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| 21$-65R-43C-21H-11 | |
| The featured position in this thread, 21$-65R-43C-21H-11, has some very tricky Nactations in the underlined E (Each) family.
In this post, I provide three tentative Nactations for the E (Each) family. They differ based on the method used by the 6pt Convention to determine the destinations (and intermediate landing spots) of blots and spares. I present these in the hope that I can explain clearly my confusion in the application of the 6pt Convention. If I have done my job, Nack should be able to point to one of these three, and say, “That’s the correct interpretation.” |
First Attempt
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The first is based on an examination of the board after all checker movements are complete. The checkers that end up as blots or spares, and only those checkers, have their destinations (and intermediate landing spots) given in the final column of the table below. Checkers that were moved, perhaps as single checkers, but which eventually ended up as part of a made point, are not included. Also, each destination is listed only once, even when a point, such as the 23pt, winds up with two spares. From left to right, blot and spares destinations are ordered by decreasing rank in the Nactation family. Note that for purposes of Nactation, the 23pt and 24pt do not count as made points, even when they hold two or more checkers. Any checker on one of these points automatically counts as a blot or spare. |
| Nactation | Traditional Notation | Far 1 3 | Down 0 0 | Jump 0 0 | Inside 3 1 | Hits on These Points (more is better) (higher is better) | Owned Points Not Held by All Members in Family (more is better) (closer to 6pt is better) (23pt and 24pt not counted) | Blot and Spare Destinations (closer to 6pt is better) (outer board: farther is better) (intermediate points count) |
|---|---|---|---|---|---|---|---|---|
| E | bar/24, 6/3 | bar/24 | 6/5 5/4 4/3 | 6pt 5pt | 5pt 4pt 3pt* 24pt | |||
| e | bar/24, 6/5, 6/4 | bar/24 | 6/5 6/5 6/4 | 6pt 5pt | 5pt 4pt* 24pt | |||
| E | bar/22, 6/5 | bar/24 24/23 23/22 | 6/5 | 6pt 5pt | 5pt 22pt* 23pt 24pt | |||
| e | bar/23, 24/23, 6/5 | bar/24 24/23 24/23 | 6/5 | 6pt 5pt* | 5pt 23pt 24pt | |||
| ... | bar/24, 6/4, 5/4 | bar/24 | 6/5 5/4 5/4 | 6pt* 4pt* | 24pt |
| *The asterisk marks the reason under Hit/More/Six that a given row ranks above the row that follows it. Other criteria are either tied or irrelevant. |
| To get a feel for how tricky this can be, consider the play in the fifth row above, bar/24 6/4 5/4. After entering from the bar, the next move is to push a single checker from the 6pt to the 5pt, where it rests momentarily as a spare. Should that part of the move be counted as involving a blot or spare? Because this checker eventually becomes part of a new made point, the 4pt, I omitted it in the chart above.
An undesirable result of this ranking is that a poor backgammon play, bar/24 6/3, rises to the top of the family. This is because of the rule that counts intermediate landing spots as “destinations.” Anytime a checker overshoots the 6pt, and continues moving within the inner board, it will tend to rack up some highly ranked destinations. All six points in the inner board rank above all other points on the board. |
Second Attempt
| The next attempt changes the assumptions in only one way. It allows duplicates in the list of blot and spare destinations. When two spares are moved from the 6pt to the 5pt, for instance, the 5pt is listed twice among blot and spare destinations. Checkers that become part of made points are still excluded. |
| Nactation | Traditional Notation | Far 1 3 | Down 0 0 | Jump 0 0 | Inside 3 1 | Hits on These Points (more is better) (higher is better) | Owned Points Not Held by All Members in Family (more is better) (closer to 6pt is better) (23pt and 24pt not counted) | Blot and Spare Destinations (closer to 6pt is better) (outer board: farther is better) (intermediate points count) |
|---|---|---|---|---|---|---|---|---|
| E | bar/24, 6/5, 6/4 | bar/24 | 6/5 6/5 6/4 | 6pt 5pt | 5pt 5pt* 4pt 24pt | |||
| e | bar/24, 6/3 | bar/24 | 6/5 5/4 4/3 | 6pt 5pt | 5pt 4pt* 3pt 24pt | |||
| E | bar/22, 6/5 | bar/24 24/23 23/22 | 6/5 | 6pt 5pt | 5pt 22pt* 23pt 24pt | |||
| e | bar/23, 24/23, 6/5 | bar/24 24/23 24/23 | 6/5 | 6pt 5pt* | 5pt 23pt 23pt 24pt | |||
| ... | bar/24, 6/4, 5/4 | bar/24 | 6/5 5/4 5/4 | 6pt* 4pt* | 24pt |
| *The asterisk marks the reason under Hit/More/Six that a given row ranks above the row that follows it. Other criteria are either tied or irrelevant. |
| Backgammon-wise, this is a little better. The top two plays have switched ranks. The best backgammon play, however, remains in third place in the Nactation rankings. |
Third Attempt
| The final possibility is that Nack intends that all checker movements take part in the ranking of blots and spares. This has the advantage that all entries for a roll of doublets will have four die movements. The fifth row of the table below can now be filled out completely. |
| Nactation | Traditional Notation | Far 1 3 | Down 0 0 | Jump 0 0 | Inside 3 1 | Hits on These Points (more is better) (higher is better) | Owned Points Not Held by All Members in Family (more is better) (closer to 6pt is better) (23pt and 24pt not counted) | Checker Destinations (closer to 6pt is better) (outer board: farther is better) (intermediate points count) |
|---|---|---|---|---|---|---|---|---|
| E | bar/24, 6/5, 6/4 | bar/24 | 6/5 6/5 6/4 | 6pt 5pt | 5pt 5pt* 4pt 24pt | |||
| e | bar/24, 6/3 | bar/24 | 6/5 5/4 4/3 | 6pt 5pt | 5pt 4pt* 3pt 24pt | |||
| E | bar/22, 6/5 | bar/24 24/23 23/22 | 6/5 | 6pt 5pt | 5pt 22pt* 23pt 24pt | |||
| e | bar/23, 24/23, 6/5 | bar/24 24/23 24/23 | 6/5 | 6pt 5pt* | 5pt 23pt 23pt 24pt | |||
| ... | bar/24, 6/4, 5/4 | bar/24 | 6/5 5/4 5/4 | 6pt* 4pt* | 5pt 4pt 4pt 24pt |
| *The asterisk marks the reason under Hit/More/Six that a given row ranks above the row that follows it. Other criteria are either tied or irrelevant. |
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The best backgammon play is still mired in third place, but notice what would happen if the underlined E family were to be split in two based on the ratios Far:Down:Jump:Inside. When the plays that move with the ratios Far:Down:Jump:Inside = 3:0:0:1 are given a family unto themselves, then the best backgammon play is also the top-ranked play in the Nactation family. To more fully understand the implications of these three alternative sets of assumptions, it may be useful to nactate the entire ten-member family, as I did in a prior post. My reason for creating this post was to identify clearly the several possible interpretations I see for the ranking of blots and spares under the 6pt Convention. I do not know which of them is the right one. But I hope I have enunciated the alternatives in a plain way so that Nack can identify the one that is correct. Mike |