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Message from discussion 15 positions that minimize waste of pips in bear-off

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More options Apr 30 1996, 3:00 am
Newsgroups: rec.games.backgammon
From: Stig Eide <sti...@james.stud.unit.no>
Date: 1996/04/30
Subject: Re: 15 positions that minimize waste of pips in bear-off

Trice:

> Your 2 2 2 3 3 3 distribution is good, given a 15 checker
> position with a pip count of 48. (In fact, subject to these two
> constraints, I believe it is optimal, but I'm going on memory.)

It's close, but 1 2 3 4 3 2 is the optimal distribution.
As pointed out by several, we can't control the pipcount. And then
our initial posting with minimal pip-waste for 1-15 checkers left
is not of much value, except for the 000357 position which should
be the aim in the bear-in.
As a consequence Eirik made a table of optimal boards for all
possible combinations of pips and checkers left.
Interested can get a postscript-file by emailing him:
th...@stud.unit.no

Here's the optimal board for each pipcount, no matter
how many checkers left. Interestingly, fewest
checkers left is often not optimal:
The 48-pip board 000008 (8 men at 6point)
wastes 1.49 pips more than 000144.

You can always transform pipwaste to expected rolls left:
Just add the pipcount, and divide it with 8.1667.
Enjoy..    8-)

1:100000(7.17)
2:200000(6.17)
3:110000(5.17)
4:101000(4.62)
5:000010(4.30)
6:000001(4.21)
7:010010(5.05)
8:001010(5.26)
9:001001(5.32)
10:000101(5.11)
11:000011(5.01)
12:000002(5.23)
13:001101(5.40)
14:001011(5.40)
15:000111(5.31)
16:000021(5.45)
17:000012(5.47)
18:001111(5.67)
19:000211(5.79)
20:001012(5.73)
21:000112(5.53)
22:000022(5.65)
23:000013(5.85)
24:001112(6.06)
25:000212(6.05)
26:000122(5.89)
27:000113(5.97)
28:000023(6.05)
29:001122(6.21)
30:000222(6.25)
31:000132(6.23)
32:000123(6.15)
33:000114(6.33)
34:000024(6.38)
35:001123(6.37)
36:000223(6.38)
37:000133(6.36)
38:000124(6.39)
39:000034(6.55)
40:001133(6.56)
41:000233(6.53)
42:000224(6.55)
43:000134(6.51)
44:000125(6.62)
45:000035(6.74)
46:001134(6.66)
47:000234(6.63)
48:000144(6.69)
49:000135(6.67)
50:001234(6.77)
51:001144(6.81)
52:000244(6.76)
53:000235(6.75)
54:000145(6.78)
55:000136(6.83)
56:001235(6.86)
57:001145(6.87)
58:000245(6.82)
59:000236(6.88)
60:000146(6.89)
61:001245(6.92)
62:000345(6.93)
63:000255(6.94)
64:000246(6.91)
65:000156(6.99)
66:000147(7.00)
67:001246(6.98)
68:000346(6.98)
69:000256(6.98)
70:000247(7.00)
71:001346(7.05)
72:001256(7.04)
73:000356(7.03)
74:000347(7.05)
75:000257(7.04)
76:000248(7.09)-001356(7.09)
77:001347(7.11)
78:001257(7.09)
79:000357(7.07)
80:000267(7.11)
81:000258(7.11)
82:000249(7.18)
83:000159(7.22)
84:00014A(7.35)
85:00005A(7.48)
86:00004B(7.68)
87:00003C(7.99)
88:00002D(8.47)
89:00001E(9.19)
90:00000F(10.17)
Where A,B,C,D,E and F represents 10,11,12,13,14 and 15 checkers.
Stig Eide
and
Eirik Milch Pedersen