Volume 24, N°3

Backgammon tools in J

2. Wastage

by Howard A. Peelle (hapeelle@educ.umass.edu)

J programs are presented as analytical tools for expert backgammon. Part 2 here develops a program to compute the number of pips expected to be wasted while bearing off pieces in one inner board (without contact).

The inner board is represented as a list of six integers. For example, 0 pieces on the 1, 2, and 3 points, 2 on the 4-point, 3 on the 5, and 4 on the 6:

   board =: 0 0 0 2 3 4  

To count pips for bearing off, sum the board times a list of the point numbers:

   ,pips =:  +/ board * 1 2 3 4 5 6  
47  

Use program N (from [1]) to compute expected number of rolls to bear off:

   ,n =: N board  
6.56681  

Expected number of pips to bear off is the expected number of rolls n times the average number of pips per roll (8 1/6):

   n * 8+1%6  
53.6289  

Wastage is this expected number of pips minus the pipcount.

   ,wastage =: (n * 8+1%6) - pips  
6.62895  

Define a program to compute wastage for any inner board:

   Wastage =: (N * 8:+1:%6:) - Pips  
      Pips =: +/ . * Points  
         Points =: 1: + i.@6:  

For the example above:

   Wastage 0 0 0 2 3 4  
6.62895  

Wastage for all possible full distributions (with all 15 pieces) in the inner board follows:

   all =: (6#16) #: i.16^6
   boards =: (15 = +/"1 all) # all

(See Appendix for an alternative way to generate these boards using partitions.)

Wastages for all full inner boards are:

   wastages =: Wastage"1 boards  

The minimum wastage is:

   <./wastages  
7.06895  

The board with minimal wastage is:

	boards #~ wastages = <./wastages  
0 0 0 3 5 7  

Other distributions can be searched similarly:

# Pieces	#boards	Minimum Wastage	Best board
15	15504	7.06895	0 0 0 3 5 7
14	11628	7.02941	0 0 0 3 5 6
13	8568	6.98028	0 0 0 3 4 6
12	6188	6.90551	0 0 0 2 4 6
11	4368	6.82085	0 0 0 2 4 5
10	3003	6.75322	0 0 0 2 3 5
9	2002	6.62895	0 0 0 2 3 4
8	1287	6.51014	0 0 0 1 3 4
7	792	6.36102	0 0 0 1 3 3
6	462	6.14734	0 0 0 1 2 3
5	252	5.88967	0 0 0 1 2 2
4	126	5.52802	0 0 0 1 1 2
3	56	5.30579	0 0 0 1 1 1
2	21	4.62037	1 0 1 0 0 0
1	6	4.20833	0 0 0 0 0 1

Reference

1. Peelle, Howard A. “Backgammon Tools in J: 1. Bearoff Expected Rolls”, Vector, Vol.24, N°2

Appendix

Program to compute wastage:

Wastage =: (N * 8:+1:%6:) – (+/ . * 1: + i.@6:)

See [1] for script with definition of N to compute expected number of rolls to bear off.

Co-recursive program to compute all partitions of an integer x into y parts:

   ELSE =: `
   WHEN =: @.

Partitions  =: ;@AllParts
   AllParts =: <@Parts"0 >:@i.
      Parts =: Ps ELSE (,.@[) WHEN (]=1:)
         Ps =: Add1 ELSE (#1:) WHEN =
            Add1 =: 1: + ] {."1 – Partitions - <. ]

For example:

   8 Partitions 3
8 0 0
7 1 0
6 2 0
5 3 0
4 4 0
6 1 1
5 2 1
4 3 1
4 2 2
3 3 2

Program to produce a table of permutations:

Perms =: i.@! A. i.

All full distributions of 15 pieces on 6 points in an inner board:

   boards =: ~. ,/ (Perms 6) {"1/ 15 Partitions 6

Number of full inner boards:

   #boards
15504

All wastages:

   wastages =. Wastage"1 boards

Average wastage:

   Average =: +/ % #
   Average wastage
15.7857

Minimal wastage:

   <./wastages
7.06895

Board with minimal wastage:

   boards #~ wastages = <./wastages
0 0 0 3 5 7

Top ten minimal boards:

   10 {. boards /: wastages
0 0 0 3 5 7
0 0 1 2 5 7
0 0 1 3 5 6
0 0 0 2 5 8
0 0 1 3 4 7
0 0 0 2 6 7
0 0 0 3 6 6
0 0 0 4 5 6
0 0 0 3 4 8
0 0 1 2 6 6