Only David Rockwell, Bob Koca and Gregg Cattanach directly addressed this puzzle. I imagine that other folks tried to solve it and gave up, though it's hard to tell because the thread broke down into a discussion of semantics (what Henrik had meant, then the definition of rolls, moves, turns, etc.) and the original poser was forgotten. I was going to let it go but fortunately Bob revived the thread, so I'm re-posting the question(s) now and the solutions.
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| Objective: Re-reach the starting position | |
The answer to the above question is five (or four if it is being reached for the third time). However, to make it easier in one way but more difficult in another, I provided the number of rolls, and divided the problem into three parts: In each case, you were to find the start-position-repeat sequences that meet the conditions below.
(a) Six rolls, if played well throughout.
(b) Five rolls, if all but one roll are played well.
(c) Four rolls, after re-reaching the starting position (doubles now legal on first roll).
I left the interpretation of "well" up to the reader, because it is tough enough as it is just to re-reach the starting position, let alone in just a few rolls, and let alone having to meet specific error size requirements. (To be clear: the number of rolls are in total; i.e., six rolls means three rolls by each player.)
Part (a): Six rolls are played well throughout. My solution (in Nactation) is
62S-41X-43H-61H-51R-51L
(In traditional notation, that's: 62: 24/18 13/11, 41: 24/20 8/7*, 43: Bar/18*, 61: Bar/24 20/14*, 51: Bar/24 18/13, 51: 14/8.)
The second and sixth plays are errors of only .003 and .013 according to Snowie evals, or .005 (full) and .019 (trunc) according to rollouts. The fifth play is almost exactly tied. The other plays -- first, third and fourth -- are clearly best. (The 43, which hits Bar/18*, can just as well be 52 or 61.)
Here is the six-roll sequence in full diagram illustration:
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| Blue will play 62S (i.e., 62: 24/18 13/11) | |
| White will play 41X (i.e., 41: 24/20 8/7*) | |
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| Blue will play 43H (i.e., 43: Bar/18*) | |
| White will play 61H (i.e., 61: Bar/24 20/14*) | |
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| Blue will play 51R (i.e., 51: Bar/24 18/13) | |
| White will play 51L (i.e., 51: 14/8) | |
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A closely related alternative sequence is 63S-41X-43H-51H-51R-61L. The total error size is overall a bit higher, but here, too, one could say the sequence is played "well."
There are many other six-roll sequences. Bob found one of the best ones: 64S-42H-61H-63H-51R-21R; the only flaw is the sixth play, which Bob posted and I replied to here. It is XG+ eval -.041 or Snowie eval -.068; it will probably roll out closer to the latter. (Also, a bot rollout amalgamation makes opening 64S an error of -.006 or so, but I have my doubts.)
Part (b): Five rolls, one big error allowed. My solution is
61B-62H-41H-51R-51L
(In traditional notation, that's: 61: 24/18 8/7, 62: 24/18* 13/11, 41: Bar/24 18/14*, 51: Bar/24 18/13, 51: 14/8.)
My allowed big error is on the first play. The other plays are played "well." According to Snowie eval, the second play is -.002 and the fifth is -.013, though the truncs show the second play best and the fifth at -.019.
Here is the five-roll sequence shown by diagrams:
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| Blue will play 61B (i.e., 61: 24/18 8/7) | |
| White will play 62S (i.e., 62: 24/18* 13/11) | |
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| Blue will play 41H (i.e., 41: Bar/24 18/14*) | |
| White will play 51R (i.e., 51: Bar/24 18/13) | |
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| Blue will play 51L (i.e., 51: 14/8) | |
There are several other five-roll sequences. Bob found 41S-44w-31H-31R-52R (where w = wild, 24/16* 13/5). His allowed big error is on the second play, and his fourth play is -.098 by XG+ eval (or -.114 by Snowie eval). We'll call the fifth play tied (slightly best by XG eval, -.003 by Snowie eval).
Bob did somewhat better with his (related) second solution: 32S-44w-51R-11R-21R. The fourth play is upgraded to -.086 by XG+ eval (or -.090 by Snowie eval). His fifth play appears to hold up as best (by a hair) in a trunc.
Part (c): Four rolls, after re-reaching the starting position (doubles now legal on first roll). I found only one solution (over thirty years ago), which greatly to his credit Bob found as well, and as far as I know it's the only one:
44w-22S-11R-51R
(In traditional notation, that's: 44: 24/16 13/5, 22: 24/20*/18 13/11, 11: Bar/24 16/14*/13, 51R: Bar/24 18/13.)
Finally, here is the four-roll sequence shown by diagrams:
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| Blue will play 44w (i.e., 44: 24/16 13/5) | |
| White will play 22S (i.e., 22: 24/20/18* 13/11) | |
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| Blue will play 11R (i.e., 11: Bar/24 16/14*/13) | |
| White will play 51R (i.e., 51: Bar/24 18/13) | |
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Summary: The sequences for six (well played), five (all but one well played) and four (for re-re-reach) rolls are:
62S-41X-43H-61H-51R-51L
61B-62S-41H-51R-51L
44w-22S-11R-51R
Nack