No Pass Go / Examples

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The list below gives the value of various eye sizes in No Pass Go. The values were generated using CGsuite V1.1. The program has been verified against the results published by Bill Spight on rec.games.go [ext] here.

1x1 eye = 1 moves, Mean 1, Temperature -1.

1x2 eye = 3/2 moves, Mean 3/2, Temperature -1/2.

1x3 eye = 2^ moves, Mean 2, Temperature 0.

1x4 eye = 3 moves, Mean 3, Temperature -1.

1x5 eye = 3* moves, Mean 3, Temperature 0.

1x6 eye = {4|15/4} moves, Mean 31/8, Temperature 1/8.

1x7 eye = {9/2|17/4} moves, Mean 35/8, Temperature 1/8.

1x8 eye = 5* moves, Mean 5, Temperature 0.

1x9 eye = {6|11/2}, Mean 23/4, Temperature 1/4.

1x10 eye = 6 moves, Mean 6, Temperature -1.

1x11 eye = 27/4 moves, Mean 27/4, Temperature -1/4.

1x12 eye = 29/4 moves, Mean 29/4, Temperature -1/4.

1x13 eye = 8 moves, Mean 8, Temperature -1.

1x14 eye = 17/2 moves, Mean 17/2, Temperature -1/2.

1x15 eye = {9|35/4} moves, Mean 71/8, Temperature 1/8.

1x16 eye = {39/4|19/2} moves, Mean 77/8, Temperature 1/8

1x17 eye = {41/4|10} moves, Mean 81/8, Temperature 1/8.

1x18 eye = {11|43/4} moves, Mean 87/8, Temperature 1/8.

1x19 eye = {23/2|45/4} moves, Mean 91/8, Temperature 1/8.

So far, a best move for black is to play in the 5th column of a 1xN eye (N>4, obviously). There are often other equally good moves. For white, playing in the middle of odd lengths eyes and in the 3rd column of even eyes works so far. It is harder to spot a pattern here because for many lengths, all white moves have equal value.

Conjecture A: John Rickard & Bill Spight, [ext] pre-2002

"John Rickard used Wolfe's CGT toolkit to analyze long linear eyes. We conjectured that there was a period of length 18, with an asymptote of 5 moves per 9 points."

Conjecture B: SiouxDenim, 2015

For N>14
1xN eye (N odd)  = 5/8*N - 1/2 + {1/8|-1/8} moves, Mean 5/8*N - 1/2, Temperature 1/8.
1xN eye (N even) = 5/8*N - 3/8 + {1/8|-1/8} moves, Mean 5/8*N - 3/8, Temperature 1/8.

Conjecture C: SiouxDenim, 2015

1x3 eye  : Mean 2
1x5 eye  : Mean 3
1x8 eye  : Mean 5
1x13 eye : Mean 8
There is an underlying relationship with the  Fibonacci numbers

Commentary SiouxDenim

The numerical evidence currently supports an asymptote of 5 moves per 8 points and a period of 2 (conjecture B), rather than 5 moves per 9 points and a period of 18 (conjecture A).
The relationship seen in Conjecture C may just be because the 5/8 multiplier in Conjecture B is a good approximation to 1/phi, where phi= (1+sqrt(5))/2 =  Golden Ratio
Conjecture B predicts the 1x21 eye will have Mean 101/8, whereas Conjecture C predicts a mean of 13. Calculating this value requires serious computing power.

Bill: Wow! Great that you have taken up the challenge. :) It has been a while, but as I recall I surmised at one point that the Golden Ratio was the key. However, the 5/9 result told against that. Maybe we made a mistake there.

2x2 eye = 2* moves, Mean 2, Temperature 0.

2x3 eye = 3 moves, Mean 3, Temperature -1.

2x4 eye = 4 moves, Mean 4, Temperature -1.

2x5 eye = 5+(1).Tiny moves, Mean 5, Temperature 0.

2x6 eye = {47/8|||23/4v,{6^|11/2,{6^||6^|5^|||{11/2|5^},{6^|5^||5^}}}||5^,{6^|5^||5^}|{5^|5vv},{5,{6v|5v}|5vv}} moves, Mean 23/4 Temperature 1/8.

2x7 eye = {27/4^|{{27/4^|13/2},{27/4^|27/4,{27/4|13/2}||27/4}||||{7^|13/2||{7^|6^||6*,{6*|11/2}},{7^|13/2||6^|11/2|||{6^|11/2},{11/2,{13/2|11/2}|11/2}}},{7^|13/2||{7^|6||11/2*},{7^|13/2||{6^|11/2},{7^|13/2||6^|11/2}|||{6+(1^).Miny|11/2v},{{13/2|11/2},{{7|13/2||6|11/2},{6,{7|6,{7|6}}|6+(1/2).Miny}|{6|11/2},{6*,{13/2|11/2}|11/2}}|11/2}}}|||6,{6,{7|6,{7|6}}|6+(1/2).Miny}|{6*|23/4},{{6,{7|6,{7|6}}|6+(1/2).Miny||6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{{7|13/2||6|11/2},{6,{7|6,{7|6}}|6+(1/2).Miny}|{6|11/2},{6*,{13/2|11/2}|11/2}}|23/4}||23/4},{{7|13/2^,{27/4v||6,{13/2*|||13/2*|11/2*||11/2*}|11/2}},{7^|13/2,{7^||7^|6^|||{7^|6^||6^},{13/2||{13/2|11/2},{{7|13/2||6|11/2},{6,{7|6,{7|6}}|6+(1/2).Miny}|{6|11/2},{6*,{13/2|11/2}|11/2}}|11/2}}}|{{7^|13/2||6*},{7^|6^,{13/2^|6^},{27/4|23/4,{13/2,{13/2|11/2}|11/2}}},{7^||||7^|6^||6*,6^|||6,6*}|{6*|23/4},{7^|6^||6*,6^|||23/4}},{6^|6,{{7^|6^||6*,6^},{{7^|6^},{7^||7^|6^|||{7|27/4||6^},{7^|6^||6^}}|{6^|23/4},{7|27/4||6^|||23/4}}|6,6*}},{13/2|||6,{6,{7|6,{7|6}}|6+(1/2).Miny}|{6*|23/4},{{6,{7|6,{7|6}}|6+(1/2).Miny||6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{{7|13/2||6|11/2},{6,{7|6,{7|6}}|6+(1/2).Miny}|{6|11/2},{6*,{13/2|11/2}|11/2}}|23/4}||23/4}}||13/2,{{{7|13/2},27/4*|6^,{13/2^|6^}},{27/4v||13/2|{25/4|11/2},{7*|6^||6v},{7^*|6^*||6*}},{27/4*|13/2,{{{7|27/4*},55/8^|6^,{13/2^|6^}},{7v*|{27/4,{7*|6*,{7*|6*}}|||6,{7*|6*||6,6*}||6+1.Miny|6},{{7^|6^},{7^,{15/2*||7*|6*}|{7*|6*},{7*|6*||6,6*}}|||6,{7*|6*||6,6*}||6+1.Miny|6}},{7*|{13/2|6^},{27/4*||{7*|6*,{7*|6*}},{7^|6^,{7^|6^}||6+(1/4).Tiny}|6,6*},{7,{15/2*||7*|6*}||7*|6*|||{7*,{7*|6*}|6*||6*,{6*,6^|6*,6v}},{{7*|6*,{7*|6*}},{7^|6^,{7^|6^}||6+(1/4).Tiny}|6,6*}}}|{27/4,{7*|6*,{7*|6*}}|||6,{7*|6*||6,6*}||6+1.Miny|6||||6+1.Miny|6},{6*,{{7|13/2||6,{6|11/2}},{6,{7|6,{7|6}}|6+(1/2).Miny}|6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}}|6v}}}|{{{7|13/2||6,{6|11/2}},{6,{7|6,{7|6}}|6+(1/2).Miny}|6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{7v|6v,{7vv|13/2v||11/2}}||{27/4,{7v|6v,{7v|6v}}|23/4},{6*,{{7|13/2||6,{6|11/2}},{6,{7|6,{7|6}}|6+(1/2).Miny}|6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}}|6v}|23/4},{{6^|6,{{7^|6^||6*,6^},{{7^|6^},{7^||7^|6^|||{7|27/4||6^},{7^|6^||6^}}|{6^|23/4},{7|27/4||6^|||23/4}}|6,6*}},{{7^|{13/2|{13/2|21/4},{6*,{13/2|11/2}|11/2}},{{7^|6^},{7^|13/2||6^|11/2}|{6^|11/2},{6*,{13/2|11/2}|11/2}}},{7|27/4||{13/2,{7^|13/2}||6^|6*,{6*,6^|6*,6v}},{7^|{27/4|6^},{7|27/4||6^}||13/2,{13/2||6,{25/4|6}|6*,6v}|23/4}},{7^|6^,{7|27/4||{7|27/4||||13/2|25/4|||13/2|25/4||6}|||6|11/2}}|||6,{{7^|6^},{7^|13/2||6^|11/2}|{6^|11/2},{6*,{13/2|11/2}|11/2}},{6^,{13/2^|6^||6}|6v,{7^|6^||6*,6^|||6,{7*|6*||6,6*}||6+1.Miny|6||||23/4},{{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{13/2|{6|6vv*},{6*,6v|||6|6vv*||6}}|23/4}}||{{6,{7|6,{7|6}}|6+(1/2).Miny||6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{{7|13/2||6|11/2},{6,{7|6,{7|6}}|6+(1/2).Miny}|{6|11/2},{6*,{13/2|11/2}|11/2}}|23/4},{{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{13/2|{6|6vv*},{6*,6v|||6|6vv*||6}}|23/4}|23/4}|{6|23/4},47/8,{{27/4,{7v|6v,{7v|6v}}|23/4},{6*,{{7|13/2||6,{6|11/2}},{6,{7|6,{7|6}}|6+(1/2).Miny}|6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}}|6v}|23/4}}},{{7|{13/2^|6^},{7^^|6^^||6^}},{{7|27/4},{7||{7|6},{7^|7,{7|6}}|6}|{13/2|6^},{7^|{13/2^|6^},{7^^|6^^||6^}|||7*|{7*|6^||6v},{7^*|6^*||6*}||6*}},{7|27/4|||13/2,{{7|27/4},{7^|13/2},{7^|7,{7|6}}|{13/2|6^},{7^,{7^|13/2}|||7^|13/2||6^|11/2||||6^}}||{7^|6^||6*,6^|||6,{7*|6*||6,6*}||6+1.Miny|23/4},{{13/2*,{7^|6^,{7^|6^}}||||7^|6^||6*,6^|||6,{7*|6*||6,6*}||6+1.Miny|23/4}|{13/2,{7*|6*,{7*|6*}}|||6,{7*|6*||6,6*}||6+1.Miny|23/4||||27/4,{7|6,{7|6}}||6+1.Miny|23/4|||6}}|23/4},{7,{7|27/4}|{{{7|27/4},{7^|6^,{7^|6^}}||||7^|6^||6*,6^|||6,{7*|6*||6,6*}||6+1.Miny|6}|||6,{7*|6*||6,6*}||6+1.Miny|6},{7^|6^,{7|27/4||{7|27/4||||13/2|25/4|||13/2|25/4||6}|||6|11/2}||6,{27/4|23/4,{{7|27/4||||13/2|25/4|||13/2|25/4||6}|23/4}}|11/2}}|{7^|6^||6*,6^|||6,{7*|6*||6,6*}||6+1.Miny|6||||6+1.Miny|6},{{6^|6,{{7^|6^||6*,6^},{{7^|6^},{7^||7^|6^|||{7|27/4||6^},{7^|6^||6^}}|{6^|23/4},{7|27/4||6^|||23/4}}|6,6*}},{{7^|{13/2|{13/2|21/4},{6*,{13/2|11/2}|11/2}},{{7^|6^},{7^|13/2||6^|11/2}|{6^|11/2},{6*,{13/2|11/2}|11/2}}},{7|27/4||{13/2,{7^|13/2}||6^|6*,{6*,6^|6*,6v}},{7^|{27/4|6^},{7|27/4||6^}||13/2,{13/2||6,{25/4|6}|6*,6v}|23/4}},{7^|6^,{7|27/4||{7|27/4||||13/2|25/4|||13/2|25/4||6}|||6|11/2}}|||6,{{7^|6^},{7^|13/2||6^|11/2}|{6^|11/2},{6*,{13/2|11/2}|11/2}},{6^,{13/2^|6^||6}|6v,{7^|6^||6*,6^|||6,{7*|6*||6,6*}||6+1.Miny|6||||23/4},{{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{13/2|{6|6vv*},{6*,6v|||6|6vv*||6}}|23/4}}||{{6,{7|6,{7|6}}|6+(1/2).Miny||6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{{7|13/2||6|11/2},{6,{7|6,{7|6}}|6+(1/2).Miny}|{6|11/2},{6*,{13/2|11/2}|11/2}}|23/4},{{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}},{13/2|{6|6vv*},{6*,6v|||6|6vv*||6}}|23/4}|23/4}|{6|23/4},47/8,{{27/4,{7v|6v,{7v|6v}}|23/4},{6*,{{7|13/2||6,{6|11/2}},{6,{7|6,{7|6}}|6+(1/2).Miny}|6},{6,{6,{7|6,{13/2|6}}|6}|6,{6|23/4}}|6v}|23/4}}}} moves, Mean 13/2 Temperature 0.

So far, the best move for black in all of the 2xN eye spaces is the (1,2) position (or its reflections).

If suicide of 2 or more stones is allowed and there is no Superko rule, then the values are as given below. (Values computed using a combination of CGSuite V1.1 and careful hand calculation. Errors are possible)

1x2 eye = 1over.

1x3 eye = {2|3/2} moves, Mean 7/4, Temperature 1/4.

1x4 eye = 2over.

1x5 eye = 5/2, Mean 5/2, Temperature -1/2.

For linear corridors, where there is a living white stone at the end, the values are given below. The values were generated using CGsuite V1.1. The program has been verified against the results published by Bill Spight on rec.games.go [ext] here.

1x1 blocked corridor = * moves, Mean 0, Temperature 0.

1x2 blocked corridor = {1|*} moves, Mean 1/2, Temperature 1/2.

1x3 blocked corridor = {3/2||1|*} moves, Mean 1, Temperature 1/2.

1x4 blocked corridor = {2^|||3/2||1|*} moves, Mean 3/2, Temperature 1/2.

1x5 blocked corridor = {3||||2^|||3/2||1|*} moves, Mean 9/4, Temperature 3/4.

1x6 blocked corridor = 3 moves, Mean 3, Temperature -1.

1x7 blocked corridor = {4|15/4||3} moves, Mean 55/16, Temperature 7/16.

1x8 blocked corridor = {9/2|17/4|||4|15/4||3} moves, Mean 125/32, Temperature 15/32.

1x9 blocked corridor = {5*||||9/2|17/4|||4|15/4||3}, Mean 285/64, Temperature 35/64.

1x10 blocked corridor = {6|11/2||{5*||||9/2|17/4|||4|15/4||3}} moves, Mean 653/128, Temperature 83/128.

1x11 blocked corridor = {6|11/2} moves, Mean 23/4, Temperature 1/4.

1x12 blocked corridor = {27/4||6|11/2} moves, Mean 25/4, Temperature 1/2.

1x13 blocked corridor = {29/4|||27/4||6|11/2} moves, Mean 27/4, Temperature 1/2.

1x14 blocked corridor = {8||||29/4|||27/4||6|11/2} moves, Mean 59/8, Temperature 5/8.

1x15 blocked corridor = {17/2|{8||||29/4|||27/4||6|11/2}} moves, Mean 8, Temperature 1/2.

1x16 blocked corridor = {9|35/4||17/2|{8||||29/4|||27/4||6|11/2}} moves, Mean 17/2, Temperature 3/8

1x17 blocked corridor = {39/4|19/2|||9|35/4||17/2|{8||||29/4|||27/4||6|11/2}} moves, Mean 145/16, Temperature 9/16.

Conjecture D: SiouxDenim, 2015
A best play for both players is at the open end of the blocked corridor.

Commentary SiouxDenim

If Conjectures B and D are true, then it follows that:

The Right option increases in length without getting simpler from length 11 onwards

For N>17
1xN corridor (N odd)  = (Long expression) moves, Mean 5/8*N - 13/8, Temperature 5/8.
1xN corridor (N even) = (Long expression) moves, Mean 5/8*N - 13/8, Temperature 1/2.

For corridors open at both ends, the first few values are given below. Note that if White manages to connect all the way through the corridor, it breaks Black into two groups. It is assumed here that both Black groups are still alive. However, as the number of Black groups has increased, Black loses an additional 2 moves due to Group Tax.

1x1 unblocked corridor = {0|-2} moves, Mean -1, Temperature 1.

1x2 unblocked corridor = 0 moves, Mean 0, Temperature -1.

1x3 unblocked corridor = * moves, Mean 0, Temperature 0.

1x4 unblocked corridor = {1|*} moves, Mean 1/2, Temperature 1/2.

1x5 unblocked corridor = {3/2||1|*} moves, Mean 1, Temperature 1/2.

Thereafter (verified up to length 17), the values of the length N unblocked corridor are the same as the N-2 blocked corridor. So far, Black and White should play at an end with the first move (sometimes there are other equally good moves).

No Pass Go / Examples last edited by SiouxDenim on July 17, 2015 - 01:47

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