--------------------------------------- IdealRacingPositions.exe, Version 1.1 Copyright 2011 Michael J. Mannon All rights reserved. --------------------------------------- Ideal Racing Positions What does it mean to say that a certain racing position is ideal? Position 1, for instance, the famous 3-5-7 position, is often described as the ideal starting position from which to begin a bear off in a race. Is Position 1 better than Position 2, another popular bear-off position? Is either one really ideal? Position 1 24 23 22 21 20 19 18 17 16 15 14 13 Player O ------------------------------------------------------- OOO | | | | Score : 0 OOO | | | | Pips : 0 OOO | | | | EPC : 0 OOO | | | | Rolls : 0 OOO | | | | | | | | | | | | | | | | | |BAR| |v Cash game | | | | | X | | | | X | | | | X X | | | | X X | | | Rolls : 10.5391 | X X X | | | EPC : 86.0689 | X X X | | | Pips : 79 | X X X | | | Score : 0 ------------------------------------------------------- 01 02 03 04 05 06 * 07 08 09 10 11 12 Player X Position 2 24 23 22 21 20 19 18 17 16 15 14 13 Player O ------------------------------------------------------- OOO | | | | Score : 0 OOO | | | | Pips : 0 OOO | | | | EPC : 0 OOO | | | | Rolls : 0 OOO | | | | | | | | | | | | | | | | | |BAR| |v Cash game | | | | | | | | | | | | | | | | | | | | Rolls : 8.13892 | X X X | | | EPC : 66.4679 | X X X X X X | | | Pips : 57 | X X X X X X | | | Score : 0 ------------------------------------------------------- 01 02 03 04 05 06 07 08 09 10 11 12 Player X It should not be a surprise that Position 2 is the better racing position. Its pip count is 57, fully 22 pips better than Position 1. If you have to choose between Position 1 and Position 2, you should pick the latter. You'll win a lot more games. What may be a surprise, however, is that Position 1 is an ideal racing position, and Position 2 is not. Computer programs have solved this problem completely. The average (i.e, mean) number of rolls remaining for any racing position has been calculated exactly. The results form what is called a one-sided bear-off database. Looking up Position 1 shows that it will require 10.5391 rolls on average to bear off all fifteen checkers. Position 2 requires only 8.13892 rolls. The programs used to create one-sided bear-off databases make calculations based on the expected number of rolls remaining. It is common, however, to multiply the number of rolls by 8 and 1/6 (the average number of pips in a roll), to produce what is called the effective pip count (EPC). For any position, the effective pip count is always larger than the actual pip count. The difference is termed wastage. In practice, wastage occurs because some checkers will be borne off using numbers higher than necessary. Every time you use a 6 to bear off a checker from a lower point, you have "wasted" part of your roll. In Position 1, where the pip count is 79, the EPC is 86.0689. On average, slightly more than 7 pips will be wasted during the bear off. So what makes Position 1 ideal? The answer is that it minimizes wastage. There is no other arrangement of checkers having a pip count of 79 that has less waste. We know this because a brute force search through all the thousands and even millions of positions in a one-sided bear-off database shows it to be true. Position 2, on the other hand, is not ideal. When the pip count is 57, and there are 15 checkers still in play, the ideal arrangement of checkers is given by Position 3. In Position 2, the EPC is 66.4679. For Position 3, it is 66.4081, a small, but nonetheless real improvement. Position 3 is an ideal racing position. Position 2 is not. Position 3 24 23 22 21 20 19 18 17 16 15 14 13 Player O ------------------------------------------------------- OOO | | | | Score : 0 OOO | | | | Pips : 0 OOO | | | | EPC : 0 OOO | | | | Rolls : 0 OOO | | | | | | | | | | | | | | | | | |BAR| |v Cash game | | | | | | | | | | | | | | | | | X | | | Rolls : 8.13161 | X X X | | | EPC : 66.4081 | X X X X X | | | Pips : 57 | X X X X X X | | | Score : 0 ------------------------------------------------------- 01 02 03 04 05 06 * 07 08 09 10 11 12 Player X As players, we have very little control over the pip count. So long as we are not wasting pips, the dice determine what our pip counts will be. What we can control is wastage. By systematically choosing positions that are as close to ideal as possible, we can minimize the average number of rolls in a bear off. To this end, I have produced the tables below. For each pip count and number of checkers still in play, they give the ideal racing position. With only a few exceptions, all of which occur when the checkers are piled onto the lower points, there is only one ideal position for any given pip count and number of checkers. The first table, where the number of checkers is given as 0, lists the ideal positions for each pip count without regard to the number of checkers still in play. There might be only one checker, there might be fifteen, or there might be any number in between. These tables are valid when all checkers are on or below the 11 point. Computations were made using IEC559 type double. Results were output in C++ using setprecision( 14 ), but it is likely that rounding and I/O conversion errors have corrupted the right-most digits. Checkers Pips EPC Rolls 01-02-03-04-05-06---07-08-09-10-11-12 GNUBG ID XGID 0 1 8.1666666666667 1 1 0 0 0 0 0 0 0 0 0 0 0 AQAABAAAAAAAAA:cAkAAAAAAAAA -A----------------------a-:0:0:1:00:0:0:3:0:8 0 2 8.1666666666667 1 0 1 0 0 0 0 0 0 0 0 0 0 AgAACAAAAAAAAA:cAkAAAAAAAAA --A--------------------a--:0:0:1:00:0:0:3:0:8 0 2 8.1666666666667 1 2 0 0 0 0 0 0 0 0 0 0 0 AwAAGAAAAAAAAA:cAkAAAAAAAAA -B----------------------b-:0:0:1:00:0:0:3:0:8 0 3 8.1666666666667 1 0 0 1 0 0 0 0 0 0 0 0 0 BAAAEAAAAAAAAA:cAkAAAAAAAAA ---A------------------a---:0:0:1:00:0:0:3:0:8 0 3 8.1666666666667 1 1 1 0 0 0 0 0 0 0 0 0 0 BQAAKAAAAAAAAA:cAkAAAAAAAAA -AA--------------------aa-:0:0:1:00:0:0:3:0:8 0 4 8.6203703703704 1.0555555555556 0 0 0 1 0 0 0 0 0 0 0 0 CAAAIAAAAAAAAA:cAkAAAAAAAAA ----A----------------a----:0:0:1:00:0:0:3:0:8 0 4 8.6203703703704 1.0555555555556 1 0 1 0 0 0 0 0 0 0 0 0 CQAASAAAAAAAAA:cAkAAAAAAAAA -A-A------------------a-a-:0:0:1:00:0:0:3:0:8 0 5 9.3009259259259 1.1388888888889 0 0 0 0 1 0 0 0 0 0 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22.469421901814 2.7513577838956 0 0 0 0 1 2 0 0 0 0 0 0 0AAAAA0AAAAAAA:cAkAAAAAAAAA -----AB------------ba-----:0:0:1:00:0:0:3:0:8 3 18 23.446877727696 2.8710462523709 0 0 0 1 1 0 0 0 1 0 0 0 KAQAgEIAAAAAAA:cAkAAAAAAAAA ----AA---A------a---aa----:0:0:1:00:0:0:3:0:8 3 19 24.423529906238 2.9906363150496 0 0 0 0 1 1 0 1 0 0 0 0 UAIAACUAAAAAAA:cAkAAAAAAAAA -----AA-A--------a-aa-----:0:0:1:00:0:0:3:0:8 3 20 25.472231067694 3.1190487021666 0 0 0 0 0 2 0 1 0 0 0 0 YAIAACYAAAAAAA:cAkAAAAAAAAA ------B-A--------a-b------:0:0:1:00:0:0:3:0:8 3 21 26.434477306416 3.2368747722142 0 0 0 1 0 1 0 0 0 0 1 0 SBAAgAQBAAAAAA:cAkAAAAAAAAA ----A-A----A--a----a-a----:0:0:1:00:0:0:3:0:8 3 22 27.509707157221 3.3685355702719 0 0 0 0 0 2 0 0 0 1 0 0 YAgAAIYAAAAAAA:cAkAAAAAAAAA ------B---A----a---b------:0:0:1:00:0:0:3:0:8 3 23 28.600465977838 3.5020978748373 0 0 0 0 0 1 0 1 1 0 0 0 IAUAAFIAAAAAAA:cAkAAAAAAAAA ------A-AA------aa-a------:0:0:1:00:0:0:3:0:8 3 24 29.550987915826 3.6184883162235 0 0 0 0 0 1 0 1 0 1 0 0 IAkAAJIAAAAAAA:cAkAAAAAAAAA ------A-A-A----a-a-a------:0:0:1:00:0:0:3:0:8 3 25 30.547992520095 3.7405705126646 0 0 0 0 0 1 0 0 1 1 0 0 IAoAAKIAAAAAAA:cAkAAAAAAAAA ------A--AA----aa--a------:0:0:1:00:0:0:3:0:8 3 26 31.515824576764 3.8590805604201 0 0 0 0 1 0 0 0 0 1 1 0 EBQAAEEBAAAAAA:cAkAAAAAAAAA -----A----AA--aa----a-----:0:0:1:00:0:0:3:0:8 3 27 32.510118841422 3.9808308785414 0 0 0 0 0 1 0 0 0 1 1 0 IBQAAEIBAAAAAA:cAkAAAAAAAAA ------A---AA--aa---a------:0:0:1:00:0:0:3:0:8 3 28 33.573900483495 4.1110898551219 0 0 0 0 0 0 1 0 0 1 1 0 QBQAAEQBAAAAAA:cAkAAAAAAAAA -------A--AA--aa--a-------:0:0:1:00:0:0:3:0:8 3 29 34.551192111164 4.2307582176936 0 0 0 0 0 0 0 1 0 1 1 0 gBQAAEgBAAAAAA:cAkAAAAAAAAA --------A-AA--aa-a--------:0:0:1:00:0:0:3:0:8 3 30 35.56116171753 4.3544279654118 0 0 0 0 0 0 0 0 1 1 1 0 ABUAAFABAAAAAA:cAkAAAAAAAAA ---------AAA--aaa---------:0:0:1:00:0:0:3:0:8 3 31 36.598621700741 4.4814638817233 0 0 0 0 0 0 0 0 0 2 1 0 ABYAAGABAAAAAA:cAkAAAAAAAAA ----------BA--ab----------:0:0:1:00:0:0:3:0:8 3 32 37.587648444847 4.6025691973282 0 0 0 0 0 0 0 0 0 1 2 0 ABoAAKABAAAAAA:cAkAAAAAAAAA ----------AB--ba----------:0:0:1:00:0:0:3:0:8 3 33 38.724762520785 4.7418076556063 0 0 0 0 0 0 0 0 0 0 3 0 ABwAAMABAAAAAA:cAkAAAAAAAAA -----------C--c-----------:0:0:1:00:0:0:3:0:8 Checkers Pips EPC Rolls 01-02-03-04-05-06---07-08-09-10-11-12 GNUBG ID XGID 4 4 14.972222222222 1.8333333333333 4 0 0 0 0 0 0 0 0 0 0 0 DwAA4AEAAAAAAA:cAkAAAAAAAAA -D----------------------d-:0:0:1:00:0:0:3:0:8 4 5 15.199074074074 1.8611111111111 3 1 0 0 0 0 0 0 0 0 0 0 FwAA4AIAAAAAAA:cAkAAAAAAAAA -CA--------------------ac-:0:0:1:00:0:0:3:0:8 4 6 15.199074074074 1.8611111111111 2 2 0 0 0 0 0 0 0 0 0 0 GwAAYAMAAAAAAA:cAkAAAAAAAAA -BB--------------------bb-:0:0:1:00:0:0:3:0:8 4 7 15.451131687243 1.891975308642 2 1 1 0 0 0 0 0 0 0 0 0 KwAAYAUAAAAAAA:cAkAAAAAAAAA -BAA------------------aab-:0:0:1:00:0:0:3:0:8 4 8 16.081275720165 1.9691358024691 1 2 1 0 0 0 0 0 0 0 0 0 LQAAoAUAAAAAAA:cAkAAAAAAAAA -ABA------------------aba-:0:0:1:00:0:0:3:0:8 4 9 16.390046296296 2.0069444444444 1 1 2 0 0 0 0 0 0 0 0 0 NQAAoAYAAAAAAA:cAkAAAAAAAAA -AAB------------------baa-:0:0:1:00:0:0:3:0:8 4 10 16.719121513489 2.0472393689986 1 1 1 1 0 0 0 0 0 0 0 0 VQAAoAoAAAAAAA:cAkAAAAAAAAA -AAAA----------------aaaa-:0:0:1:00:0:0:3:0:8 4 11 18.325988797439 2.2439986282579 1 1 1 0 1 0 0 0 0 0 0 0 lQAAoBIAAAAAAA:cAkAAAAAAAAA -AAA-A--------------a-aaa-:0:0:1:00:0:0:3:0:8 4 12 19.200838763146 2.3511231138546 1 0 1 2 0 0 0 0 0 0 0 0 aQAAIA0AAAAAAA:cAkAAAAAAAAA -A-AB----------------ba-a-:0:0:1:00:0:0:3:0:8 4 13 19.813381550704 2.4261283531474 1 0 1 1 1 0 0 0 0 0 0 0 qQAAIBUAAAAAAA:cAkAAAAAAAAA -A-AAA--------------aaa-a-:0:0:1:00:0:0:3:0:8 4 14 20.41598595552 2.4999166476147 0 1 1 1 1 0 0 0 0 0 0 0 qgAAQBUAAAAAAA:cAkAAAAAAAAA --AAAA--------------aaaa--:0:0:1:00:0:0:3:0:8 4 15 21.65849773599 2.6520609472641 0 1 1 1 0 1 0 0 0 0 0 0 KgEAQCUAAAAAAA:cAkAAAAAAAAA --AAA-A------------a-aaa--:0:0:1:00:0:0:3:0:8 4 16 22.388399447211 2.7414366670054 0 1 1 0 1 1 0 0 0 0 0 0 SgEAQCkAAAAAAA:cAkAAAAAAAAA --AA-AA------------aa-aa--:0:0:1:00:0:0:3:0:8 4 17 23.086574076279 2.8269274379117 0 1 0 1 1 1 0 0 0 0 0 0 UgEAQCoAAAAAAA:cAkAAAAAAAAA --A-AAA------------aaa-a--:0:0:1:00:0:0:3:0:8 4 18 23.668050503695 2.8981286331056 0 0 1 1 1 1 0 0 0 0 0 0 VAEAgCoAAAAAAA:cAkAAAAAAAAA ---AAAA------------aaaa---:0:0:1:00:0:0:3:0:8 4 19 24.794995059472 3.036121844017 0 0 0 2 1 1 0 0 0 0 0 0 WAEAACsAAAAAAA:cAkAAAAAAAAA ----BAA------------aab----:0:0:1:00:0:0:3:0:8 4 20 25.727819734874 3.1503452736581 0 0 1 0 1 2 0 0 0 0 0 0 pAEAgDQAAAAAAA:cAkAAAAAAAAA ---A-AB------------ba-a---:0:0:1:00:0:0:3:0:8 4 21 26.528018903402 3.2483288453146 0 0 0 1 1 2 0 0 0 0 0 0 qAEAADUAAAAAAA:cAkAAAAAAAAA ----AAB------------baa----:0:0:1:00:0:0:3:0:8 4 22 27.648232475843 3.3854978541849 0 0 0 0 2 2 0 0 0 0 0 0 sAEAADYAAAAAAA:cAkAAAAAAAAA -----BB------------bb-----:0:0:1:00:0:0:3:0:8 4 23 28.756866266942 3.5212489306459 0 0 0 1 1 1 0 1 0 0 0 0 qAQAAJUAAAAAAA:cAkAAAAAAAAA ----AAA-A--------a-aaa----:0:0:1:00:0:0:3:0:8 4 24 29.812493607622 3.6505094213415 0 0 0 1 1 1 0 0 1 0 0 0 qAgAABUBAAAAAA:cAkAAAAAAAAA ----AAA--A------a--aaa----:0:0:1:00:0:0:3:0:8 4 25 30.805477427362 3.7720992768198 0 0 0 1 1 1 0 0 0 1 0 0 qBAAABUCAAAAAA:cAkAAAAAAAAA ----AAA---A----a---aaa----:0:0:1:00:0:0:3:0:8 4 26 31.821797400736 3.8965466204983 0 0 0 0 2 1 0 0 0 1 0 0 sBAAABYCAAAAAA:cAkAAAAAAAAA -----BA---A----a---ab-----:0:0:1:00:0:0:3:0:8 4 27 32.779747292722 4.0138466072721 0 0 0 0 1 2 0 0 0 1 0 0 0BAAABoCAAAAAA:cAkAAAAAAAAA -----AB---A----a---ba-----:0:0:1:00:0:0:3:0:8 4 28 33.852008856074 4.14514394156 0 0 0 0 1 1 1 0 0 1 0 0 UBEAACoCAAAAAA:cAkAAAAAAAAA -----AAA--A----a--aaa-----:0:0:1:00:0:0:3:0:8 4 29 34.778383307942 4.2585775479113 0 0 0 0 1 1 0 1 0 1 0 0 UBIAAEoCAAAAAA:cAkAAAAAAAAA -----AA-A-A----a-a-aa-----:0:0:1:00:0:0:3:0:8 4 30 35.799214428263 4.3835772769302 0 0 0 0 1 1 0 0 1 1 0 0 UBQAAIoCAAAAAA:cAkAAAAAAAAA -----AA--AA----aa--aa-----:0:0:1:00:0:0:3:0:8 4 31 36.848473236746 4.5120579473567 0 0 0 0 1 1 0 0 0 2 0 0 UBgAAAoDAAAAAA:cAkAAAAAAAAA -----AA---B----b---aa-----:0:0:1:00:0:0:3:0:8 4 32 37.800508286875 4.6286336677806 0 0 0 0 1 1 0 0 0 1 1 0 UCgAAAoFAAAAAA:cAkAAAAAAAAA -----AA---AA--aa---aa-----:0:0:1:00:0:0:3:0:8 4 33 38.848748358686 4.7569895949412 0 0 0 0 0 1 0 1 1 1 0 0 IBUAAKQCAAAAAA:cAkAAAAAAAAA ------A-AAA----aaa-a------:0:0:1:00:0:0:3:0:8 4 34 39.850810545993 4.8796910872645 0 0 0 0 0 1 0 1 0 2 0 0 IBkAACQDAAAAAA:cAkAAAAAAAAA ------A-A-B----b-a-a------:0:0:1:00:0:0:3:0:8 4 35 40.828439695948 4.9994007790957 0 0 0 0 0 1 0 1 0 1 1 0 ICkAACQFAAAAAA:cAkAAAAAAAAA ------A-A-AA--aa-a-a------:0:0:1:00:0:0:3:0:8 4 36 41.812321415728 5.1198760917218 0 0 0 0 0 1 0 0 1 1 1 0 ICoAAEQFAAAAAA:cAkAAAAAAAAA ------A--AAA--aaa--a------:0:0:1:00:0:0:3:0:8 4 37 42.819318670796 5.2431818780567 0 0 0 0 0 1 0 0 0 2 1 0 ICwAAIQFAAAAAA:cAkAAAAAAAAA ------A---BA--ab---a------:0:0:1:00:0:0:3:0:8 4 38 43.846085637676 5.3689084454298 0 0 0 0 0 1 0 0 0 1 2 0 IDQAAIQGAAAAAA:cAkAAAAAAAAA ------A---AB--ba---a------:0:0:1:00:0:0:3:0:8 4 39 44.886334943764 5.4962859114813 0 0 0 0 0 0 0 1 0 2 1 0 gCwAAJAFAAAAAA:cAkAAAAAAAAA --------A-BA--ab-a--------:0:0:1:00:0:0:3:0:8 4 40 45.869163762882 5.6166322974958 0 0 0 0 0 0 0 0 1 2 1 0 AC0AAKAFAAAAAA:cAkAAAAAAAAA ---------ABA--aba---------:0:0:1:00:0:0:3:0:8 4 41 46.849145917317 5.7366301123245 0 0 0 0 0 0 0 0 1 1 2 0 ADUAAKAGAAAAAA:cAkAAAAAAAAA ---------AAB--baa---------:0:0:1:00:0:0:3:0:8 4 42 47.859831796497 5.860387566918 0 0 0 0 0 0 0 0 0 2 2 0 ADYAAMAGAAAAAA:cAkAAAAAAAAA ----------BB--bb----------:0:0:1:00:0:0:3:0:8 4 43 48.885586242425 5.9859901521337 0 0 0 0 0 0 0 0 0 1 3 0 ADoAAEAHAAAAAA:cAkAAAAAAAAA ----------AC--ca----------:0:0:1:00:0:0:3:0:8 4 44 50.011400658133 6.1238449785469 0 0 0 0 0 0 0 0 0 0 4 0 ADwAAIAHAAAAAA:cAkAAAAAAAAA -----------D--d-----------:0:0:1:00:0:0:3:0:8 Checkers Pips EPC Rolls 01-02-03-04-05-06---07-08-09-10-11-12 GNUBG ID XGID 5 5 22.00462962963 2.6944444444444 5 0 0 0 0 0 0 0 0 0 0 0 HwAAwAcAAAAAAA:cAkAAAAAAAAA -E----------------------e-:0:0:1:00:0:0:3:0:8 5 6 22.00462962963 2.6944444444444 4 1 0 0 0 0 0 0 0 0 0 0 LwAAwAsAAAAAAA:cAkAAAAAAAAA -DA--------------------ad-:0:0:1:00:0:0:3:0:8 5 7 22.00462962963 2.6944444444444 4 0 1 0 0 0 0 0 0 0 0 0 TwAAwBMAAAAAAA:cAkAAAAAAAAA -D-A------------------a-d-:0:0:1:00:0:0:3:0:8 5 7 22.00462962963 2.6944444444444 3 2 0 0 0 0 0 0 0 0 0 0 NwAAwA0AAAAAAA:cAkAAAAAAAAA -CB--------------------bc-:0:0:1:00:0:0:3:0:8 5 8 22.029835390947 2.6975308641975 4 0 0 1 0 0 0 0 0 0 0 0 jwAAwCMAAAAAAA:cAkAAAAAAAAA -D--A----------------a--d-:0:0:1:00:0:0:3:0:8 5 8 22.029835390947 2.6975308641975 3 1 1 0 0 0 0 0 0 0 0 0 VwAAwBUAAAAAAA:cAkAAAAAAAAA -CAA------------------aac-:0:0:1:00:0:0:3:0:8 5 9 22.073945473251 2.7029320987654 3 1 0 1 0 0 0 0 0 0 0 0 lwAAwCUAAAAAAA:cAkAAAAAAAAA -CA-A----------------a-ac-:0:0:1:00:0:0:3:0:8 5 10 22.163565957933 2.7139060356653 2 2 0 1 0 0 0 0 0 0 0 0 mwAAwCYAAAAAAA:cAkAAAAAAAAA -BB-A----------------a-bb-:0:0:1:00:0:0:3:0:8 5 11 22.437328532236 2.7474279835391 1 3 0 1 0 0 0 0 0 0 0 0 nQAAQCcAAAAAAA:cAkAAAAAAAAA -AC-A----------------a-ca-:0:0:1:00:0:0:3:0:8 5 12 22.460783893461 2.7503000685871 1 2 1 1 0 0 0 0 0 0 0 0 rQAAQCsAAAAAAA:cAkAAAAAAAAA -ABAA----------------aaba-:0:0:1:00:0:0:3:0:8 5 13 22.742851143753 2.7848389155617 2 1 1 0 0 1 0 0 0 0 0 0 KwIAwIoAAAAAAA:cAkAAAAAAAAA -BAA--A------------a--aab-:0:0:1:00:0:0:3:0:8 5 14 23.052438573261 2.8227475803993 1 1 2 0 1 0 0 0 0 0 0 0 NQEAQE0AAAAAAA:cAkAAAAAAAAA -AAB-A--------------a-baa-:0:0:1:00:0:0:3:0:8 5 15 23.350084384367 2.859194006249 1 1 1 1 1 0 0 0 0 0 0 0 VQEAQFUAAAAAAA:cAkAAAAAAAAA -AAAAA--------------aaaaa-:0:0:1:00:0:0:3:0:8 5 16 23.766742274777 2.9102133397687 1 1 1 1 0 1 0 0 0 0 0 0 VQIAQJUAAAAAAA:cAkAAAAAAAAA -AAAA-A------------a-aaaa-:0:0:1:00:0:0:3:0:8 5 17 24.570057602672 3.0085784819599 1 0 1 2 1 0 0 0 0 0 0 0 aQEAQFoAAAAAAA:cAkAAAAAAAAA -A-ABA--------------aba-a-:0:0:1:00:0:0:3:0:8 5 18 25.12263297969 3.0762407730233 1 1 0 1 1 1 0 0 0 0 0 0 pQIAQKkAAAAAAA:cAkAAAAAAAAA -AA-AAA------------aaa-aa-:0:0:1:00:0:0:3:0:8 5 19 25.843912589828 3.1645607252851 1 0 1 1 1 1 0 0 0 0 0 0 qQIAQKoAAAAAAA:cAkAAAAAAAAA -A-AAAA------------aaaa-a-:0:0:1:00:0:0:3:0:8 5 20 26.459597183476 3.2399506755277 0 1 1 1 1 1 0 0 0 0 0 0 qgIAgKoAAAAAAA:cAkAAAAAAAAA --AAAAA------------aaaaa--:0:0:1:00:0:0:3:0:8 5 21 27.506047391042 3.3680874356378 0 0 2 1 1 1 0 0 0 0 0 0 rAIAAKsAAAAAAA:cAkAAAAAAAAA ---BAAA------------aaab---:0:0:1:00:0:0:3:0:8 5 22 28.226696486312 3.4563301819974 0 0 1 2 1 1 0 0 0 0 0 0 tAIAAK0AAAAAAA:cAkAAAAAAAAA ---ABAA------------aaba---:0:0:1:00:0:0:3:0:8 5 23 29.075112054103 3.5602178025433 0 0 1 1 2 1 0 0 0 0 0 0 1AIAALUAAAAAAA:cAkAAAAAAAAA ---AABA------------abaa---:0:0:1:00:0:0:3:0:8 5 24 30.061878919307 3.6810463982825 0 0 1 1 1 2 0 0 0 0 0 0 VAMAANUAAAAAAA:cAkAAAAAAAAA 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----BEF--AA----aa--feb----:0:0:1:00:0:0:3:0:8 15 89 96.101814983789 11.767569181688 0 0 0 2 4 7 0 0 1 1 0 0 2PujAADY+6MAAA:cAkAAAAAAAAA ----BDG--AA----aa--gdb----:0:0:1:00:0:0:3:0:8 15 90 97.098782610125 11.889646850219 0 0 0 2 5 6 0 0 0 1 1 0 2PdDAQDY90MBAA:cAkAAAAAAAAA ----BEF---AA--aa---feb----:0:0:1:00:0:0:3:0:8 15 91 98.099556853922 12.012190635174 0 0 0 2 4 7 0 0 0 1 1 0 2PtDAQDY+0MBAA:cAkAAAAAAAAA ----BDG---AA--aa---gdb----:0:0:1:00:0:0:3:0:8 15 92 99.104267931434 12.1352164814 0 0 0 1 5 6 0 1 1 1 0 0 6PupAADo+6kAAA:cAkAAAAAAAAA ----AEF-AAA----aaa-fea----:0:0:1:00:0:0:3:0:8 15 93 100.10146884301 12.25732271547 0 0 0 2 4 6 0 1 0 1 1 0 2PtJAQDY+0kBAA:cAkAAAAAAAAA ----BDF-A-AA--aa-a-fdb----:0:0:1:00:0:0:3:0:8 15 94 101.09883305509 12.379448945521 0 0 0 2 4 6 0 0 1 1 1 0 2PtRAQDY+1EBAA:cAkAAAAAAAAA ----BDF--AAA--aaa--fdb----:0:0:1:00:0:0:3:0:8 15 95 102.10150180303 12.502224710576 0 0 0 2 4 6 0 0 0 2 1 0 2PthAQDY+2EBAA:cAkAAAAAAAAA ----BDF---BA--ab---fdb----:0:0:1:00:0:0:3:0:8 15 96 103.10264598914 12.624813794588 0 0 0 1 4 7 0 0 1 1 1 0 6P1RAQDo/VEBAA:cAkAAAAAAAAA ----ADG--AAA--aaa--gda----:0:0:1:00:0:0:3:0:8 15 97 104.10525219435 12.747581901349 0 0 0 2 3 6 0 1 1 1 1 0 2P1UAQDY/VQBAA:cAkAAAAAAAAA ----BCF-AAAA--aaaa-fcb----:0:0:1:00:0:0:3:0:8 15 98 105.10431421252 12.869916026023 0 0 0 1 4 6 0 1 1 1 1 0 6P1UAQDo/VQBAA:cAkAAAAAAAAA ----ADF-AAAA--aaaa-fda----:0:0:1:00:0:0:3:0:8 15 99 106.10261820289 12.992157330966 0 0 0 2 3 6 0 0 1 2 1 0 2P1oAQDY/WgBAA:cAkAAAAAAAAA ----BCF--ABA--aba--fcb----:0:0:1:00:0:0:3:0:8 15 100 107.10249200488 13.114590857741 0 0 0 1 4 6 0 0 1 2 1 0 6P1oAQDo/WgBAA:cAkAAAAAAAAA ----ADF--ABA--aba--fda----:0:0:1:00:0:0:3:0:8 15 101 108.10620810168 13.237494869594 0 0 0 2 3 6 0 0 0 2 2 0 2P2wAQDY/bABAA:cAkAAAAAAAAA ----BCF---BB--bb---fcb----:0:0:1:00:0:0:3:0:8 15 102 109.1059341701 13.359910306543 0 0 0 1 4 5 0 1 1 2 1 0 6H1qAQDofWoBAA:cAkAAAAAAAAA ----ADE-AABA--abaa-eda----:0:0:1:00:0:0:3:0:8 15 103 110.10398171663 13.482120210199 0 0 0 1 3 6 0 1 1 2 1 0 6H5qAQDofmoBAA:cAkAAAAAAAAA ----ACF-AABA--abaa-fca----:0:0:1:00:0:0:3:0:8 15 104 111.10488372942 13.604679640337 0 0 0 1 3 6 0 1 1 1 2 0 6H6qAQDofqoBAA:cAkAAAAAAAAA ----ACF-AAAB--baaa-fca----:0:0:1:00:0:0:3:0:8 15 105 112.10597564688 13.727262324108 0 0 0 2 2 6 0 0 1 2 2 0 2H60AQDYfrQBAA:cAkAAAAAAAAA ----BBF--ABB--bba--fbb----:0:0:1:00:0:0:3:0:8 15 106 113.1050711478 13.849600548711 0 0 0 1 3 6 0 0 1 2 2 0 6H60AQDofrQBAA:cAkAAAAAAAAA ----ACF--ABB--bba--fca----:0:0:1:00:0:0:3:0:8 15 107 114.10701641486 13.972287724268 0 0 0 1 3 5 0 1 1 3 1 0 6D51AQDoPnUBAA:cAkAAAAAAAAA ----ACE-AACA--acaa-eca----:0:0:1:00:0:0:3:0:8 15 108 115.1062342056 14.094640923135 0 0 0 1 3 5 0 1 1 2 2 0 6D61AQDoPrUBAA:cAkAAAAAAAAA ----ACE-AABB--bbaa-eca----:0:0:1:00:0:0:3:0:8 15 109 116.10614492122 14.217078969945 0 0 0 1 2 6 0 1 1 2 2 0 aD+1AQBoP7UBAA:cAkAAAAAAAAA ----ABF-AABB--bbaa-fba----:0:0:1:00:0:0:3:0:8 15 110 117.10667418498 14.339592757344 0 0 0 1 3 5 0 0 1 3 2 0 6D66AQDoProBAA:cAkAAAAAAAAA ----ACE--ACB--bca--eca----:0:0:1:00:0:0:3:0:8 15 111 118.10694267129 14.462074612811 0 0 0 1 2 6 0 0 1 3 2 0 aD+6AQBoP7oBAA:cAkAAAAAAAAA ----ABF--ACB--bca--fba----:0:0:1:00:0:0:3:0:8 15 112 119.10764992349 14.584610194713 0 0 0 1 2 5 0 1 2 2 2 0 aJ+2AQBon7YBAA:cAkAAAAAAAAA ----ABE-ABBB--bbba-eba----:0:0:1:00:0:0:3:0:8 15 113 120.10681938092 14.706957475214 0 0 0 1 2 5 0 1 1 3 2 0 aJ+6AQBon7oBAA:cAkAAAAAAAAA ----ABE-AACB--bcaa-eba----:0:0:1:00:0:0:3:0:8 15 114 121.10723239869 14.829457028411 0 0 0 1 2 5 0 1 1 2 3 0 aJ/aAQBon9oBAA:cAkAAAAAAAAA ----ABE-AABC--cbaa-eba----:0:0:1:00:0:0:3:0:8 15 115 122.10744091081 14.951931540099 0 0 0 0 2 6 0 1 1 3 2 0 sJ+6AQCwn7oBAA:cAkAAAAAAAAA -----BF-AACB--bcaa-fb-----:0:0:1:00:0:0:3:0:8 15 116 123.10784390997 15.074429866527 0 0 0 1 2 5 0 0 1 3 3 0 aB/dAQBoH90BAA:cAkAAAAAAAAA ----ABE--ACC--cca--eba----:0:0:1:00:0:0:3:0:8 15 117 124.10911385801 15.19703434996 0 0 0 0 2 5 0 2 1 3 2 0 sM+6AQCwz7oBAA:cAkAAAAAAAAA -----BE-BACB--bcab-eb-----:0:0:1:00:0:0:3:0:8 15 118 125.1076228278 15.319300754424 0 0 0 0 2 5 0 1 2 3 2 0 sE+7AQCwT7sBAA:cAkAAAAAAAAA -----BE-ABCB--bcba-eb-----:0:0:1:00:0:0:3:0:8 15 119 126.10795637088 15.441790576027 0 0 0 0 2 5 0 1 1 4 2 0 sE+9AQCwT70BAA:cAkAAAAAAAAA -----BE-AADB--bdaa-eb-----:0:0:1:00:0:0:3:0:8 15 120 127.10790967127 15.564233837299 0 0 0 0 2 5 0 1 1 3 3 0 sE/dAQCwT90BAA:cAkAAAAAAAAA -----BE-AACC--ccaa-eb-----:0:0:1:00:0:0:3:0:8 15 121 128.10861537342 15.686769229398 0 0 0 0 2 5 0 0 2 3 3 0 sI/dAQCwj90BAA:cAkAAAAAAAAA -----BE--BCC--ccb--eb-----:0:0:1:00:0:0:3:0:8 15 122 129.1091605909 15.809284970314 0 0 0 0 2 5 0 0 1 4 3 0 sI/eAQCwj94BAA:cAkAAAAAAAAA -----BE--ADC--cda--eb-----:0:0:1:00:0:0:3:0:8 15 123 130.10872757378 15.931680927401 0 0 0 0 2 4 0 1 2 3 3 0 sKfdAQCwp90BAA:cAkAAAAAAAAA -----BD-ABCC--ccba-db-----:0:0:1:00:0:0:3:0:8 15 124 131.10831187509 16.054079005113 0 0 0 0 1 5 0 1 2 3 3 0 0KfdAQDQp90BAA:cAkAAAAAAAAA -----AE-ABCC--ccba-ea-----:0:0:1:00:0:0:3:0:8 15 125 132.10862789587 16.176566681127 0 0 0 0 1 5 0 1 1 4 3 0 0KfeAQDQp94BAA:cAkAAAAAAAAA -----AE-AADC--cdaa-ea-----:0:0:1:00:0:0:3:0:8 15 126 133.10888557012 16.299047212668 0 0 0 0 1 5 0 0 2 4 3 0 0MfeAQDQx94BAA:cAkAAAAAAAAA -----AE--BDC--cdb--ea-----:0:0:1:00:0:0:3:0:8 15 127 134.10941010901 16.421560421512 0 0 0 0 1 5 0 0 2 3 4 0 0MfuAQDQx+4BAA:cAkAAAAAAAAA -----AE--BCD--dcb--ea-----:0:0:1:00:0:0:3:0:8 15 128 135.10904224926 16.543964357052 0 0 0 0 1 4 0 1 2 4 3 0 0NPeAQDQ094BAA:cAkAAAAAAAAA -----AD-ABDC--cdba-da-----:0:0:1:00:0:0:3:0:8 15 129 136.10893827879 16.666400605567 0 0 0 0 1 4 0 1 2 3 4 0 0NPuAQDQ0+4BAA:cAkAAAAAAAAA -----AD-ABCD--dcba-da-----:0:0:1:00:0:0:3:0:8 15 130 137.10905809307 16.788864256294 0 0 0 0 0 5 0 1 2 3 4 0 4NPuAQDg0+4BAA:cAkAAAAAAAAA ------E-ABCD--dcba-e------:0:0:1:00:0:0:3:0:8 15 131 138.10925897976 16.911337834257 0 0 0 0 1 4 0 0 2 4 4 0 0GPvAQDQY+8BAA:cAkAAAAAAAAA -----AD--BDD--ddb--da-----:0:0:1:00:0:0:3:0:8 15 132 139.10950283726 17.033816673951 0 0 0 0 0 5 0 0 2 4 4 0 4GPvAQDgY+8BAA:cAkAAAAAAAAA ------E--BDD--ddb--e------:0:0:1:00:0:0:3:0:8 15 133 140.10958184118 17.156275327492 0 0 0 0 0 4 0 1 3 3 4 0 4OnuAQDg6e4BAA:cAkAAAAAAAAA ------D-ACCD--dcca-d------:0:0:1:00:0:0:3:0:8 15 134 141.1092140415 17.278679270387 0 0 0 0 0 4 0 1 2 4 4 0 4GnvAQDgae8BAA:cAkAAAAAAAAA ------D-ABDD--ddba-d------:0:0:1:00:0:0:3:0:8 15 135 142.10941457089 17.401152804598 0 0 0 0 0 4 0 1 2 3 5 0 4Gn3AQDgafcBAA:cAkAAAAAAAAA ------D-ABCE--ecba-d------:0:0:1:00:0:0:3:0:8 15 136 143.1096602171 17.523631863318 0 0 0 0 0 4 0 0 2 5 4 0 4LHvAQDgse8BAA:cAkAAAAAAAAA ------D--BED--deb--d------:0:0:1:00:0:0:3:0:8 15 137 144.10974266998 17.646090939181 0 0 0 0 0 4 0 0 2 4 5 0 4LH3AQDgsfcBAA:cAkAAAAAAAAA ------D--BDE--edb--d------:0:0:1:00:0:0:3:0:8 15 138 145.11002065905 17.768573958251 0 0 0 0 0 3 0 1 2 5 4 0 4LTvAQDgtO8BAA:cAkAAAAAAAAA ------C-ABED--deba-c------:0:0:1:00:0:0:3:0:8 15 139 146.10967351148 17.890980429977 0 0 0 0 0 3 0 1 2 4 5 0 4LT3AQDgtPcBAA:cAkAAAAAAAAA ------C-ABDE--edba-c------:0:0:1:00:0:0:3:0:8 15 140 147.10999365747 18.013468611119 0 0 0 0 0 3 0 1 1 5 5 0 4NT3AQDg1PcBAA:cAkAAAAAAAAA ------C-AAEE--eeaa-c------:0:0:1:00:0:0:3:0:8 15 141 148.10999950168 18.135918306328 0 0 0 0 0 3 0 0 2 5 5 0 4Nj3AQDg2PcBAA:cAkAAAAAAAAA ------C--BEE--eeb--c------:0:0:1:00:0:0:3:0:8 15 142 149.11031798995 18.258406284484 0 0 0 0 0 3 0 0 2 4 6 0 4Nj7AQDg2PsBAA:cAkAAAAAAAAA ------C--BDF--fdb--c------:0:0:1:00:0:0:3:0:8 15 143 150.11055403513 18.380884167566 0 0 0 0 0 2 0 1 2 5 5 0 YNr3AQBg2vcBAA:cAkAAAAAAAAA ------B-ABEE--eeba-b------:0:0:1:00:0:0:3:0:8 15 144 151.11050744375 18.503327442092 0 0 0 0 0 2 0 1 2 4 6 0 YNr7AQBg2vsBAA:cAkAAAAAAAAA ------B-ABDF--fdba-b------:0:0:1:00:0:0:3:0:8 15 145 152.11079716124 18.625811897294 0 0 0 0 0 2 0 1 1 5 6 0 YOr7AQBg6vsBAA:cAkAAAAAAAAA ------B-AAEF--feaa-b------:0:0:1:00:0:0:3:0:8 15 146 153.11078264428 18.7482590993 0 0 0 0 0 2 0 0 2 5 6 0 YOz7AQBg7PsBAA:cAkAAAAAAAAA ------B--BEF--feb--b------:0:0:1:00:0:0:3:0:8 15 147 154.11142701011 18.87078698083 0 0 0 0 0 2 0 0 1 6 6 0 YPT7AQBg9PsBAA:cAkAAAAAAAAA ------B--AFF--ffa--b------:0:0:1:00:0:0:3:0:8 15 148 155.11156862618 18.993253301165 0 0 0 0 0 1 0 1 2 5 6 0 IO37AQAg7fsBAA:cAkAAAAAAAAA ------A-ABEF--feba-a------:0:0:1:00:0:0:3:0:8 15 149 156.11205320047 19.115761616384 0 0 0 0 0 1 0 1 2 4 7 0 IO39AQAg7f0BAA:cAkAAAAAAAAA ------A-ABDG--gdba-a------:0:0:1:00:0:0:3:0:8 15 150 157.11215380743 19.238222915195 0 0 0 0 0 1 0 0 2 6 6 0 IPb7AQAg9vsBAA:cAkAAAAAAAAA ------A--BFF--ffb--a------:0:0:1:00:0:0:3:0:8 15 151 158.11233502118 19.360694084226 0 0 0 0 0 1 0 0 2 5 7 0 IPb9AQAg9v0BAA:cAkAAAAAAAAA ------A--BEG--geb--a------:0:0:1:00:0:0:3:0:8 15 152 159.1127975427 19.483199699106 0 0 0 0 0 1 0 0 1 6 7 0 IPr9AQAg+v0BAA:cAkAAAAAAAAA ------A--AFG--gfa--a------:0:0:1:00:0:0:3:0:8 15 153 160.11332360085 19.605713093982 0 0 0 0 0 1 0 0 1 5 8 0 IPr+AQAg+v4BAA:cAkAAAAAAAAA ------A--AEH--hea--a------:0:0:1:00:0:0:3:0:8 15 154 161.11412291286 19.728259948513 0 0 0 0 0 0 0 1 1 6 7 0 gPr9AQCA+v0BAA:cAkAAAAAAAAA --------AAFG--gfaa--------:0:0:1:00:0:0:3:0:8 15 155 162.11406720136 19.850702106289 0 0 0 0 0 0 0 0 2 6 7 0 APv9AQAA+/0BAA:cAkAAAAAAAAA ---------BFG--gfb---------:0:0:1:00:0:0:3:0:8 15 156 163.11492110872 19.973255645966 0 0 0 0 0 0 0 0 2 5 8 0 APv+AQAA+/4BAA:cAkAAAAAAAAA ---------BEH--heb---------:0:0:1:00:0:0:3:0:8 15 157 164.11516480216 20.095734465571 0 0 0 0 0 0 0 0 1 6 8 0 AP3+AQAA/f4BAA:cAkAAAAAAAAA ---------AFH--hfa---------:0:0:1:00:0:0:3:0:8 15 158 165.1162452604 20.218315746172 0 0 0 0 0 0 0 0 1 5 9 0 AH3/AQAAff8BAA:cAkAAAAAAAAA ---------AEI--iea---------:0:0:1:00:0:0:3:0:8 15 159 166.11755226787 20.340924767494 0 0 0 0 0 0 0 0 0 6 9 0 AH7/AQAAfv8BAA:cAkAAAAAAAAA ----------FI--if----------:0:0:1:00:0:0:3:0:8 15 160 167.11954779174 20.463618096947 0 0 0 0 0 0 0 0 0 5 10 0 AL7/AQAAvv8BAA:cAkAAAAAAAAA ----------EJ--je----------:0:0:1:00:0:0:3:0:8 15 161 168.12360471283 20.586563842388 0 0 0 0 0 0 0 0 0 4 11 0 AN7/AQAA3v8BAA:cAkAAAAAAAAA ----------DK--kd----------:0:0:1:00:0:0:3:0:8 15 162 169.13155406403 20.709986211922 0 0 0 0 0 0 0 0 0 3 12 0 AO7/AQAA7v8BAA:cAkAAAAAAAAA ----------CL--lc----------:0:0:1:00:0:0:3:0:8 15 163 170.14587698966 20.834189019142 0 0 0 0 0 0 0 0 0 2 13 0 APb/AQAA9v8BAA:cAkAAAAAAAAA ----------BM--mb----------:0:0:1:00:0:0:3:0:8 15 164 171.17012897103 20.959607629106 0 0 0 0 0 0 0 0 0 1 14 0 APr/AQAA+v8BAA:cAkAAAAAAAAA ----------AN--na----------:0:0:1:00:0:0:3:0:8 15 165 172.20972289023 21.086904843701 0 0 0 0 0 0 0 0 0 0 15 0 APz/AQAA/P8BAA:cAkAAAAAAAAA -----------O--o-----------:0:0:1:00:0:0:3:0:8 End of Table Here are the pip counts for which there is more than one ideal racing position. Checkers Pips EPC Rolls 01-02-03-04-05-06---07-08-09-10-11-12 GNUBG ID XGID 0 2 8.1666666666667 1 0 1 0 0 0 0 0 0 0 0 0 0 AgAACAAAAAAAAA:cAkAAAAAAAAA --A--------------------a--:0:0:1:00:0:0:3:0:8 0 2 8.1666666666667 1 2 0 0 0 0 0 0 0 0 0 0 0 AwAAGAAAAAAAAA:cAkAAAAAAAAA -B----------------------b-:0:0:1:00:0:0:3:0:8 0 3 8.1666666666667 1 0 0 1 0 0 0 0 0 0 0 0 0 BAAAEAAAAAAAAA:cAkAAAAAAAAA ---A------------------a---:0:0:1:00:0:0:3:0:8 0 3 8.1666666666667 1 1 1 0 0 0 0 0 0 0 0 0 0 BQAAKAAAAAAAAA:cAkAAAAAAAAA -AA--------------------aa-:0:0:1:00:0:0:3:0:8 0 4 8.6203703703704 1.0555555555556 0 0 0 1 0 0 0 0 0 0 0 0 CAAAIAAAAAAAAA:cAkAAAAAAAAA ----A----------------a----:0:0:1:00:0:0:3:0:8 0 4 8.6203703703704 1.0555555555556 1 0 1 0 0 0 0 0 0 0 0 0 CQAASAAAAAAAAA:cAkAAAAAAAAA -A-A------------------a-a-:0:0:1:00:0:0:3:0:8 2 6 11.115740740741 1.3611111111111 1 0 0 0 1 0 0 0 0 0 0 0 IQAACAEAAAAAAA:cAkAAAAAAAAA -A---A--------------a---a-:0:0:1:00:0:0:3:0:8 2 6 11.115740740741 1.3611111111111 0 1 0 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