"Nack 57" rule -- new race formula (for both traditional and Naccel)
Posted By: Nack Ballard
Date: Wednesday, 3 February 2010, at 4:30 p.m.
Nack 57 rule
The so-called gold standard chart provides the best estimate we have of the take points at each leader count (in a straight race, assuming minimal wastage).
For example, if the leader has a total pipcount of 65, the last (highest) pipcount at which the trailer has a proper take is 73, at a difference of 8. The way I phrase it (though I'm willing to be corrected), without all the extra verbiage, is: "The take point of 65 is 8."
Until today, the best race formulae that I am aware of still have several "misses" in trying to match the gold standard table. For example, Zare/Weaver/Shaw have the rest of the table covered down to 61 except they are a pip high at 61, 79, 89, 110, 120, 121 and 122 (and some higher numbers). Trice misses at all those numbers except for 120 (and has slightly fewer misses above that), and I think additionally misses at 80, 90 and 100.
In striving to create the best Naccel race formula possible, I became so familiar with the "gold standard" table, that I have been able to work on possible solutions with my eyes closed, a good use of time while I'm falling asleep.
About 3 hours ago, at 3:57 a.m., I discovered a way to match the gold standard table, for both traditional and Naccel counts. I stumbled across a solution so elegant that I practically bounced out of bed in my excitement.
I'm calling it the "Nack 57" rule, for three reasons:
1) I was born in 1957.
2) It works for all traditional pipcounts down to 57 (actually 55, but forget about that).
3) "+57" is in the Naccel formula! -- it works for all counts down to -33 (actually -35).
Here's how it works. To find the take point for TRADITIONAL pipcounts:
Subtract 33, double, and find the nearest (integer) square root.
For example, If the pipcount is 65, subtract 33, getting to 32, and double it, giving you 64. The square root is 8, so that's your take point.
Another example: the pipcount is 85. Minus 33 is 52, double = 104. The nearest square root is 10, so that's your take point.
[Below 57, use the lower half of Trice's rule, which is: subtract 5, divide by 7 and round down. Note that his miss at 61 is now bypassed.]
I hope people will verify these examples and others. I believe that the Nack 57 rule works for all pipcounts from 55 to 122, except at 111 it's a pip low. (AFAIK, the official table only goes up to 122, though it can be reasonably extrapolated beyond.)
The Nack 57 rule works exactly the same way for NACCEL, but with the 57 and 33 flip-flopped (i.e., it works down to a count of -33, and the 57 is added). Specifically:
Convert to pips, add 57, double, and find the nearest square root.
For example, if your count is 2(2), first convert to 2*6 + 2 = 14. Plus 57 is 71, double = 142, and the nearest square root is 12.
[Note that you don't add 90 -- it is a wasted step to convert to traditional pipcount.]
Another example: Your count is -5. Convert to -30, plus 57 is 27, double = 54, nearest square root is 7.
In case your Naccel count is lower than -33:
Add 1, divide by 7, round down, and add 12.
[Academic note: The Nack 57 rule is neither as accidental nor as brilliant as it sounds. It came about because I noticed that the central part of the gold standard table has a run of eight 8s, nine 9s, ten 10s, and eleven 11s. To that set, I added the last seven of the eight 7s, and the eleven 12s, to effectively capture a range of 56 pips, then retrofitted a version of the n^2 + n formula.
The existing eighth 7 (and the seventh 6 and the sixth and seventh 5, etc., are irrelevant to this scheme because they correspond to a pipcount below 57 -- the lower Trice rule has them covered. The non-existing twelfth 12 (at 111) is the only miss for pipcounts up to at least 122. To cover that one, just remember: "at 111, add 1" -- or for Naccel, "at 21, add 1."]
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