001,00010,01000,10010,00100,00000].  This encoding has1C as a big number and aesthetically we want the reverse. So encode right to left. [00000,00100,00001,00010,01000,00010,00100]. 

Here is the tough bit-There are  21 in-between-bid call sequences (see graphic) but, for the moment, consider that there are only 9.  All the (1)s, except the leading leftmost, are  replaced with the correct IBB number that shows the call sequence to the next bid.   N.B. There are no “in-betweens” after the leftmost (1) , because the calls that complete the bidding  and the passes that start the bidding are encoded as a separate amble digit to be incorporated later.

The leftmost (1) is, from an IBB perspective, always a (1). That means our consecutive  bidding sequences starting at 1C are not numerically continuous. They  proceed as shown with jumps or steps because no sequence starts with a 2 or 3 etc only 1 (leftbox) .

We have to rationalise these number patterns  into a regular continuous  numerical sequence(rightbox).

The method is write  1s under all the digits bar the 0s left of the leading 1 and the leading 1 itself. Then do an addition with carry forward.

There are however 21 possible IBB sequences for each bid  used but method is the same although the arithmetic is base 22, as on the left  and now the whole encoding is a valid  base 22 number in a numerically continuous series . The graphic shows the complete IBB choice, and the corresponding alphanumeric code. NB that ‘bids without in-between sequences’ ie where there is a immediate opposition overcall use the alphanumeric (1).

An actual 35 bid sequence would look something like

 [00000,00100,0000H,000K0,0I000,000B0,00900]_base22

This 35 digit, base 22 number is, in principle, converted to a decimal (base 10) number then  multiplied by 28(4x7)  to create the ‘room’ to add in the encoding  of the start and end call sequences. These sequences are enumerated with the formula [4 x end + start] =ambledigit, a number between 0 and 27. 

The routines actually  convert the base 22 number to a base 28 number, shifts 1 digit left and inserts the start/end (amble) digit in the rightmost empty slot. That base 28 number is converted to a decimal page number albeit with 1C PPP as page zero. If that offends add 1 to page number.