The question was first posed by Ray:
What is the maximum gain in pip difference (X's pips minus O's pips) that X can achieve from one roll ? (To be clear, this is not one roll each; it is just one roll by X only. Any legal position is permissible.)
I examined this question a long time ago (as apparently did Julian Fetterlein as well). The answer is 94 pips, and it can be achieved by any doublet.
The simplest is double 1s: One checker on the roof enters with Bar/24*/23*/22*/21*. For other doublets, you need up to three more hitters; e.g., for double 6s, Bar/19* 24/18* 23/17* 22/16* likewise gains 94 pips. Double 6s is 20 pips bigger than double 1s, but the checkers being hit are further from home (they're not sent back as far) by a combined 20 pips, which exactly compensates. That is, 94 pips is the maximum gain however you slice it.
Ray then followed up with...
The next question is the maximum gain if you roll a non-double.
... to which the answer is 50 pips. The point number on which any checker hits plus the number of pips on the die always sums to the source point number (in this case the roof, which is the 25pt). Thus, two checkers entering from the bar hitting two blots gain 25*2 = 50.
Back to doublets. Bob expanded the arena by proposing...
So next is
1) Maximum pip gain in 1 move (2 rolls)
2) Maximum pip gain in 2 move (4 rolls)
3) Maximum pip gain in 3 move (6 rolls)
(He used the word "sequence" instead of "move," but I prefer "sequence" to describe a continuous string. For example, I think of the third row above as referring to a 3-move or 6-roll sequence.)
If Blue is the player making the gain, his best result is if White fans for her part (which, therefore, we will henceforth assume). So, the answer to (1) is 94 pips, the same as the answer to Ray's original question.
For (2), I believe the maximum gain is 182 pips, achievable (for example) from the position below:
| 159 | |
|---|---|
| |
| Blue rolls 33 and 66 | 159 |
For the three-move sequence, the best I've found is 253 pips.
| 150 | |
|---|---|
| |
| Blue rolls 22, 55 and 66 | 150 |
Finally, Stein suggested the possibility of adding a fourth (final) move to the sequence. Stein and I independently concluded (though he was first) that 308 pips is maximal.
Because there are only 15 blots to hit, and four doublets yield 16 move subparts, Blue can afford one non-hit finesse with one of the subparts. It turns out that the only way he can succeed is to squander his precious finesse on the first roll with a mere deuce!
| gnubg | 135 |
|---|---|
| |
| Blue rolls 22, 66, 55 and 44 | 185 |
22: Bar/23*/21 24/22*(2)!
66: 24*/18* 22/16* 21/15*
55: 24*/19*/14* 22/17* 16/11*
44: 24/20* 17/13*/9* 14/10*
Blue starts down 50 pips in the race (135-185) and after the four-move sequence he is up 258 pips (117-375), a gain of 308 pips.
Nack