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Thanks Joe Joyce and Ben Reiniger. Great there are more than one way to win a matt, schachmatt. Here is another argument the same as or overlapped with yours. TRIANGLES OF THE MIND. Of course the complete triangular grid is already there inside the square. Just take Betza's first step, rotating the utility board 45 degrees. Do not perform Recta-hex-ing, stop at Betza's first sentence. Give each Square/Diamond a top-half and bottom-half. That object then has all the features of equilateral triangular grid linked, http://www.chessvariants.org/index/displaycomment.php?commentid=25282, since different angles 90, 45 and line lengths topological equivalence dissolves. Then require each piece entered into the space be positioned close to either an ''up-vertex'' or a down-vertex slot, so the two dimensions, conjuring infinitesmals, do not ramify to anything as significant as ''triangle.'' Each cell to play, insofar as it has minimal practical room for piece-marker, must needs have/be an up-vertex area or else a down-vertex and never both. No triangles are involved or ever required to be drawn or even pictured. Play within squares from vertex area to vertex area. Keep looking at squares only and play Chess (Variants). That way the Anti-Triangular board is made only of the everlasting same 64 2^6 squares (and Checkers 100 10^2 squares for enlargement). __________________________________Since we are only using squares, the regular 64 of them, movement has to be carefully defined. Each move begins and ends into up-vertex slot or down-vertex slot, never something like phony ''side-vertex.'' The latter does not exist. Remember to keep the board with diamond orientation. Thus 64 ''squares'' are 128 vertex-locales, more than enough. Each up-vertex is unique, so it has a number like '9', Cell 9. From '9' there are four equidistant other up-vertices lettered 'a' to 'd' CW from top. They are a square side-length away. The first one of them, 'a', is actually top-off-center right. And so on with down-vertex locations we want to fulfill from a starting cell. Play from up-vertex cell to down-vertex and vice versa, and up to up and down to down, completing all the possibilities relatively nearby, for defined piece-types. Each sub-space/cell is as that infinitesmal building towards a point(corner of Square) and has no dimension except created by convenience in marking. Bi-colouration of the board may tend to distract and can be eliminated. So long as the divisible, exactly-delineated space is elementary/fundamental, Rules then can achieve the full degree of decadent over-refinement customary to CVers. Watch your step. These proofs are not semantic; is not all space everywhere one-dimensional-acquired directionality? Summary: Hexagonal -> Rectahex -> Square -> Triangle (reversible). The only trick is the Rectahex slide a la Betza. http://www.chessvariants.org/index/displaycomment.php?commentid=19657

**Ben Reiniger**wrote on Sat, Nov 13, 2010 02:51 AM UTC:

Another way to get triangular is by playing on the corners of a hex grid, so you could use the rectahex board in the same fashion; you just have to be able to remember that the intersections of a vertical edge with what appears to be a square's edge is actually another corner (since the apparent edge of the square is in fact two separate edges in the rectahex sense). Looking a bit more closely though, this won't work if the board is turned around in the 'proper' orientation. Notice that the corner of the hexes d4,d5,e4 should be adjacent to the corner of the hexes d5,e4,e5; but on the usual orientation these are the same corner! You could get around this I suppose, but it would be ugly.

**Joe Joyce**wrote on Sat, Nov 13, 2010 12:42 AM UTC:

To answer your question about squares used for triangular movement: You remove some of the squares in a particular pattern. Each removed square is at the center of a hexagonal arrangement of squares that are always present. Pick any square from that always-present hex arrangement of squares, and remove neighboring squares such that any is always surrounded by an alternating pattern of present and absent squares. This pattern allows triangular movement, some of which may pass through the absent squares to the present squares on the other side. This pattern leaves the board with these features: 1 - all absent squares are at the center of a hexagonal ring of present squares. 2 - All present squares are members [junction points] of 3 rings of present squares. 3 - All present squares have 3 absent neighbors, and 3 present neighbors.

Fergus Duniho illustrates the 12 directions of movement on a hexagon board and inteprets them for Shogi pieces on his Hex Shogi page. In Hex Shogi 81 he copies the traditional Shogi setup to a 'tilted rectangle' made up of 81 hexagons. A few weeks ago I was looking at Duniho's game and thinking that it could be also played on a square board, with a little mathematical magic.

It should be possible to use Ralph Betza's work to accomplish this task. Start with a traditional Shogi board and pieces. Replace the Rooks with 'Rectahex Rooks' and the Bishops with 'Rectahex Bishops'. Looks like the Shogi Knight can be replaced by a Rectahex Knight, restricted to four forward Bison moves. In the final analysis, pieces are completely defined by their movement rules - the geometry of the board is merely a convenient aid to play. But I am not seriously recommending that anyone try to play a game of **Rectahex Shogi 81**.

I like this game !! I have an observation. If we merge the board rotated to the right with the board rotated to the left, We get Queens for Rooks, Unicorns (BNN) for Bishops, and NJZ (Knight + Camel + Zebra) for Knights, Queen of the Night (BRNN) for Queen. Sounds like a nice variant. If we subtract the original pieces movements from Rooks, Knight, and Bishops, we get Bishops for Rooks, Bisons for Knights, Nightriders for Bishops, and Unicorn for Queen. This makes a nice CwDA army. Don't you think ? I will post this into a new page, since it is a very different variant.

In mentioning a three-geometry game, I meant a game which incorporated piece movements derived from hexagon-tiled, square-tiled and triangle-tiled chess variants. The board itself would involve one geometry. Consider a 12x12 board tiled with equilateral triangles (all have a one horizontal side, a1 points at W, a2 at B, a3 at W ..., b1 points at B, c1 at W, etc.). Triangle chess movement (as in Dekle's Triangular Chess - see Pritchard's Encyclopedia of Chess Variants) would be based on the shapes, standard chess movement on the ranks and files (N leaps 2 ranks, 1 file or 2 files, 1 rank; R slides along rank or along file; B leaps in 1r,1f steps in same direction, etc.) and hexagonal movement would follow a scheme similar to that in Hexoid Chess (/Rectahex Chesss). A hex-style three-coloring of the board (a1-blue, a2-yellow, a3-red, a4-blue, ... b1-yellow, b2-red, b3-blue, b4-yellow, ..., c1-red, c2-blue, c3-yellow, ...) would help a little for the hex movements. On such a board we could have Standard Knights, Triangular Queens and Hexagonal Rooks, etc. cheerfully (?) coexisting. Getting the hang of the game would be a little tough -- but then again, who ever said unified (playing) field theory would be easy?

If I have inspired Trigon Chess, then I count RectaHex as a success, a veritable succectahex. I first read of Glinski's game in my tattered copy of Boyer, with the diagram of 'Echecs Hexagonaux de W. Glinski' right there on the cover; and I always felt there was something wrong about the game. The rationale for why the pieces move as they do is so logical and convincing that it is hard to question the game; but something is wrong somewhere. Perhaps Glinski is fine as is, but Rectahex should spur some new thinking.

Should compare this to my Hexoid Chess and Rex Chess (both hexagonal on 8x8 standard chessboard) or Tetragonal Chess (more an inspired by; rotate the board, but retain hex-type movement) -- all 'modest chess' variants (piece variants) buried somewhere on CVP. One difference: I use Q as a third bishop to offset bishops only covering 2/3 of board in Hexoid Chess (and also because Glinski-type hex-Q is a killer on 8x8 board). I also felt that directions are much easier if board is rotated 45 degrees, so a1 points at White. By the way, a similar translation can be made for 'Triangular Chess': imagine all dark squares are equivalent to equilateral or isosceles triangles with edges horizontal and 'pointing' at Black. Pieces move differently on dark squares than on light squares -- on dark squares on type of N is >Black as [v2, h1], >White as [v1, h2] and horizontally as [h3, v0], and reverses White-Black move pattern if on a light square. Probably no more than one Trigonite type per game is best (Trigonic Knight Chess?). Reversing imagined direction of 'pointing' gives a second type of crabbish Knight. This opens up the possibility of a three-geometry chessboard game -- if only as a sort of unified (playing) field theory.

For the sake of argument, I'll take the opposing point of view. :) 'Hexagonal Chess can be played quite simply on a normal rectangular board' is a statement not justified by the article. First the player is expected to either rotate a board 45 degrees and remember that corners are now edges and vice versa, or they need to memorize a new army unusual-moving pieces. For the author of the article these may be simple tasks, but I would venture to say that for the casual CV player it is difficult. The author even implies this himself when describing the moves of the pieces: 'This is confusing' and 'This is a cumbersome piece' are used, and the 'normal' description of how the pieces move are complex. The advantage of playing with hex-moving pieces on a readily available rectangular board is outweighed by the complexities of 'biasing' the board to match the connections of a hexagonal one. It would be to a player's advantage to buy an inexpensive set of poker chips and arrange them as a hexagon and use the 'standard' hexchess pieces. The article is useful, in that it shows how one type of board and pieces can be mapped to another type. It can provide the starting point for further hex/rect explorations, and possible new pieces for the rectangular board. 'Biased' pieces as described in the article are vaguely reminiscent of left- and right-handed pieces in shogi variants. The rectahex knight could be matched with a mirrored one to make a pair of 'ufos' on a large-board variant. Thanks for the article!

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