Ray pointed out that Phil's parenthetical conjecture, "you can't have all your checkers on one side of the board in 3 rolls" (unless oddly he meant 2 rolls by one player and 1 by the other) is false. In fact, the two players can both do so in three rolls each, as independently demonstrated by Ray and Daniel.
Ray then wrote: A slightly more challenging problem is the minimum number of pips I must roll to get all my checkers on my side of the board in three rolls. I make it 36 pips (with opponent rolling at least 13 pips):
This is the first time I read any of the posts in this subthread (inside the thread started by Phil)... thirteen and a half years later! This is one of the advantages of Bgonline; for comparison, try navigating the archives on other backgammon sites (e.g., the primary one on FB).
For this supplemental puzzle, Daniel is correct that the fewest number of pips with which "Me" can accomplish the feat is 31 (beating Ray's 36).
For reference, Daniel's solution is: Opp 32 (13/11 13/10), Me 21 (13/10), Opp 32 (13/11 13/10), Me 66 (24/12*(2)), Opp 21 (bar/22), Me 11 (13/12(4)). The opponent, going first, rolls and plays the same 13 pips given in Ray's solution.
Ray's wording of the puzzle did not specify that minimizing Opp's pips was part of the goal. However, since Ray and Daniel both made a point of listing Opp's 13 pips in their solution summary, I'll point out that if Me goes FIRST, he can still achieve the stated goal in 31 pips, while the opponent has rolled only 11 pips, as follows:
Me 21 (13/10), Opp 22 (13/11(4))
Me 66 (24/12*(2)), Opp 21 (Bar/22)
Me 11 (13/12(4))
Here, Opp rolls only twice. It was never stated that she has to roll three times. The trick is that Opp can clear her midpoint in one 8-pip roll (double 2s) instead of two 5-pip rolls (32, 32). If you believe that three rolls each was implied, ignore my solution.
It would seem that nobody responded to Ray's second puzzle:
Considerably more challenging is the minimum number of pips rolled in 6 rolls (3 for each player) for each player to have all his checkers on his side of the board.
I submit this 66-pip solution (Me = 31 pips, Opp = 35 pips).
Opp 21 (13/11 6/5), Me 21 (13/11 6/5)
Opp 22 (13/11(4)), Me 11 (13/12(4))
Opp 66 (24/12(2)), Me 66 (24/12(2))
The final position is diagrammed below.
Thematically, there's nothing new in this 3x2 roll solution (it uses the same double 2s resource), except that 13/11 6/5 is a stronger opener and a stronger reply backgammon-wise than 13/10. :)
Nack
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| XGID=-----AD-C--EBfa--c-da-----:0:0:-1:00:0:0:3:0:10 | |||||
| Roll sequence (White first): 21, 21, 22, 11, 66, 66 | |||||
eXtreme Gammon Version: 2.10
An incidental sidelight is that White (now on roll) and Blue have roughly even racing chances.