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【野子】UFO系列之Sando Ring [复制链接]

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发表于 2011-4-5 13:15:23 |只看该作者 |倒序浏览

英文名：Sando Ring，谷歌翻译叫：桑多环，我管它叫：四球ufo
感谢鲁尼的帮忙，几经波折终于拿下。
英文资料：
There are several puzzles called Ufo, but this one is made by Netblock. It consists of a disc which is split into two layers which can rotate about the central axis. Spaced evenly along the rim of the disc are 3 spheres of different colours. These are split into octants, so that when the layers of the disc are turned, one half of each sphere (4 pieces) travels with each layer. The spheres can also rotate in two halves about an axis along the disc edge. Although this puzzle is hard to describe, it is fairly easy to solve.
An earlier version is called King Ring, or Sando Ring. This is much larger, uses bright colours, and has four spheres. It looks very much like a toddler's toy.
The Ufo was patented on 13 May 1997 by Wai K. Chan, US 5,628,512. The King Ring was patented on 21 March 1996 by Zoltan Pataki, Istvan Varadi, and Attila Kovacs, WO 96/08297.

The number of positions:Ufo: There are 24 pieces, 8 of each colour. They can therefore be arranged in at most 24!/8!3=9,465,511,770 ways. This limit is not reached because:

The pieces in the left halves and the right halves of the balls never intermingle.
The orientation of the puzzle itself is unimportant.

The first restriction means that there are at most 12!2 / 4!6 = 1,200,622,500 possible positions. The second restriction means that the real number of positions is about 1/6th of that number because the puzzle can be held in 6 different ways (due to the three-fold symmetry around the centre, and because it can be turned over). As some positions are themselves symmetric, the exact number can best be calculated with Burnside's Lemma, and this gives 200,121,075 positions.

King Ring / Sando Ring: There are 32 pieces, 8 of each colour. Using the same reasoning as above, we get 16!2 / 4!8 = 3,976,941,969,000,000 positions. The real number of positions is about 1/8 of this, and the exact number iven by the Burnside Lemma is 497,117,746,919,592.