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| XGID=-b----E-C---eE---c-e----B-:4:-1:1:00:0:0:3:0:10 | |||||
.....(1) Standard money rules were in effect. Crikey, was I in a madhouse?
Or... Could Smith's and Jones' claims actually be true? If so, HOW?
Let's recap what we were told about this amazing game (in progress):
.....(2) The checker play and cube action were perfect on both sides.
.....(3) Jones rolled 55 times (counting the opening roll), and Smith rolled 54 times.
.....(4) The cube was turned 64 times (and thus not turned only 44 times!). It thereby reached 2^64.
.....(5) The opening position was showing on the board.
The key posts to review are: (a) The Smith and Jones story, and (b) The post under the same thread with "Correction" in the subject header. You might also want to review (c) an explanation of the FARE position, though the position itself is repeated below.
It's been a while since I kicked off the "Smith and Jones" thread. Depending on your settings, the original thread might be buried where you are unlikely to run across it. I am therefore continuing the thread here.
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| XGID=aa-a-ADACA--bB--acada-A-AA:1:1:1:00:0:0:3:0:10 | |||||
The Smith-and-Jones "FARE" position | |||||
Smith and Jones encountered the above position about midway through their game. Not a single bgonliner came up with a suggestion of how their position might have been reached, even in the backtracking of a roll or two. Granted, the concept may not come easily without practice.
General Brown and Admiral White sent me a record of Smith's and Jones' perfectly played masterpiece, which I have reproduced below. Not every move of this game is represented with its own diagram, but I believe you will find it easy enough to fill in the gaps.
Secretly, we know that Smith and Jones are fictional characters. I have added notes in a typeface of a regular size and color, so that you can glimpse some of the more interesting aspects that can be encountered in the creation of a retro-sequence.
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| XGID=-a----E-C-A-fD---cAe----A-:0:0:-1:43:0:0:3:0:10 | |||||
White to play 43 | |||||
White (Dr. Jones) opened with 65R and Blue (Sir Smith) replied with 63S. Now, with 43 to play, White's best move is T (sTack/Tower, 13/6).
There are equally short sequences with Blue's reply being 62S (instead of 63S), that I cannot use because for 65R-62S-43, T seems not to be the best play. XGR++ evaluates T as best by .012, but a 58k rollout has it losing to Z (though by only .001, with a 17% chance to overturn). By contrast, for the featured start of 65R-63S-43, T rolls out as best by .035. Inching the Blue builder forward from 11pt to 10pt discourages White from splitting with his 3.
I generally avoid creating "perfect" sequences with opening 64, because it is not yet clear which play is best for Money. Below are the most relevant rollouts of which I am aware (strongest on top). The results are TCTC, and a bit of variance remains even after 100+k trials.
.....[S P2 R4] "+46.....(v2.00, NeilK, Jan 2011)... Details
.....[R=S P4] '^46.......(v1.12, NeilK, Mar 2010).. Details
.....[P S4 R10] '<107..(v1.12, NeilK, Feb 2010)... Details
.....[P R2 S2] "<62.....(v2.xx, DavidR, 2011?)
.....[P S3 R6] "^82.....(v2.10, PaulW, Aug 2015) ..Details (not Money)
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| XGID=-a----E-C-A-eD---cAf----A-:0:0:1:43:0:0:3:0:10 | |||||
Blue to play 43 | |||||
Our friend Smith has his own 43 to play. The two strong candidates both involve 18/14 with the 4. With the 3, Smith could have continued on with his blot to partial safety (14/11), but he correctly decided it is (slightly) better to cover the 10pt.
Having figured out that I needed one player's 6pt to end up with at least seven checkers on it, I initially focused on sequences whereby (say) Blue began the game with 41S or 32S, made an advanced anchor, got hit in his outer board, and stayed on the roof until White had been forced to stack a seventh (or even an eighth) checker on his 6pt before leaving a shot. (For example, 41S-65R-21@-62H-F-64T-F-43T-F-54D(shot)-51H-64R-65X-F-61$-62R...)
Looser starts such as 32S-64H-63r make it difficult to force a seventh checker onto the 6pt, as White so often prefers to hit some blot, even on the 1pt. This led me to experiment with the quick-stack alternative of 51S-43T-52S-43T. (I didn't bother with 65R-51S-43T-52S-43T because I doubt the second T's advertised eval edge of .008 would hold up in a rollout.) However, Blue's 13/8 merely deferred the hit exchange I strived to save (as Blue has no fourth 8pt checker in the target position). In the labyrinth of variations generated from these starts, I was able (albeit in the not-yet-rollout-verified stage) to improve on the anchor-and-fan strategy by as many as three rolls and nearly four, but a feeling persisted that I hadn't quite hit the mark.
I struck paydirt when I decided to try 65R-63S-43T (see above diagram) and work out White's second stack-move on the fly. This approach saves five rolls over the sequence implied in my original post!
To induce White to play a second 43T, Blue's only real try here is 21P (13/10). Indeed, -21P-43T-21R?-41D-21E-62R-61H comes out two rolls ahead of the featured sequence (i.e, seven over the original one). The unfortunate flaw is that for the second of the three rolls of 21, W = 18/16 6/5 is the best play (by far), not R = 18/15 (marked with a ?).
Yet to abbreviate the sequence to that extent, if three 21's don't work, nothing will. White cannot exceed 9 pips: the 8 from 18/10 and the 1 dedicated to 6/5. Once he moves a checker beyond the 10pt or a second checker beyond the midpoint, it is necessary to recycle a checker (i.e., get it hit, in order to bring it back around to a viable location), the cost on that being two White turns (four rolls for the sequence).
In short, recycling is inevitable, and I was reluctantly prepared to settle for something like -21P-43T-41R-51Z-21$-21H... that is, until I discovered a way to suck a seventh checker onto White's 6pt without him playing 43T. With this elegant resource available (to be revealed presently), and given that Blue must pass the 10pt/13pt threshold by his third turn anyway, having him pass it a turn sooner salvages back two of the four "lost" rolls. Hence 43S instead of 21P.
(You may well be mystified by my apparent foreknowledge of goals. Actually, I started the process by rolling out candidate FARE positions. From the successful subset, I chose the one I thought could be reached and returned from in the fewest number of rolls, and worked backwards. To make a long story short, I determined that I was basically forced to reach the cube-turn position that you'll see a dozen rolls from now, as a stepping stone to reaching the chosen FARE position. Limiting my options made it easier to figure out economical paths.)
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| XGID=-a----E-C-B-cCA-ad-f----A-:0:0:1:41:0:0:3:0:10 | |||||
Blue to play 43 | |||||
After Smith (Blue) played 43S (18/14 13/10) in the previous diagram, Jones brought down two checkers with 54D. It is Smith's turn again and he rolls 41. Trailing in the race and noting the duplication of 4s, Smith makes his best play, which is $ (14/10 6/5).
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| XGID=-a---AD-C-C-cC--ad-f----A-:0:0:-1:43:0:0:3:0:10 | |||||
White to play 43 | |||||
White (Jones) has a roll of 43. He correctly hits with the 4. With the 3, he wisely selected the lesser of evils by stacking it on his 6pt. The combined effect of stripping the midpoint and leaving so many blots makes the 13/10 alternative unacceptable. In fact, making his own 5pt is the second best play.
Did you see it coming? Upon completion of his play, White will have moved the checker from his midpoint to the 6pt (stacking a seventh there) after all, over the span of two moves. We can now see how assigning Blue 43S instead of 21P (two diagrams ago) has gained two rolls: (a) Blue played 43S and 41$, and White 54D and 43C, instead of the slower alternative of (b) Blue 21P, 41R and 21$, and White 43T, 51Z and 21H.
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| XGID=-----aD-C-C-cC---d-g----AA:0:0:1:54:0:0:3:0:10 | |||||
Blue to play 54 | |||||
You often hear, "Just make the golden anchor, don't think about it." However, Smith sees that he is heavily trailing in the race, and that anchoring would let his super-stacked opponent off the hook. Smith shrewdly elects to hit.
Indeed, we are looking for ways to have Blue hit even more blotfully, as a necessary precursor to reaching his side of the FARE position.
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| XGID=-----AD-CaB-cC---d-g-A--A-:0:0:1:61:0:0:3:0:10 | |||||
Blue to play 61 | |||||
After Smith hit on his 5pt (from the previous diagram), Jones (White) ran from the bar with 63 (having no real alternative), and now Smith (Blue) rolls 61.
In for a penny, in for a pound; thus Smith hits with on his 9pt with the ace. He also makes his best choice with the 6, which is to come out with 21/15.
It may seem "too bad" that 13/7 is not the best 6. But never fear, Blue's (resulting) 15pt checker will join the near-side fray in a moment.
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| XGID=-----ADaCAA-cC-A-d-g----A-:0:0:1:53:0:0:3:0:10 | |||||
Blue to play 53 | |||||
After Smith hit on his 9pt (from the previous diagram), Jones (White) came out with 61, which was scary-looking but the best of a bad lot.
As for Smith and his roll of 53: In for a pound, in for a blot-prime. There is no better play than his swashbuckling hit of 15/7*!
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| XGID=-----ADACAa-cB---d-gA---AA:0:0:-1:00:0:0:3:0:10 | |||||
White on roll. Cube action? | |||||
There have been a couple of hit exchanges since the previous diagram. After Blue hit with 15/7*, White hit back with bar/15*, Blue hit back with bar/20 13/10*, White hit back again with bar/15*, and Blue fanned. Ouch, not a good time for Smith to dance!
Jones, however, thought it was a great time, and immediately pondered the cube. This position is a strong double for White and an easy take for Blue, and both players did act properly.
Alternatively, the fourth checker on White's 8pt can be on his 10pt (though I haven't investigated whether that position would be reachable by perfect play) or on his 6pt (where the position still ekes out a double by .009). In those cases, I'd assign a post-cube roll of 65 or 61 (respectively) instead of 63, thereby transposing.
After Jones (White) doubled, he rolled 63 (above) and correctly hit with 15/9 8/5*.
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| XGID=-----ADACA--cB--ac-ga---AB:1:1:-1:65:0:0:3:0:10 | |||||
White to play 65 | |||||
Jones, who was hoping to cover his 5pt, rolls 65. Taking into account his stacked distribution and the blottiness of his opponent's reception committee, Jones makes the most of his poor roll by coming down and hitting.
XGR++ evaluates this decision as a dead heat, but actually making the 3pt is wrong by .024 (2k rollout). IF the cube were still in the center, making the 3pt would be right by .030 (2k rollout) and the sequence would be busted.
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| XGID=a----ADACA--bB--acafA---aB:1:1:-1:00:0:0:3:0:10 | |||||
White on roll | |||||
Smith (Blue) rolled 65. He entered only one of his three checkers, but he was able to hit.
There is a reason I am showing this diagram. If White owned the cube here, he wouldn't be quite good enough to play on and would have to cash, thereby spoiling the sequence. Therefore, I had to make sure not only that the cube was turned earlier, but that it ended up in the right player's hands.
It is possible to construct a sequence that reaches the above diagram with White's 1pt blot instead on his 2pt, with White owning the cube, and the lead-up cube decisions all having been not good enough or too good. In that case, White would be too good to redouble. However, the earlier machinations required to have gotten the cube to White's side cost additional rolls.
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| XGID=a----ADACA--bB--acafAAA-a-:1:1:-1:11:0:0:3:0:10 | |||||
White to play 11 | |||||
After Smith entered with a 5 (hitting), it was Jones' turn to fan. Ugh, what a setback! Smith took the opportunity to enter his other two checkers with 43, but now Jones (White) makes up for his fan by rolling 11 (see diagram).
It's hard to resist pointing on the 5pt and making a four-point prime. However, Jones surveyed all the blots lying around the board and came up with an even better play -- much better in fact. Can you see it?
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| XGID=aa---ADACA--bB--acaeA-A-aA:1:1:-1:31:0:0:3:0:10 | |||||
White to play 31 | |||||
Smith had taken two rolls to bring his three checkers in, and with a single blow Jones' put all three back on the roof! From the previous diagram, Jones played the devastating enter-and-triple-hit of bar/24 6/5*/4*/3*. However, Smith was not to be outdone. He reentered two of his checkers and hit again, with 53. That brings us current, to the diagram above, where Jones (White) has rolled 31. What should he do?
This is no time to mamby-pamby around by covering the 7pt. With all these juicy blots floating around, White should continue to attack, and damn the torpedos. (It's safer, anyway!) Attack Jones did, and in the most aggressive fashion: not by anchoring with half his move (bar/24 8/5*), but rather by spreading his back checkers with bar/22 6/5*.
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| XGID=-a-a-ADACA--bB--acada-A-aB:1:1:1:00:0:0:3:0:10 | |||||
Blue on roll | |||||
Smith and Jones are now just one roll away from reaching their famed FARE position, which is shown in the second diagram of this post, or you can see it here. Can you figure out what Smith (Blue) must have rolled in the diagram above to achieve the FARE position?
The answer is...
Smith (Blue) rolled 61, fanning with one checker and entering the other, which hits White on his 1pt. The result is that each player has one checker on the bar and the position is perfectly symmetrical.
Alternatively, it worked to have Blue enter with 51 (instead of 53) for his previous roll, and this roll enter a checker with 63 (instead of 61).
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| XGID=aa-a-ADACA--bB--acada-A-AA:1:1:1:00:0:0:3:0:10 | |||||
FARE position | |||||
And here we are, with 26 rolls (i.e., 13 for each side) having been played. In this position, White fans and Blue redoubles, then Blue fans and White redoubles, then White fans and Blue redoubles, and in theory this can one-cube-turn-per-roll phenomenon can go on ad infinitum. Hence the appellation "Fan And Redouble Eternally." It might even be said, "All is FARE in love and backgammon."
In reference to my notes from a few rolls ago (starting with "There is a reason I am showing this diagram"), it may be easier to understand why I bothered looking for a same-length sequence that ended up with the side on roll owning the cube. If White owns the cube here (i.e., after Blue's 63 enter-one-checker-and-hit play), he can redouble immediately, which saves a roll. Instead, we wait one roll for White to fan, and Blue is the first to proffer a redouble.
Getting the cube into the other player's hands (even including the too-good-or-not-good-enough nuances of keeping it there) is not as far-fetched as it might sound. A good try is to invest a four-roll opening with (Blue) 62S-43Z-62Z-F-C followed by a reversal of fortune, though I was unable to make up the lost ground.
Poor Smith and Jones were tortured a long time by the repeated fan-and-redouble sequence. It happened 63(!) times, propelling the cube to 2^64 (i.e., 2 to the power of 64). If we write it longhand, the cube in this game reached
(which is "well into the quintillions," as Smith had estimated) before one of them finally rolled a number other than double 6s and they started moving the checkers again.
There is an amusing old fable about a wily artisan who concocted a scheme to swindle the Emperor of India out of his fortune. To pay for his handsome customized chessboard, the grand ruler agreed to the crasftsman's price, which sounded very reasonable: a single grain of rice for the first square of the chessboard, two grains for the second square, four grains for the third square, and so on, each time doubling the amount until that had been done for the 64th square. You can imagine the monarch's distress when he learned from his treasurer that he owed far more than all the wealth he possessed or could ever hope to raise. If the dumbfounded emperor could somehow obtain this many grains of rice and they were placed end-to-end, they would reach from Earth to the star Alpha Centauri and back again. His debt expressed in grains (plus one) was the 20-digit number you see emblazoned above.
(There are many versions of the chessboard-and-rice story. This version has a clever twist at the end.)
If we take Sir Smith and Dr. Jones at their word (which has been good so far), we know that continuing from the FARE position (diagrammed above), they subsequently re-reached the opening position while still playing perfectly the whole way. Is that really possible? What's a good way to start heading in that direction? Any ideas?
Nack