I'll respond to both of the questions you framed, and to #2 first only because my answer is much shorter. (I realize this is a rehash for you, but others are reading.)
2. What method such as Trice or N57 should be used to reproduce the table at the board without memorizing all of the entries?
For the time being, the best we have is:
For counts of 57 and above, use N57 -- Subt 33, double, find nearest sqrt.
For counts at 56 and below, use Trice's lower formula -- Subt 5, div 7, rd down.
The N57 formula perfectly reproduces the gold standard table (with the exception of a 111 leader count, which teeters on the edge and adding 1 is a tad better). If a modern table (yet to be constructed, and for which the name "platinum standard table" has been proposed) is used instead of the gold standard table, it is likely (as Neil has evidenced) that the subtract 33 part of the formula will be changed to subtract 32.
I may be able to improve on Trice's lower formula as well, or even integrate the two, but of course it would be silly to waste my time trying until we know all the correct take points at all pipcounts.
1. What table should be used as the point of last take?
Minimal wastage should be used when the count is low, in part because we already know exactly what those positions are, but more importantly because those minimal wastage positions (along with all the rest) are positions that will actually arise in practice. Your excellent examples are all at the low end of the count; we're in complete agreement/alignment on those.
OTOH, when the count is high, minimal wastage positions will become unnatural. For example, suppose we want to know if 123-137 is a take. Which of the two positions below would we rather have represented in a platinum standard table?
| 137 | |
|---|---|
| |
| 123 | |
| 137 | |
|---|---|
| |
| 123 | |
In the second position, both players have checkers on their midpoints, and their 8pts are purposely be made to be a little heavier than the other points. Big numbers don't play conveniently onto the 6pt, 5pt and 4pt, and yet... This is more representative of a 123-137 position that might actually arise in a game.
If the trailer's chances of winning the first position are the same as his chances of winning the second position, then it doesn't matter which we use. However, if the trailer's chances are significantly different in the two positions, isn't there a reasonable case for using the second? That way, the player has to do less adjusting for wastage.
If the platinum standard table represents positions like the second one, then in a normal game, without even knowing what the representative position is he can be confident to be close without any need to adjust. He can just apply the N57 formula (by subtracting 33, or 32, or whatever the rollouts prove) directly, and if he happens to reach a position like the first one once in a blue moon, he knows it is unusual and can make some sort of efficiency adjustment the other way.
Finding the key representative position at each leader count will require testing, whether we apply minimal wastage rules or not. But if we apply more restrictive rules, such as at counts of 110+ both players have the midpoint, and the 8pt always has at least as many checkers as other outer board points (or has at least one more than...) -- and then apply minimal wastage within those parameters, it reduces the number of positions that need to be tested, providing an additional benefit.
Whatever we use, I believe that the rules for constructing the positions and the positions themselves (for use in the platinum standard table) should be documented, so that anyone can look them up.
Nack