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BGonline.org Forums
Jeff Ward's 'Magic Triangle'
Posted By: Timothy Chow In Response To: Jeff Ward's 'Magic Triangle' (Albert Steg)
Date: Thursday, 10 March 2016, at 11:47 p.m.
Albert Steg wrote:
I guess now I'm wondering whether for some reason one is never faced with a choice between say 2 and 5 or 4 and 1? My head is starting to hurt.
First, you need to convince yourself that it never makes sense to waste pips when you roll something that leaves you with 2 checkers. So for example, if you have three checkers remaining on the 6pt, 4pt, and 3pt, and you roll 62, there's no reason to play 6/4/off instead of 6/off 4/2 or 6/off 3/1. This is intuitively obvious although I think it may be a pain to prove it rigorously.
If no pips are wasted, then the spacing between your two checkers will always have the same "parity"—i.e., if one play leaves an odd spacing (1, 3, or 5) then every other play will also leave an odd spacing, and if one play leaves an even spacing then every other play will also leave an even spacing. The reason is that if one checker is moved one pip, then the other checker must move one pip in the opposite direction to keep the pip count the same, so the spacing must change by 2 pips.
Therefore, all that matters is that in your "magic number", 3 comes before 1 comes before 5, and 2 comes before 4 comes before 0.
Having said all that, I'll agree with ah_clem that the easiest way to remember the rule is that you should make the spacing as close to 2.7 as possible.
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