The x + y backgammon puzzle, which I originally posed here, is repeated below.
In a position that you construct: If Blue is on roll, he can win a backgammon (i.e., triple game) in x rolls. If White is on roll, he can win a backgammon in y rolls. (Sequence "Blue-White" counts as 2 rolls, not 1.)
Conditions: Blue never rolls larger than a deuce on either die. White never rolls smaller than a deuce on either die.
What is the minimum value of x + y?
Extra Credit: From the set of positions that satisfy that minimum value, what is (a) the minimum pipcount difference, and (b) the maximum pipcount difference?
Evidently, nobody else who attempted a solution was able to beat the submissions of Michael and Pierre.
My solution: x + y = 16. Blue, who can't roll larger than a deuce on either die, wins a backgammon in 7 rolls (Blue 4, White 3). White, who can't roll smaller than a deuce on either die, wins a backgammon in 9 rolls (White 5, Blue 4).
Extra credit: (a) Minimum pipcount difference I found is 7, shown in the left-hand diagram, below. (b) Maximum pipcount difference I found is 89, shown in the right-hand diagram.
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On the left, White on roll gets 44 and plays Bar/21*/17/13/9; then Blue fans with 11 repeatedly while White bears off with four doublets (the first of which is 66 or 55) and wins a bargammon. [Variation: Blue limps with 21 on her last one or two rolls.]
On the right, Blue on roll gets 11 three times, bearing off twelve checkers, and 22 on his last roll, bargammoning White who keeps fanning with 22. [Variation: Blue can roll 22 on his second-to-last-roll and/or White rolls 32 (or sometimes 42) on her last roll.]
On the right, White on roll gets 44 and plays Bar/21*/17/13/9. Blue rolls 22 repeatedly while White rolls 66 the rest of the game, winning a BG. [Variations: Blue can enter with 21 (played Bar/22) then reenter with 22, etc.; or he can enter with 11 (played Bar/21) and/or he can roll 11 later as long as his remaining rolls are also 11.]
These solutions (including variations) work whether or not a best-checker-play restriction is imposed.
Nack