The original question was posed here.
- From the starting position of backgammon, what is the least number of moves by which both players can obtain THREE checkers on each of FIVE different points?
Extra credit: Given that least number of moves,
(a) Find the maximum pipcount difference possible (between the two players).
(b) Find the minimum pipcount difference.
Main answer: The least number of moves is three (meaning three full moves, or six rolls).
For the extra credit, let's knock off (b) first -- the minimum pipcount difference. Stein didn't spell out his exact answer, but I'm pretty sure he found the same 0-pip solution I did:
| |
|---|---|
| 41: Blue plays 6/5 6/2 | |
| 41: White plays 24/23* 6/2 | |
|---|---|
| |
| |
|---|---|
| Blue plays Bar/23* 24/23(2) | |
| White plays Bar/23 24/23 6/5 | |
|---|---|
| |
| |
|---|---|
| Blue and White both roll 44 and play 13/5(2) | |
| Both sides: 3 (checkers) x 5 (points) | 165 |
|---|---|
| |
| Final position: 0-pip difference | 165 |
1) 41: 6/5 6/2... 41: 24/23* 6/2
2) 11: Bar/23* 24/23(2)... 11: Bar/23 24/23 6/5
3) 44: 13/5(2)... 44: 13/5(2)
There's an elegance in the recouped symmetry.
For part (a), Stein was again able to match my solution. The maximum pipcount difference is 57, which is achieved as follows:
| |
|---|---|
| 43: Blue plays 13/10 13/9 | |
| 33: White plays 24/21 6/3(3) | |
|---|---|
| |
| |
|---|---|
| 42: Blue plays 10/6 9/7 | 160 |
| 66: White plays 24/18* 13/7(3) | |
|---|---|
| |
| |
|---|---|
| 11: Blue plays Bar/24 6/5(3) | |
| 55: White plays 21/6 18/13 | |
|---|---|
| |
| Both sides: 3 (checkers) x 5 (points) | 111 |
|---|---|
| |
| Final position: 57-pip difference | 168 |
1) 43: 13/9 13/10... 33: 24/21 6/3(3)
2) 42: 10/6 9/7... 66: 24/18* 13/7(3)
3) 11: Bar/24 6/5(3)... 55: 21/6 18/13
The 33 and 66 rolls can be transposed. If the 11 follows the 66 (as here), the non-doublets can be any two rolls that sum to 11 pips. If the 11 follows the 33, the non-doublets must be a 9 and a 7.
Later, I challenged Stein to find the maximum pipcount difference (with the condition that the bar/25pt is considered a "point") after an extra move -- FOUR moves (eight rolls) altogether. The greatest pipcount difference I've found is 153, and I'll post the sequence for that solution soon (probably tomorrow) unless Stein (or someone else) tells me he is still working on it.
Nack