Upon meeting Sir Smith and Dr. Jones (full story here), it appeared that they were about to start their game, but with Jones holding a 16-cube. I was told that in fact they had RE-reached the opening position after 55 rolls by Jones and 54 by Smith, in a sequence in which they both played the checkers and cube perfectly, yet somehow in the 109-roll sequence the cube had been turned 64 times!!
On the first (cube) turn, the cube showed "2" and on the redouble "4," and so on to "64." Next turn, the cube showed "2" again (= 128), which Jones called the first turn of the second "cycle" (each cycle being comprised of six turns, one for each face of the cube). After the cube reached 64 the second time (which was really 10^12 = 4096), the next cube-turn was to "2" again (= 8192) in the third cycle. Smith and Jones also kept overall track on an electronic device, but to help clarify the cube value, I've typed "CYCLE #11" to the right of the diagrams. Thus, when I first saw their game position, the cube Jones owned was into its fourth turn of the 11th cycle, or 2^64 (which exceeds 18 quintillion).]
The first part of the mystery has been revealed. Smith and Jones reached the position diagrammed below after playing 13 rolls apiece (blow-by-blow account). Here, Jones fanned and Smith (Blue) redoubled. Then Smith fanned and Jones redoubled. This fan-and-redouble loop repeated 63 times, until Jones owned the cube on 2^64. It was Smith who then (finally) varied the incredible streak by rolling a number other than double 6s.
The remainder of this post is an account of Smith's and Jones' (still-perfectly-played) 20-roll game continuation back to the opening position; their "journey home," if you will. At the end is a recap of the entire game sequence, with an inventory of margins by which each of their plays was best (i.e., better than the second best play). Editor notes are in a typeface of regular size and color (black).
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| XGID=aa-a-ADACA--bB--acada-A-AA:4:-1:1:00:0:0:1:0:10 | |||||
FARE position | |||||
From the position above, Smith (Blue) rolled 63. He entered with the 3, anchoring, and hit on his 18pt with the 6. Objectively, it is a poor roll for this position, but to Smith, after fanning on a one-point board for 31 turns, it must have felt like a master stroke, an extraordinary conquest.
It is important that Blue's roll be relatively weak. We cannot allow him to cover his 5pt (which he should if he legally can), because to subsequently force that point to break is problematic, requiring an expenditure of extra rolls. (Remember, we aim to re-reach the opening position.) Thus, one of Blue's dice must be a 6. You'll see in a moment the reason I made the other die a 3.
Anyway, hitting on the opponent's 5pt and 7pt is or will be vital, for both players. The 5pt, obviously, because it has already gone past the 6pt. The 7pt, because a spare ace will naturally be played 8/7 (which we don't want) rather than 7/6. The ideal coup is for each player to be induced to play 9/6.
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| XGID=ba-a-ADACA--bB--acAda-B---:4:-1:-1:55:0:0:3:0:10 | |||||
White on roll | |||||
Jones (White) grabs a point in the outfield. There is no reasonable alternative with his roll of 55. It's a great catch-up number, which Jones needed, given that Smith was first to awake from the sixty-three-roll nightmare.
After looking down some less promising alleys, I determined that the quickest progress can me made here if White brings both checkers in and makes the 15pt, which is well positioned for the end play. To allow this to happen, it was necessary to eliminate the stronger play of C (bar/20(2) 8/3*(2)). That is why on the previous roll Blue was assigned a 3: to create a temporary anchor on White's 3pt.
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| XGID=-a-a--DACAb-bB--acAda-B--A:4:-1:1:41:0:0:3:0:10 | |||||
Blue on roll | |||||
Once again, Smith limps in, but like a man recently released from prison he is gleeful just to be able to move a checker. As this roll does not cover his own 5pt (preempted), he naturally hits on the opponent's 5pt.
Inescapably patient. Certainly we can't permit Blue to cover his 7pt, and giving him a double-hit number would be counterproductive. We don't want to recycle White's 9pt checker if we can make it to go directly to the 6pt.
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| XGID=aa-a--DACAb-bB--acAdA-B---:4:-1:-1:41:0:0:3:0:10 | |||||
White on roll | |||||
Both players continue to fight for the key points. To that end, Jones (White) could hit on his 5pt here (which is more important than either 7pt), but there is too great a risk he could lose the battle for both. Anyway, there is a general pull for hitting on the opponent's side of the board, for the greater gain in pips. In short, Jones hits straightforwardly with bar/24 22/18*.
This takes care of the last of the unavoidable hits. There are many rolls we can assign for White's 18pt-hit here. Holding two checkers back on the 24pt typically provides less flexibility, which is not a plus for the player trying to win the game, but it is often preferable for the retrofitter! Given that two checkers will need to end up on the 24pt in the target position, that is the recommended bias until proven otherwise.
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| XGID=-b----DaCAb-bB--acAdA-B--A:4:-1:1:53:0:0:3:0:10 | |||||
Blue on roll | |||||
Naturally, Smith comes in with the 5, anchoring. With the 3, he prudently lifts the 9pt blot to the 6pt, though (the Nackgammon-like) 18/15 is not a bad alternative.
If somehow Blue's 18/15 or 8/5 were better, rest assured I would have given White a different hitting number last roll so that one of his 24pt checkers would instead be split (say) to the 23pt or 21pt, to make sure that stacking is now Blue's best 3.
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| XGID=-b----EaC-b-bB--acAdB-B---:4:-1:-1:53:0:0:3:0:10 | |||||
White on roll | |||||
It is Jones' turn to stack, and he does so with 18/13 9/6. Slotting with 18/15 9/4 is not quite as strong, due to the superiority of gaining a spare on the 13pt instead of the 15pt.
Thankfully, slotting the 4pt is not best. If it were, I would have needed to (a) vary the sequence earlier, or (b) have White lift with an ace now and find a way to induce 8/6 later, or if neither worked well, then (c) have the 9pt checker hit/recycled, typically adding roll(s) to the sequence.
From the standpoint of retro-strategy, keeping White's 24pt checkers together has been vindicated. If White had been split to the 22pt or 20pt, then Blue would have used his 53 to hit (instead of stack). Or if White had been split to the 23pt or 21pt, he would have covered his 18pt blot (instead of lifting it) with 53. Either way, we would clearly be left with more hoops to jump through than we do now. In order for the 53 stacking plays to be best moves, there needed to be an absence of attractive alternatives.
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| XGID=-b----E-C-b-cB---cAeB-B---:4:-1:1:65:0:0:3:0:10 | |||||
White on roll | |||||
Smith wouldn't mind rolling a number with which he can conveniently unstack his 6pt for a better distribution, but clearly he can't do that with a roll of 65. Well, his midpoint could use a spare anyway, hence he just makes the obvious plodding move forward.
Is this all there is to re-reaching the opening position? Just get both players to stack the checkers back on their original points, in which case all that is apparently left to do now is rush the remaining checkers headlong towards the midpoints?
No. Even if Blue owned his 24pt instead of his 22pt, it's not that simple -- not with perfect play as a prerequisite. It's much more difficult to induce the players to stack their last few checkers, because that tends to sacrifice their sole vestige of versatility. Special maneuvers are required. As it stands, with the completion of this 65 move we've barely passed the third-way mark.
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| XGID=-b----E-C-b-cC--Ac-eB-A---:4:-1:-1:65:0:0:3:0:10 | |||||
White on roll | |||||
Jones correctly concludes that the double hit is way too creative. White will be up 11 pips after the lover's leap, which he quietly makes.
There is an old saying from chess: "Simple moves are best; put them in your treasure chest." Except then they're not! Did I hear a voice from the crowd? Oh, it's Tim. :)
Conveniently, a 24pt checker is within slingshot of the midpoint (where we want to stack); it doesn't need to be any closer to make the trip in a single roll. Indeed, if it were closer, any roll that would bring it to the midpoint would automatically make it possible to diversify one of the back checkers to the 15pt or play some other candidate that XGR++ rates as superior.
We're setting up a pair of hit exchanges for the finale, which will give Blue checkers on his 24pt (and replenish the one that is now leaving White's 24pt). It would be one beat early to launch those hit exchanges now. Read on.
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| XGID=-a----E-C-b-dC--Ac-eB-A---:4:-1:1:54:0:0:3:0:10 | |||||
Blue on roll | |||||
Unlike last turn, Blue's midpoint is no longer stripped; there is less value in adding a second spare there. With his 54, Smith takes a hard look at 16/7, but he sensibly decides that the 13pt-spare factor is not enough of a reason to change course. With no good play available on the near side, Smith runs another checker to his midpoint. After all, backgammon is in essence a race.
Here it comes...
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| XGID=-a----E-C-b-dD--Ac-eB-----:4:-1:-1:62:0:0:3:0:10 | |||||
White on roll | |||||
White is ahead, but if he runs out, he'll be hit with half the deck. In fact, running is the third best play. (Double-slotting with 13/7 6/4 is second.) Jones' actual play of breaking the 15pt to hit on the 9pt is a shrewd gambit (though not best by much), keeping in mind that with the majority of numbers that hit back there Blue will break his golden anchor.
There is only one more point that needs to be broken, and we just gave ourselves the justification (bait) to do it.
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| XGID=-a----E-C---eD--ac-eB----A:4:-1:1:41:0:0:3:0:10 | |||||
Blue on roll | |||||
Smith's invitation needs no engraving. Down 19 pips, he is happy to hit, even giving up his advanced anchor in the process.
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| XGID=aa----E-C---eD--Ac-eA---A-:4:-1:-1:43:0:0:3:0:10 | |||||
White on roll | |||||
Jones (White), showing exemplary judgment once again, enters with the 3 and hits with the 4. Anchoring on his 21pt is only a touch worse.
To make this sequence work, we need White to hit back on the 9pt, so that he can get hit back himself. That will give him an extra back checker he can run back to the midpoint, thereby re-reaching his starting position exactly one roll before or after Blue (hopefully) does the same, though Blue's task is somewhat trickier.
As for White's specific roll, he should hit on the 9pt only with 41, 42 or 43. (He should anchor with 54, or make his 4pt with 22.) I assigned White a roll of 43, even though hitting wins by only .004 (55k rollout). For 41 or 42, hitting on the 9pt is much clearer, but the resulting position -- with the second checker back on the 23pt or 24pt -- might cause the sequence to fail later. (More on that in a minute.)
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| XGID=-a-a--E-C---dD--ac-eA---AA:4:-1:1:41:0:0:3:0:10 | |||||
Blue on roll | |||||
Another tough decision with 41, but Smith (Blue) is up to the challenge. Hitting is a little better than re-anchoring.
Aha, after Blue hits back here, then is it a straight shot to the midpoint? Nope, I'm afraid not. Even if 61R (bar/24 22/16 were White's best play after that, Blue would have better options than 21R (as the rollout emphasizes). And even if Blue's 21R were best, White's final 21R would certainly not be (evaluates fifth best, and .053 behind 21$).
Well then, what does White roll? The surprise answer is double 6s: she fans!
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| XGID=aa-a--E-C---dD--Ac-e----B-:4:-1:1:65:0:0:3:0:10 | |||||
Blue on roll | |||||
Jones fanned, and with a roll of 65 it is now Smith's (Blue's) turn to apply the axiom, "Simple moves are best..." After pondering the three dynamic alternatives of D (16/10 13/8), H (16/10 8/3*) and S (24/18 16/11) for some time, he clicks a fifth checker into place on the midpoint.
It may seem counterintuitive that we would cause Blue's 24pt to break after working so hard to make it. Actually, though, for our plan to work, we need to have just ONE Blue checker on the 24pt and add a second one at the end. Having White lose a tempo (by fanning) was inevitable, because in any case we needed Blue to use up a roll stacking a fifth checker on his midpoint. Doubling up on the 24pt was the only way we could make Blue move to the 13pt by best play! For example, if last turn Blue had hit with 43 (instead of 41) so that his second back checker were now on the 22pt (instead of 24pt), his best play with 54 would be 24/20 13/8 (by .012, says ++), thereby defeating the sequence.
It was in looking ahead to this position that I decided against White's hitting with 41 or 42 (picture White's 22pt checker back on his 24pt or 23pt) instead of 43 on his previous turn. In the former case 65R beats D or S by a mere .001 and .002 (25k), and in the latter case R loses to D by .001 (12k). I haven't bothered to extend these rollouts (though I will eventually, in the interest of completeness) because it seems very unlikely that R will win either one by more than .004 (the margin by which 43H beat 43@ in the other position). My goal was to choose the sequence with the greatest chance of withstanding the test of time (rollouts of future XG versions or other bots).
In the featured sequence, where White's hit with a roll of 43 has led to the current diagram above, in a 15k rollout Blue's 65R beats D by .008, H by .009, and S by .014, which is a relatively comfortable result.
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| XGID=-a-a--EaC---dE--Ac-e----A-:4:-1:1:51:0:0:3:0:10 | |||||
Blue on roll | |||||
After Smith (Blue) ran with 65, Jones rolled a 61 and made the obvious play of coming out to his 18pt, hoping to generate an exchange of hits and/or to anchor there.
It is Smith's turn again and he rolls 51. He quickly rejects K -- the kamikaze hit-two play that breaks his 8pt, but he strongly considers the more balanced double hit of k (8/7* 6/1*), as well as the single hit of h (13/7*), which keeps a spare on his 8pt and duplicates 4s. In the end, he selects a different play altogether, which is the coordinated H (15/11 8/7*). Again, with magical consistency, Smith opts for the best move!
Let's have a final look at the parallel variations of White hitting with 41 or 42 (instead of 43) a few rolls ago: If White's 22pt checker were on the 24pt here, H would be an easy decision. With it on the 22pt as it is, H is only .007 better than k (16k rollout) and .014 better than h. However, even with such a tight result, the 43-hit was the safest roll assignment overall. By the way, with that checker on the 23pt, the evaluation shows k and K only .004 behind H, which is another strike against it -- a secondary reason for electing not to go with a roll of 42. (No need to do a long rollout that couldn't qualify 42 by itself.)
A useful reopening principle is hereby revealed: the restoration of a stripped 8pt is much more accommodating than the stacking of a fifth checker on the non-receptive midpoint. We therefore induce Blue to strip his 8pt. The checker he moves from there will get hit and replace the one he moved off his 24pt, and the "floating checker" (on his 11pt, though it could instead be on his 14pt, 13pt or 10pt) will be the one that restocks his 8pt.
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| XGID=-b----EaB--AdE---c-e----AA:4:-1:1:31:0:0:3:0:10 | |||||
Blue on roll | |||||
After Smith hit on his 7pt, Jones (White) rolled 41 and naturally hit back (bar/24 22/18*). It is Smith's turn again and he rolls 31. What should he play? Smith was stumped by the myriad of decent candidates, most notably H (bar/22 8/7*) and U (bar/21), but in the end he sagely chose to lift his blot and resupply his 8pt with L (bar/24 11/8).
Can you guess White's next roll?
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| XGID=-b----EaC---dE---c-e----B-:4:-1:-1:41:0:0:1:0:10 | |||||
White on roll | |||||
Jones (White) rolled 41, and suddenly both players burst out laughing, realizing what was about to happen. (Or so I was informed by General Brown and Admiral White, the game recorders.)
Nevertheless, Jones was determined to play the best move. He managed to narrow down his list of candidates to: R (18/13), S (24/23 13/9), and D (13/8), then further narrowed it to R and S. This decision was so close that he really couldn't make up his mind by backgammon expertise alone, so he went with the move that would make it a game for the ages (and the one we're all rooting for), which is R.
Having rereached the opening position in spectacular fashion (i.e., with 63 cube turns), the players took a break for lunch. And so we've come full circle. As you know, or may have heard through the grapevine, I wandered into the club not long afterwards and saw their board position (diagrammed below).
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| XGID=-b----E-C---eE---c-e----B-:4:-1:1:00:0:0:3:0:10 | |||||
Lunch break (Blue/Smith on roll) | |||||
In 2010 or 2011(?), with a late version of XG1 or early version of XG2, Ken Bame kindly contributed mucho CPU time by doing dozens of lengthy 4-ply rollouts for me. Among other things, we tested to see which moves were best when played as the final move in re-reaching the opening position. The conclusions were:
With the cube in the center, best plays are: ......61Z, 51Z, 51R, 32R, 65R, 63R, 62R.
For the side owning the cube, best plays are:... 61Z, 51Z, 51R, 41R, 65R.
For the side not owning the cube, best are:...... 61Z, 51Z.
(Nactation: 61Z = bar/24 14/8, 51Z = bar/24 13/8, 51R = bar/24 18/13. The rest are one-checker plays to the midpoint.)
According to my notes, the closest successful decision was for the roll of 41, R beating its closest rival S by .001. As I recall, cube centered, 32R had a margin of .002, the other R plays were around .005, and the Z plays (which replenish the 8pt) dominated. It is interesting that 32R works only with the cube centered, while 41R works only for the cube owner.
If future versions/bots overturn the final play of 41R (and 32R still falls short with this cube placement), I shall have to make the sequence a little longer and alter that part of the story. :)
Below is a recap of the Smith-Jones game sequence (including the first part, from opening position to FARE position), the entirety of which is perfectly played: all moves are best. The moves of Sir Smith (who played Blue) are in blue, and the moves of Dr. Jones (who played White) are in gray. (You might need to widen your window a bit.)
The red number denotes the number of thousandths by which that best play beat its closest rival according to XGR++ evaluation. The (more important) green number shows how that play fared in a rollout. Numbers greater than 50 (i.e., .050 margin) are considered to be slam-dunks and therefore omitted, except as an accompaniment.
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.....11. 34 ..5 .48 .32..46 .........43 .39 .21 ........................0
.....20. 35 ..4 .32 .39..36 .........31 .37 .53 .......................24
65R-63S-43T-43S-54D-41$-43C-54H-63R-61X-61U-53H-64H-53H-64H-F-C-63H-F-65H-5-F-43-11e.51-31H-1-
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..44 ..........................27 .14 ........13 ..16 ......5 .13 ....0 .20 .6 ......11 ..1
..12 ..........................39 ..7 ........10 ..20 ......4 ..8 ....8 .23 .7 .......3 ..1
{F-R, x 63!} -63U-55R-41H-41H-53C-53C-65R-65R-54r-62Z-41H-43H-41H-F-65R-61U-51H-41H-31L-41R
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I rolled out all plays on XG2, 3-ply checker play and XGR cube, that evaluated within .050 (plus any that I thought should), according to XGR++. I dropped off plays when they reached a 0.0% chance (but with a minimum of 216 trials, and a 1296 minimum for the top two plays). 43S and 31L (against their closest rivals) were rolled to 99.7% and 99.8%, respectively (cut off at 75k), and the rest of the plays to 100.0% or 0.0%. That is, except for the final position with 41R, which was rolled at 4-ply (with XG1 or XG2, not sure which) and probably to at least 100k. (I no longer have that rollout file, but Ken might.)
Nack