White is Player 2 score: 0 pip: 167 Unlimited Game Jacoby Beaver pip: 216 score: 0 Blue is Player 1
XGID=-b----C-C---eC---c-eAAAAAA:0:0:1:00:0:0:3:0:10

PROBLEM 1: From the opening position, with perfectly play (according to XGR++) how can a player can end up with a single checker on each of six consecutive points? (The bearoff tray counts as the 0pt, and the bar/roof counts as the 25pt.) The player's other nine checkers can be anywhere. [Here, I encouraged the reader to focus on reaching the 25pt-through-20pt blot-formation above, though s/he was still welcome to target any other six consecutive points.]

PROBLEM 2: What is the fewest number of rolls that accomplish this feat?

PROBLEM 3: Same, but White's ending pipcount is a multiple of 8.

SOLUTIONS

My initial thought was to set up a triple hit, such as represented by the position below. (Either or both player's 13pt/8pt/6pt distributions can vary, and another set of positions can be formed by Blue's 9pt and 20pt blots being replaced by 16pt and 23pt blots.)

		White is Player 2 score: 0 pip: 187 Unlimited Game Jacoby Beaver pip: 170 score: 0 Blue is Player 1
XGID=ba--AAC-BA--eD---c-dA-A-A-:0:0:-1:44:0:0:3:0:10
21$-64K-53-41H-54@-63R-65K-54-32K, and from here 44R-42 will finish the solution

This specific position can be reached by the sequence captioned, but there are many other (perfectly played) 11-roll solutions that culminate with the same (or similar) triple-hit. [That is, 9 rolls to get to here, plus 2 more rolls post-diagram, White playing 44R (bar/21*(2) 24/20*/16*) and Blue entering two of his three hit checkers with 42.]

The triple-hit solution can almost be achieved (perfectly) in just 9 rolls. A couple of close failures start with: (1) 41S-41K-21-53U (error of .006, 53S is best), and (2) 52S-52S-43H (error of .008, 43x is best).

It turns out the "triple-hit" theme, while cute, is not the most efficient -- it requires too many moves to set up. Numerous 10- and 9-roll solutions can be produced by (simpler) "repeat-hit" themes. In fact, there are even a few 8-roll solutions, with the repeat hit occurring on Blue's 11pt or 9pt/8pt. I'll describe one of each, using abbreviated (single-diagram) film:

		White is Player 2 score: 0 pip: 168 Unlimited Game Jacoby Beaver pip: 168 score: 0 Blue is Player 1
XGID=aa--B-C-C--AeC---c-e-A-AA-:0:0:-1:65:0:0:3:0:10
32S-64H-22Y_, White rolls 65

Blue played opening 32S, White ran/hit with 64, and Blue followed up with 22Y_ (bar/23 13/11* 6/4(2)). This is an efficacious start. Blue has managed to split his two back checkers and acquire a third, while creating/maintaining a blot in the outfield. In addition, the inside point he just made will help ensure that his upcoming rehits will be best (even when he has the option of anchoring on his 20pt).

From here, White hits with 65 (bar/14*), Blue hits back with 52, White hits again with 65, Blue hits back with 32, and White hits one more time with 65. The checkers that Blue feeds from the midpoint end up as back checker blots; all told he has six (one each on his 25pt, 24pt, 23pt, 22pt, 21pt and 20pt) -- not counting the irrelevant blot left on his midpoint. [The order of Blue's 52 and 32 can be reversed.]

The other 8-roll repeat-hit solution is summarized below.

		White is Player 2 score: 0 pip: 168 Unlimited Game Jacoby Beaver pip: 162 score: 0 Blue is Player 1
XGID=aa--B-E-AA--eC---c-e-A-AA-:0:0:-1:62:0:0:3:0:10
41S-62H-44M_, White rolls 62

Blue played opening 41S, White hit in the outfield with 62, and Blue played 44M_ (bar/21 13/9* 8/4(2)). Note how similar this is to the diagram of the 32S solution. Blue has an extra near-side blot here, which isn't needed but adds flexibility.

From here, White hits (bar/17*) with a 53 or 62 (or can branch to bar/16* with 63 or 54), Blue hits back with 53 (bar/22 13/8*), White rehits with an 8, Blue hits back with 51 (bar/20 9/8*), and White hits with yet another 8. As before, Blue ends up with six blots (including the one on the 25pt/roof) on consecutive points.

Remarkably, it is possible to shave one more roll off the solution, bringing it down to the minimum of 7 rolls. Read on.

		White is Player 2 score: 0 pip: 167 Unlimited Game Jacoby Beaver pip: 164 score: 0 Blue is Player 1
XGID=-b---AD-C--AeD---c-e----B-:0:0:-1:54:0:0:3:0:10
Blue played opening 21$, White rolls 54

21$ is the only XGR++ best opening play that leaves a direct shot in the player's own inner board. The significance of this is that White will be able to hit twice (instead of once) on a well-timed future roll, allowing Blue to finish the sequence a roll before White's final hit would normally end it.

Granted, the accounting doesn't seem to reconcile. What about the half roll Blue needs to split his back checkers? The answer is that he will be able to make up that extra roll portion with a wily set of doublets that enter and split and hit, as you will soon see.

For this sequence to work, White has to reply to the opening move by hitting one checker with a 4. I assigned White a roll of 54 at this stage, though rolls of 31 and 41 also work. [Counting all variations, there are eleven legitimate 7-roll sequences. I'll list them all at the end.]

		White is Player 2 score: 0 pip: 158 Unlimited Game Jacoby Beaver pip: 184 score: 0 Blue is Player 1
XGID=-a---aD-C--AdD---d-e----BA:0:0:1:00:0:0:3:0:10
Blue played 21$, White played 54S, Blue now rolls what number?

A couple of paragraphs ago, I mentioned a "wily set of of doublets that enter and split and hit." Based on that description, can you figure out what Blue rolls here?

		White is Player 2 score: 0 pip: 163 Unlimited Game Jacoby Beaver pip: 168 score: 0 Blue is Player 1
XGID=aa---AD-C--AdC---d-eAA--A-:0:0:-1:32:0:0:3:0:10
Blue played 44X, White rolls 32

Pat yourself on the back if you guessed that Blue's wily roll was 44. The best move was to enter (bar/21), split (24/20), and hit (13/5*) -- as played. With this move, Blue has managed to keep his 11pt checker loose, too, thereby ensuring the gain of a full roll over the repeat-hit sequences previously shown.

[Earlier, I excluded 21$-43U (better than S) because 44C would point on the 4pt, making it no longer quite possible to complete the sequence in 7 rolls. One continuation is very nearly successful: -65H-31H-53K, which XGR++ rates inferior to 53k by a mere .0002.]

In this position, I assign White a roll of 32, which can also be 41 -- though when a roll generates the identical checker play I count it as the same solution. Two other (slightly different) solutions continue -42H-31H and -42H-21H.

		White is Player 2 score: 0 pip: 158 Unlimited Game Jacoby Beaver pip: 188 score: 0 Blue is Player 1
XGID=-a---aD-C--AdC---d-eAA--AA:0:0:1:31:0:0:3:0:10
White played 32H, Blue rolls 31

Blue is assigned a roll of 31. It would also work (though by only .002) to assign him a roll of 21.

		White is Player 2 score: 0 pip: 163 Unlimited Game Jacoby Beaver pip: 184 score: 0 Blue is Player 1
XGID=aa---AC-C--AdC---d-eAAA-A-:0:0:-1:65:0:0:3:0:10
Blue played 31H, White rolls 65

And now the payoff roll. White hits twice with 65R (bar/20*/14*).

Following that, Blue rolls 62, entering with the 2 and fanning with the 6, which gives us the final position:

		White is Player 2 score: 0 pip: 152 Unlimited Game Jacoby Beaver pip: 216 score: 0 Blue is Player 1
XGID=-a----C-C--adC---d-eAAAAAA:0:0:1:00:0:0:3:0:10
21$-54S-44X-32H-31H-65K-2

Here are all eleven 7-roll solutions:

.....Sequence...........................................White pipcount

.....21$-31H-44X-42H-31H-65K-2........156
.....21$-31H-44X-42H-21H-65K-3........156
.....21$-31H-44X-32H-31H-65K-2........157
.....21$-41U-44X-43H-31H-65K-2........154
.....21$-41U-44X-43H-21H-65K-3........154
.....21$-41U-44X-31H-31H-65K-2........157
.....21$-41U-44X-32H-31H-65K-2........156
.....21$-41U-44X-32H-21H-65K-3........156
.....21$-54S-44X-42H-31H-65K-2.........151
.....21$-54S-44X-42H-21H-65K-3.........151
.....21$-54S-44X-32H-31H-65K-2.........152.......(Feature sequence)

Problem 3 (think of it as extra credit) requested a White pipcount that is a multiple of 8. The filmed sequence, leading to the final diagram, is the only one culminating in such a pipcount (19 x 8 = 152).

Nack

Messages In This Thread

• Mystery of the Six Blots
  Nack Ballard -- Monday, 18 April 2016, at 3:03 p.m.
  • Mystery of the Six Blots -- HINT
    Nack Ballard -- Tuesday, 19 April 2016, at 8:57 p.m.
  • Mystery of the Six Blots -- SOLUTIONS
    Nack Ballard -- Friday, 22 April 2016, at 6:06 a.m.

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