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George Duke wrote on 2011-02-13 UTC
More on cv#44, Cyclohex. (1) Pawns interact with only 5 other Pawns and never even ''see'' the other 20 Pawns, including 5 of their own side. (2) How to calculate quickly root-7 distance for Sennight? Each Rook move hex center to hex center is 1.0. Three steps are 3.0, and call that straight distance AB, B being the center of hexagon-b, or hex-b. There are two arrival hexagons adjacent to hex-b that Sennight reaches from hex-a, and just choose one of them, hex-c. Draw the line segment from B through center C and continue it terminating at D on the far side of hex-c. D will be midpoint of its side in hex-c and BD perpendicular to that side. In triangle ABD, AB is 3.0, and BD is 1.5, and since angle ADB is right, AD = (3)(root-3)/2. We want Sennight distance AC, so now switch to another right triangle, ACD. Triangle ACD has AD = (3)(root-3)/2 and CD = 0.5, so therefore AC = root-7. Square root of 7, 2.6457.... It is from the same class having Ferz moving not 1.0 but 1.414.... (3) That is where Sennight gets the name, and real value root-7 applies most of the time in hexagonal because there is not need for topological distortion. However, the Cyclohex board is special and has to be twisted around, so only some Sennight distances might really preserve root-7 after the stretching and bending. Cyclohex board will not really have root-7 (and other) distances predominate exactly, the way a real flat symmetrical hexagonal board does. To or from an outer-file hexagon the distance is greater than when an inner-file hexagon is involved. (4) As move-description for a piece-type in hexagonal connectivity, root-7 is perfectly well understood as ideal.

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