Puzzle 3. Challenge Problem
Find a perfectly played sequence that leads to the position below. Blue is on roll. The cube is centered.
  White is Player 2
score: 0 pip: 159  Unlimited Game  pip: 139 score: 0
Blue is Player 1  
XGID=aBBaCBdDdeAA:0:0:1:00:0:0:0:0:10 
Blue on roll, cube action? 
The first solution I found for this puzzle has 14 moves. Later on, I found a pair of 9move solutions. Finally, Nack found a couple of solutions requiring only 7 moves. Armed with a confirmation that a pair of 7move solutions existed, I eventually discovered them.
Given that there are 7 moves, and remembering that Blue must be on roll at the end of the sequence, it is easy to see that White rolls first. White gets 4 turns; Blue, 3.
Now, Blue has only moved 28 pips. If Blue is going to roll 55 as part of the solution, that accounts for 20 pips already. Barring a hit, Blue's other two moves must total only 8 pips. In no particular order, that means Blue must have rolled 21 along with either 32 or 41, or else he rolled a pair of 31s. I discarded the possibility that Blue rolled 11 rather than 31. If he had, he probably would have made his 5pt.
White, of course, has moved 9 pips forward, and 1 pip backwards, netting him a total of 8 pips. Obviously we will have to put him on the bar early in the solution, and hit him whenever he makes any progress. It is likely that White's opening roll was 51, 52, or 54.
Although I spent some time experimenting with all three of those possibilities, from the start, 51S = 24/23 13/8, was the most likely. That's because a subsequent roll of 41 allows Blue to hit twice, thus preventing White from playing any more checkers down from his midpoint.
  White is Player 2
score: 0 pip: 161  Unlimited Game Jacoby Beaver  pip: 167 score: 0
Blue is Player 1  
XGID=aaECdEdeB:0:0:1:41:0:0:3:0:10 
Blue to play 41 
1.  Book^{1}  6/2* 2/1*  eq: 0.1448 
 Player: Opponent:  46.58% (G:13.01% B:0.40%) 53.42% (G:15.59% B:0.80%)  Conf.: ±0.0027 (0.1475...0.1421)  [100.0%] 

2.  Book^{2}  13/8  eq: 0.1601 (0.0153) 
 Player: Opponent:  46.37% (G:12.05% B:0.47%) 53.63% (G:15.50% B:0.73%)  Conf.: ±0.0017 (0.1619...0.1584)  [0.0%] 


^{1} Generated by Steven Carey on 3/31/2013 using eXtreme Gammon 2.10 46656 Games rolled with Variance Reduction. Dice Seed: 90680458 Moves: 3ply, cube decisions: XG Roller
^{2} Generated by David Rockwell on 6/16/2012 using eXtreme Gammon 2.00 62208 Games rolled with Variance Reduction. Dice Seed: 90056687 Moves and cube decisions: 4ply

Unfortunately, this turned out to be a dead end. In most of the variations I tried, Blue gets stuck attacking on his side of the board, and never finds the right time to split his rear checkers.
That's when I remembered that 51S21 is best played up, U = 24/21. That's a second roll position where XG has overturned the older GnuBg rollouts that favored slotting, $ = 13/11, 6/5. This was a promising sequence, because 21 was one of the rolls I expected might be part of the solution.
  White is Player 2
score: 0 pip: 161  Unlimited Game Jacoby Beaver  pip: 167 score: 0
Blue is Player 1  
XGID=aaECdEdeB:0:0:1:21:0:0:3:0:10 
Blue to play 21 
1.  Book^{1}  24/21  eq: 0.1648 
 Player: Opponent:  46.37% (G:11.14% B:0.47%) 53.63% (G:15.16% B:0.59%)  Conf.: ±0.0026 (0.1674...0.1623)  [100.0%] 

2.  Book^{1}  13/11 6/5  eq: 0.1740 (0.0092) 
 Player: Opponent:  46.72% (G:12.25% B:0.50%) 53.28% (G:17.21% B:1.24%)  Conf.: ±0.0029 (0.1769...0.1711)  [0.0%] 


^{1} Generated by Steven Carey on 3/29/2013 using eXtreme Gammon 2.10 46656 Games rolled with Variance Reduction. Dice Seed: 90680458 Moves: 3ply, cube decisions: XG Roller

In the resulting position, I searched for rolls that White would play from the rear. Any roll that moved on his side of the board (landing between the 1pt and 12pt) could not have been a part of any solution. As it happens, there is only one possibility. White must have run when he rolled 62.
  White is Player 2
score: 0 pip: 161  Unlimited Game  pip: 164 score: 0
Blue is Player 1  
XGID=aaECdEdeAA:0:0:1:62:0:0:0:0:10 
White to play 62 
1.  XG Roller++  23/15  eq: 0.0047 
 Player: Opponent:  50.30% (G:12.00% B:0.48%) 49.70% (G:13.27% B:0.50%)  

2.  XG Roller++  24/16  eq: 0.0226 (0.0179) 
 Player: Opponent:  50.07% (G:10.72% B:0.38%) 49.93% (G:12.51% B:0.44%)  

eXtreme Gammon Version: 2.10
And now it is time to pounce. Let's give Blue his 55. After that, it is clear that White must enter. Otherwise, Blue will have a cube. 32 is a good entering roll to give White. It puts his checker on his 20pt, right where we want it.
  White is Player 2
score: 0 pip: 154  Unlimited Game  pip: 144 score: 0
Blue is Player 1  
XGID=aBBCAadEdeAA:0:0:1:32:0:0:0:0:10 
White to play 32 
1.  XG Roller++  Bar/20  eq: 0.3754 
 Player: Opponent:  43.10% (G:9.42% B:0.48%) 56.90% (G:24.83% B:0.42%)  

2.  XG Roller++  Bar/23 13/10  eq: 0.4947 (0.1193) 
 Player: Opponent:  40.79% (G:9.75% B:0.49%) 59.21% (G:27.55% B:0.94%)  

eXtreme Gammon Version: 2.10
If Blue has a cube, we are in trouble. As things stand, however, White is not on the bar, and Blue only has 8 checkers in the zone. Furthermore, Blue's 3 is triplicated. This is a nodouble.
  White is Player 2
score: 0 pip: 149  Unlimited Game  pip: 144 score: 0
Blue is Player 1  
XGID=BBaCAadEdeAA:0:0:1:00:0:0:0:0:10 
Blue on roll, cube action? 
Analyzed in XG Roller++  No double  Double/Take 
Player Winning Chances:  56.616% (G:25.080% B:0.429%)  56.601% (G:25.169% B:0.419%) 
Opponent Winning Chances:  43.384% (G:9.449% B:0.478%)  43.399% (G:9.609% B:0.493%) 
Cubeless Equities  +0.2881  +0.5738 
Cubeful Equities 
No double:  +0.3693  
Double/Take:  +0.2524 (0.1169)  
Double/Pass:  +1.0000 (+0.6307) 

Best Cube action: No double / Take 
Percentage of wrong pass needed to make the double decision right: 13.5% 
eXtreme Gammon Version: 2.10
The rest is obvious. Having used 21 and 55 already, Blue must roll either 32 or 41 next. Otherwise, he won't reach the 28pip total he needs. Conveniently, 32 allow him to hit and cover. After that, all that remains is for White to dance.
  White is Player 2
score: 0 pip: 149  Unlimited Game  pip: 144 score: 0
Blue is Player 1  
XGID=BBaCAadEdeAA:0:0:1:32:0:0:0:0:10 
Blue to play 32 
1.  XG Roller++  13/10* 10/8  eq: +0.4680 
 Player: Opponent:  58.68% (G:23.20% B:0.43%) 41.32% (G:9.04% B:0.36%)  

2.  XG Roller++  13/10* 8/6  eq: +0.3685 (0.0995) 
 Player: Opponent:  56.01% (G:22.83% B:0.44%) 43.99% (G:10.44% B:0.46%)  

eXtreme Gammon Version: 2.10
The complete solution is
51S21U62R55A32U32PF
A variant has White playing 41U rather than 32U:
51S21U62R55A41U32PF
  White is Player 2
score: 0 pip: 154  Unlimited Game  pip: 144 score: 0
Blue is Player 1  
XGID=aBBCAadEdeAA:0:0:1:41:0:0:0:0:10 
White to play 41 
1.  Rollout^{1}  Bar/20  eq: 0.3659 
 Player: Opponent:  43.12% (G:9.77% B:0.53%) 56.88% (G:25.51% B:0.43%)  Conf.: ±0.0019 (0.3678...0.3640)  [87.6%] Duration: 5 hours 18 minutes 

2.  Rollout^{1}  Bar/21 15/14  eq: 0.3675 (0.0016) 
 Player: Opponent:  43.23% (G:9.29% B:0.46%) 56.77% (G:23.84% B:0.39%)  Conf.: ±0.0019 (0.3695...0.3656)  [12.4%] Duration: 5 hours 33 minutes 


^{1} 62208 Games rolled with Variance Reduction. Dice Seed: 15715320 Moves: 3ply, cube decisions: XG Roller

Rollout by Taper_Mike 2015Nov08 eXtreme Gammon Version: 2.10.199.2658 51S21U62R55A41ntm.xgp n[U R2] "<=62 n[R U3] "&e 
eXtreme Gammon Version: 2.10
As a bonus question, do you know the proper cube action in the final position?