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BGonline.org Forums
NACK'S POSITION
Posted By: Nack Ballard In Response To: NACK'S POSITION (Ray Kershaw)
Date: Monday, 24 May 2010, at 9:12 p.m.
Let's say that you replace one of White's second-through-twelfth rolls of 44, 55, or 66, with any two other rolls from the pool of 33/36 numbers (which is very generous, because so many combos fail to sum to enough replacement pips or meet crossover requirements, etc.). Because you also have to give Black the extra 11, that variation is at most (33/36)^2 / 18 = only 4.7% as likely. You can do that with any of the eleven rolls, and you can tack on a few three-for-two rolls replacements, but that still comes to only (and generously) 55 percent. To put that in perspective, the difference between the low and high of the range I guesstimated is a matter of three zeros, which is 100,000 per cent!
There are a huge number of three-for-one and four-for-two rolls replacements, etc., but those each have to be divided by 324 (i.e., 18 squared). I'd guess they don't even add up to what the first round of replacements yielded. And so on. Continue adding the pieces until you reach the point that you're giving Black all rolls of 21, but with the (1/18)^n dilution factor it seems to me you'll be dealing with small potatoes compared to the effect of allowing all combinations of double 4s, 5s and 6s in the main variation (the quick and dirty method I used to reckon the optimistic end of the range).
Having said that, I can't think of any good shortcuts to estimate what you're proposing (sorry), so feel free to have at it!
Nack
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