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Cyclohex. 3-player round hex variant. (24x5, Cells: 120) [All Comments] [Add Comment or Rating]13.February.2011 the 44th day generates cv#44, Cyclohex. Take the author's word that root-7 is the algebraic distance Sennight reaches right beyond Duchess. Mutually exclusive are Rook, Unicorn of 3 bindings, Sennight. Rook should allow 23 not 22 only to avoid null move. There are 19 pieces per side, leaving 21 spaces empty in array position between each army. Pieces move both ways, Pawns one way only. Grandduke, one-stepping Duchess, is King. Three player always lends itself to diplomacy if it works. Because of the narrow board, there is no interest to split up the Rooks as in AltOrth Hex. Unicorn only goes up to 4 steps, because the narrowness prevents a fifth step.
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.
What this site needs is individualized filters so people don't have to see the posts of people who are a waste of time. You set your filter to block all posts of whomever, and then come and read the rest of the stuff free from annoyance.
Up for a year, Cyclohex was never commented until now. Probably half of 'Gilmans' are uncommented and all worth looking at, therefore this gradual survey. Crooked Board was uncommented all its 5 years until couple days ago, and already in a re-write Crooked Board is the only cv game posted this week. Crouching Stepper Hidden Rider also got its first comment yesterday in this survey, never noticed 5 years, and C.S.H.R. ought to be played somewhere, it looks so good a concept. //// In case a sentence about Pawns is misunderstood, the rules of Cyclohex say Pawn promotes on the 7th step. One double move could mean promotion in six moves. That creates interesting interaction of 5 pawns against 5 pawns. Any five pawns will never be able to pass, sit adjacent, or capture any of the other 20 Pawns. Interesting, since there has never been another case like that, sort of isolating the Pawns into three clusters. Cyclohex is also worth this extra attention because good three-player chesses are rare. There may be only about 10 three-player cvs of note so far. //// Gilman's math can be intimidating in duals and root-leapers, but the math can be made understandable. Chess variants are not a new field any more, and there are levels of appreciation. Novices ought to start with Grand Chess or Pocket Knight and work up.
Ben Reiniger wrote on Sun, Feb 13, 2011 10:33 PM UTC:Just to comment on the root-7 idea, note that this board is the same (as far as adjacency and connectivity go) to a cylinder that is tiled with hexagons. Then the distance in that cylinder is exactly the same as in flat hex, and so root-7 needn't even be understood as an idealization. (I've wanted a toroidal board that is actually built toroidal for a while; with magnetic pieces I think it would be amusing.)
(Also, the calculation that the distance is sqrt(7) can be shortened if one is willing to use the law of cosines.)
Actually you don't need to take my word for it about root 7, just the word of whichever mathematician it was - I cannot remember offhand - who extrapolated Pythagoras' theorem to a²+b²-2ab cosine(internal angle between a and b)=c² for any sides of any triangle. For an internal angle of 180° - straight on - this renders c² as a²+b²+2ab which many will recognise as indeed the formula for (a+b)². Likewise for 0° - doubling back - it gives a²+b²-2ab which many will recognise as the formula for (a-b)². For 90° it gives Pythagoras' original a²+b². For 120° it gives a²+b²+ab. In the case of the Sennight this is 2²+1²+(2x1)=7.
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