Minimal number of moves.
The difference between Jemimah's dice decreased by 1 each roll.
How did the game go?

It is fairly easy to see that 5 moves are not enough, but there are a number of 6-move solutions. In order to have 4 men sent back these must follow this general outline:

1. Player A leaves a forward blot
2. Player B hits from the 24-point and also leaves a forward blot
3. A hits B's forward blot when entering and again hits in his own board while leaving another forward blot on the 2 point
4. B hits A's forward blot when entering his 2 men and again hits in his own board while leaving yet another forward blot on the 2 point
5. A enters 2 men, hits and tidies up
6. B enters and tidies up

It is clear that both players must also hit in their own boards on moves 3 or 4. If the hitting man from moves 2 or 3 is left on the 3,4 or 5 points it can not go backwards in the last move to form the double anchor, so it must be hit to reenter. If it is on the 2 point it will necessarily be hit as the last blot must be left there so that we may hit and tidy up on the last moves. (Actually, the 1 point is also a contender for placing the last blot, but I've checked that there are no variations where this work).

This means that moves 4 and 5 must be doubles. Then we can already deduce that Jemimah was player A moving first, and that her dice differed by 2, 1 and 0 respectively.

To be able to enter two blots and hit and blot on the 2-point the 4th roll must be 22, and this move must be either bar/23*(2) 6/2* or bar/23*(2) 6/4*/2, meaning that Jemimah entered in move 3 with a 2 or a 4. This means that her number that hit and left the blot on the 2-point must have been odd, so she cannot have moved from the 6- or 8-points. The only possibility is that she moved with a 5 from the 7-point, again meaning that she must have moved 13/7 in her first move.

Putting it all together the game played went:

Jemimah	Harvey
64: 13/7 6/2	41: 24/23* 8/4
54: b/21*  7/2*	22: b/23*(2) 6/4*/2
11: b/23*(2)	51: b/24 13/8

Show game

The following table shows all the other variations that go from backgammon to Nackgammon in 6 moves:

	Player 1, roll 1	Player 2, roll 1	Player 1, roll 2	Player 2, roll 2	Player 1, roll 3	Player 2, roll 3
a	63/54: 13/4	43: 24/21* 8/4	22: b/21* 6/4*/2	22: b/23*(2) 6/4*/2	11: b/23*(2)	51: b/24 13/8
b	63/54: 13/4	63: 24/21* 8/2	22: b/23*/21 6/4*/2	22: b/23*(2) 6/4*/2	11: b/23*(2)	51: b/24 13/8
c	63/54: 13/4	33: 24/21* 13/10 8/2	22: b/23*/21 6/4*/2	22: b/23*(2) 6/4*/2	11: b/23*(2)	21: b/24 10/8
d	65: 13/2	41: 24/23* 8/4	22: b/21* 6/2*	22: b/23*(2) 6/4*/2	11: b/23*(2)	51: b/24 13/8
e	65: 13/2	61: 24/23* 8/2	22: b/23*/21 6/2*	22: b/23*(2) 6/4*/2	11: b/23*(2)	51: b/24 13/8
f	65: 13/2	61: 24/23* 8/2	42: b/23* 6/2*	22: b/23*(2) 6/2*	11: b/23*(2)	51: b/24 13/8
g	43: 13/10 8/4	63: 24/21* 8/2	22: b/23* 10/8 6/4*/2	22: b/23*(2) 6/2*	11: b/23*(2)	51: b/24 13/8
h	43: 13/10 8/4	33: 24/21* 13/10 8/2	22: b/23* 10/8 6/4*/2	22: b/23*(2) 6/2*	11: b/23*(2)	21: b/24 10/8
i	63: 13/10 8/2	61: 24/23* 8/2	22: b/23* 10/8 6/2*	22: b/23*(2) 6/2*	11: b/23*(2)	51: b/24 13/8
j	32: 13/10 6/4	43: 24/21* 8/4	22: b/21* 6/4*/2	22: b/23*(2) 6/4*/2	22: b/23*(2) 10/6	51: b/24 13/8
k	32: 13/10 6/4	63: 24/21* 8/2	22: b/23*/21 6/4*/2	22: b/23*(2) 6/4*/2	22: b/23*(2) 10/6	51: b/24 13/8
l	32: 13/10 6/4	33: 24/21* 13/10 8/2	22: b/23*/21 6/4*/2	22: b/23*(2) 6/4*/2	22: b/23*(2) 10/6	21: b/24 10/8
m	43: 13/10 6/2	41: 24/23* 8/4	22: b/21* 6/2*	22: b/23*(2) 6/4*/2	22: b/23*(2) 10/6	51: b/24 13/8
n	43: 13/10 6/2	61: 24/23* 8/2	22: b/23*/21 6/2*	22: b/23*(2) 6/4*/2	22: b/23*(2) 10/6	51: b/24 13/8
o	54: 13/8 6/2	61: 24/23* 8/2	62: b/23*  8/2*	22: b/23*(2) 6/2*	11: b/23*(2)	51: b/24 13/8
p	54: 13/8 6/2	41: 24/23* 8/4	64: b/21*  8/2*	22: b/23*(2) 6/4*/2	11: b/23*(2)	51: b/24 13/8
q	54: 13/8 6/2	61: 24/23* 8/2	22: b/23*  8/2*	22: b/23*(2) 6/2*	11: b/23*(2)	51: b/24 13/8
r	64: 13/7 6/2	61: 24/23* 8/2	52: b/23*  7/2*	22: b/23*(2) 6/2*	11: b/23*(2)	51: b/24 13/8
s	64: 13/7 6/2	41: 24/23* 8/4	54: b/21*  7/2*	22: b/23*(2) 6/4*/2	11: b/23*(2)	51: b/24 13/8