I would think slotting would do OK here after 41S since it does OK after 51S noting that slotting is duping some 3's and 4's after 41S. Apparently, it just doesn't gain much to split when the opp has played his early 5 from 13/8 adding more fire power but not creating a blot.
It is natural that you (and others) are wondering why $ loses to S (by about .020) for 41S-21, but $ beats S (by about .020) for 51S-21. That's quite a swing.
I know your comments are only offhand but the underlined your key phrase above demonstrates a somewhat common misconception. I thought many might benefit by the following explanation. The four after-positions are diagrammed at the end of this post, for reference.
The only roll with a 4 that is duped better with 41S-21$ (than with 51S-21$) is 41. In either non-slot position, 64 54 would hit on the 14pt, 43 would make the 20pt (rather than the 5pt), and 42 would make the 4pt.
The 44 roll argues the opposite: for 51$-21$ (more than for 41$-21$). With 51$-21S, you get pointed on while Opp unloads two stacks anyway. Conversely, with 41$-21$, 44 becomes a highly efficient roll for Opp, hitting and using her 9pt builder to make her own 5pt; 41S-21S manages a lot better against this roll.
Granted, 41 is a bigger swing (than 44), but it’s only one roll (i.e., we’re not talking “4s” in the plural) and the effect of 44 offsets about 30% of it (yes, I cut the swing in half to account for doublets being half as likely to arise).
As for 3 duplication, 3s are duped better with 51S-21$ than with 41$-21$ because 3s unstack. Double 3s in particular is better duplicated, but also 53 31 are duped better. Slotting deflects Opp from unstacking her fourth 8pt checker with these rolls.
Let's count 21 as a 3. That is the only roll other than 41 that definitely argues for 41S-21$ more than 51S-21$, because it would otherwise make Opp’s 7pt. Other rolls that use the 9pt builder to make a point (62 52) aren't duplicated/deflected -- they can't hit the slotted 5pt.
Based on my 36-ing, the 41 21 duplication that argues for 41S-21$ is more than offset by the 44 33 53 31 duplication that argues for 51S-21$.
Another factor is that when you leave indirect shots (in this case 64 63 54) and are hit, you have more play against a 9pt blot rather than an 8pt spare; the efficiency of splitting (rather than slotting) is increased because the 23pt checker might roll a 7 (or inch up and then hit in a scramble).
Other rolls, too, speak favorably to 41S-21S compared to 51S-21S: Consider Opp's 61 11 65. You gain more by having split because you have immediate 7s as well as 8s to hit the 16pt blot, and if your 5pt is slotted you're conflicted about hitting on the 16pt anyway (except with 44). That aspect doesn’t exist for 51S-21S-xx (where xx is 61 11 65 and others) because there is no 16pt blot to hit.
You were on the right track when you touched on the value of splitting against a 9pt blot, and some of my analysis addresses the manifestation of that value. It is the issue of duplication that is often radically different than it appears on the surface.
Hope that helps.
Nack
| 162 | |
|---|---|
| |
| 41S-21$: Wrong | 164 |
Position ID: 0HPkATDgc+EBKA Match ID: MAEAAAAAAAAA | |
| 162 | |
|---|---|
| |
| 41S-21S: Right | 164 |
Position ID: 4HPkASjgc+EBKA Match ID: MAEAAAAAAAAA | |
| 161 | |
|---|---|
| |
| 51$-21$: Better than the S alternative | 164 |
Position ID: 0HPkATDg8+ABKA Match ID: MAEAAAAAAAAA | |
| 161 | |
|---|---|
| |
| 51S-21S: Wrong | 164 |
Position ID: 4HPkASjg8+ABKA Match ID: MAEAAAAAAAAA | |