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Ninety-one and a Half Trillion Falcon Chess Variants

Notes

Ninety-one and a Half Trillion Falcon Chess Variants

Chess Variant Page has approximately 3000 chess-like games as of mid-2006 listed alphabetically with the four-square square icon. Most of those write-ups of game rules include alternate versions, some being close cousins, others widely divergent. So, let us approximate that there are presently ten thousand(10,000) CVs available there. Is it possible to have 100,000 CVs as an attainable object? Or 1,000,000? Or even one trillion?

To develop those potentialities, Falcon Chess (protected 1997 USP5690334) lends itself to ready multiplication of forms after the following numbered statements (Rules). For example-- as will be put into plain words after the first 7 numbered statements(Rules) below -- '1c2b3a4e5c6a7c' represents one very specific CV using a Falcon(actually four(4) Falcons, two for each side in all cases).

RULE NUMBER 1: Falcon Chess as embodied on board size 8x10 comprises 10 Pawns, 2 Rooks, 2 Knights, 2 Bishops, 2 Falcons, 1 Queen and 1 King per side. Articles on Falcon Chess recommend one preferred CV with 'free castling' explained there and no Queen promotion. The methods of movement of PRNBQK are well known, and the Falcon move described therein creates a complete and consistent set of game rules. However, there are alternate board sizes in both original FC Patent 5690334 and follow-up articles. The ten files being constant and the pieces positioned on back ranks as in the drawings, we can extend this argument to designate '1a' as representing 8x10, '1b' as 9x10, and '1c' as 10x10. Cumulative Total: 3 Chess Variations.

RULE NUMBER 2: The first new CVs herein permit instead only fixed castling two steps over for the King, or allow Queen promotion, making four possibilities. 'Default', so to speak, is the standard free castling and promotion to RNBF only, and we must list and label it too. So, RN2 here requires designations '2a' as fixed castling, '2b' as QRNBF promotion, '2c' as both fixed castling and Queen promotion, and '2d' as default. It should be clear enough that these four possible configurations regarding promotion and castling can go with any of the 3 board sizes of RN1. Therefore, multiplying as we go, three times four gives the result so far. Cumulative Total: 12 Chess Variations.

RULE NUMBER 3: Vary the Rook's move with '3a' standing for normal Rook(default). '3b' is Ralph Betza's Charging Rook, which is Rook-like except backwards like a King. '3c' is Betza's Different Armies' Short Rook moving up to four squares. '3d' is Betza's Half-Duck moving more like a Rook than like any other basic piece, namely, 2 or 3 spaces leaping orthogonally or else one square diagonally(Ferz). '3e' will stand for normal Rook enhanced by Ferz's move by choice. Cumulative Total: 60 Chess Variations.

RULE NUMBER 4: Enhance the Knight's move with that of Camel(1,3 leaper) at option, Ferz, Wazir (one square orthogonally), or King, respectively labelled 4a, 4b, 4c, 4d, 4e, where '4a' is the (default) normal Knight. Cumulative Total 300 Chess Variations.

RULE NUMBER 5: Next we think of five good choices in varying implementation of Bishop's move. '5a' normal Bishop. '5b' Bishop plus Wazir(so-called Conversion), '5c' Betza's Bede as Bishop plus Dabbabah(two-square orthogonal leaper), '5d' Crooked Bishop (aka Boyscout) making 90-degree turns in the same directions, and '5e' Reflecting Bishop incident 90 degrees off the edges of the board. Cumulative Total: 1500 Chess Variations.

RULE NUMBER 6: Some reasonable alternative Falcons appear to be as follows: 6a normal Falcon, 6b Falcon plus Wazir (wherein player chooses whether to move as Falcon or W in a given turn), 6c Falcon plus Ferz, 6d Falcon plus King, 6e Falcon plus Knight.

Cumulative Total: 7500 Chess Variations.

RULE NUMBER 7: For the Pawn here are ten types to differentiate: 7a standard Pawns, 7b Berolina Pawns, 7c Cannon Pawns as in Rococo, 7d Chaturanga one-step Pawns, 7e Quadra-Pawns as in Centennial Chess, 7f Pawns allowed 1-, 2-, or 3-step opening, as in many ten-rank examples, 7g Berolina Pawns with no two-step, 7h the Modern FIDE Pawns without en passant, 7i Ultima Pawns, 7j Rococo Pawns enhanced by Quadra-Pawn diagonal capture. Cumulative Total 75,000 Chess Variations.

This is no joke. There are precisely 74,999 clearly demarcated CVs spelled out without ambiguity so far, plus the one preferred embodiment, designated '1a2d3a4a5a6a7a', totalling 75,000 CVs. The more or less random example in the second paragraph above '1c2b3a4e5c6a7c' would be, therefore, a variant of Falcon Chess played with Cannon Pawns on 10x10, permitting QRNBF promotion, Knight's moving like King too, Bishop's like Bede and normal Rook and Falcon -- one specific way of 75,000, fully described by following these numbered Rules, so far to play Chess. Incidentally the very example 1c2b3a4e5c6a7c, or simply numerically 3,215,313, looks pretty good, because on 10x10 stronger Pawns and enhanced Knight, Bishop and promotion make sense.

Why stop at 75,000 CVs? Rules other than movements themselves of the basic chess pieces may be altered and factored systematically into the recommended board sizes, castling, promotion and move choices of Rules Number 1 through 7. In each category below we shall continue to take five alternatives for convenience in comparison, although obviously more could easily be adopted, as just done with the ten Pawn alternatives under RN7. Before that, Rules 8 and 9 here deal with the remaining chess pieces, Queen and King.

RULE NUMBER 8: 8a usual 'mad Queen'. 8b 'short Queen' up to four spaces. 8c 'Medium Queen' up to five squares in the eight directions. 8d 'restricted Queen' up to three spaces. 8e restricted Queen one or two squares only (recently used in Mastodon Chess Preset within CVPage). Cumulative Total: 375,000 Chess Variations.

RULE NUMBER 9: 9a standard King. 9b King allowed Knight's move once per game, as mediaeval custom. 9c King plus 'lame Dabbabah' (straight non-jumping) optionally. 9d King plus 'lame Alfil' (non-jumping two steps diagonally in one direction) optionally. 9e King plus 'lame Knight' (along either of the two two-step pathways non-jumping) at choice. Cumulative Total 1,875,000 Chess Variations.

RULE NUMBER 10: Move-turn order factors start with conventional White moves once, then Black once, and so on, marked '10a', default. 10b simple two-move Chess, removing check on the first and checking only on the second; 10c progressive Chess with 1 White, 2 Black, 3 White and so on; 10d progressive 1 White, 1 Black, 2 White, 2 Black and so on; '10e' permits moving a piece or else a Pawn or else two Pawns per turn. Cumulative Total 9,375,000 Chess Variations

RULE NUMBER 11: It is interesting to add new piece(s) or constraints on individual squares. 11a, no new Immobilizer. '11b' adds one capturable Immobilizer (as in Rococo et al.), initially in front of Queen's Pawn, that moves like a Queen, cannot capture, but immobilizes adjacent enemy pieces. 11c, such Immobilizer moves Knight-like instead. 11d, Immobilizer moves only Bishop-like. 11e, Immobilizer moves King-like. Cumulative Total: 46,875,000 Chess Variations.

RULE NUMBER 12: The new piece added in front of King's Pawn is uncapturable Promoter(Promoter's Chess), moving adjacent to which any enemy piece may be immediately exchanged for any same-coloured captured piece. '12a' no such piece. '12b' such piece moves King-like. '12c' Promoter moves Knight-like only. '12d' Promoter moves Falcon-like only. '12e' Promoter moves Rook-like. Cumulative Total: 234,275,000 Chess Variations.

RULE NUMBER 13: Triangular transference. This is a new idea, slightly related to coordinate pieces, never used before and thrown into this development. Three same-side pieces(Kings too but not Pawns, except for 13d) that form a right triangle positionally, even if two or more are adjacent(except for 13e), mutually transfer the moving powers of those two pieces that define the hypoteneuse. 13a, no such effect. 13b, the effect strictly exchanges in the transfer so that each piece loses its normal power. 13c, the effect adds to each of the two pieces' moving ability, so for example a B and N at vertices of a hypoteneuse may each move as either B or N, so long as the geometry holds. 13d, a Pawn may create the effect for pieces only by being situated at the 'right angle'. 13e, the transference, as exchange in 13b, only holds if all three pieces are non-adjacent. Cumulative Total: 1,171,875,000 Chess Variations.

We have reached a billion CVs without much difficulty. The billion-plus games happen to be based on standard Falcon Chess(USP5690334). We submit that the difference between even closely-resembling ones among these billion is greater than the difference between, for example, most any two of twenty-odd separately-recognized variants modelled after 17th-century Carrera's Chess, the likes of Bird's, Capablanca's, Aberg's, Embassy, Gothic, a fair number of which differ only in the set-up. Yet that characteristic presents quite another way to generate a long list, simply shuffling the back rank in almost any mix of chess pieces with accompanying rules. For instance, most of the very 3000 CVs, mentioned as identified by the square logos in the CVPage index, are amenable to multiplicity of form taking as unique each single potential start-up array. Relatedly, Nova Chess claims almost 39 million possible piece sets with 30 units (piece-types), with no factoring of shuffled arrays. Likewise without such rather trivial distinguishment as moving around the array, our RN1 through RN20 make each and every solitary combination from 1 into the billions its unique, coherent rules-set, consistent and complete.

For practice, '1b2d3a4a5b6d7e8a9b10a12d', or number 2,411,245,121,141, actualizes one perfectly clear and distinct game to play. It is Falcon Chess on 9x10 permitting the usual free castling, with standard Rook, Knight and Queen, Bishop's 'conversion rule' allowing one-step straight at option, Falcon also permitted full King one-step, Quadra-Pawns, and King allowed one Knight move per game; also Falcon-moving Promoters situated at f3 and f7 respectively. The fact that there are no 'eleven(11)' or 'thirteen(13)' among the descriptors points to their defaults '11a' and '13a', and there are no additional pieces besides the two Promoters.

RULE NUMBER 14: 14a No (selectively-used)Warp Points. 14b, the two (uncapturable) Warp points moving like a King are postioned initially to any available empty square on the player's side of board, and they are usable by Rook only, whereby Rook passes through in course of legal move one WP out the other one in the same direction. 14c, such same two (per side)Warp Points are usable by Bishop only. 14d, WPs used by both Rook and Bishop. 14e, WPs used by Rook, Bishop, and Queen. Cumulative Total: 5,859,375,000 Chess Variations.

RULE NUMBER 15: 15a, no Cylindrical Chess powers. In the remaining cases pieces in turn have power to move across the board as if files a and j were adjacent in continuation of their legal movements. 15b, Falcon moves cylindrically. 15c, Bishop moves cylindrically. 15d, Knight moves cylindrically. 15e, Rook moves cylindrically. Cumulative Total: 29,296,875,000 Chess Variations

RULE NUMBER 16: Switching adjacent pieces stand in next, there being many other possible implementations than these four. 16a, no such effect. 16b, any two adjacent pieces including Pawns of either side may switch places counting as a turn(at least one a friendly colour). 16c, such a switch may not remove check. 16d, King may not switch at all. 16e, all three 16b,16c, and 16d.

RULE NUMBER 17: Black Hole(s), or square(s) removed and unavailable. 17a, no such cases. 17b, one Black Hole initially placed in lieu of any still-empty square on player's half of board; it cannot be moved. 17c, required two non-adjacent Black Hole 'squares' so placed. 17d, required two adjacent Black Hole squares(per side). 17e, King alone can use either side's (non-adjacent)Black Holes(4 all together) as a Warp Point. Cumulative Total: 732,421,875,000 Chess Variations.

For another sample Chess form, what is number 31,124,125,112,113,512? It has 17 places for the 17 Rules and is easy to describe. The game is Falcon Chess on 10x10, fixed castling and no Queen promotion for its Berolina Pawns, only Rooks moving cylindrically at option, Knight's adding Camel to its repertoire, Crooked Bishops alone able to use King-moving warp points, restricted one- and two-step Queen, one Queen-like Immobilizer and one fixed-at the-start Black Hole per side. The other nuances of rule register their defaults here and so do not apply. Maybe not so nice a game as number 3,215,313 above, yet understandable and playable without contradictions.

RULE NUMBER 18: 18a, no added Jack, based on Jacks & Witches. 18b, add in sequence with those elements of RNs 11,12,14, and 17 one capturable Jack to an available square same side of board, that moves and captures like a King out of its "Eye," the symmetric square along its rank. 18c, Jack moves (out of Eye) like Knight instead. 18d, Jack so moves (but captured in situ in all cases) Alfil-like only. 18e, Jack so moves Dabbabah-like. Cumulative Total: 3,662,109,375,000 Chess Variations.

RULE NUMBER 19: 19a, no Altair-like rank-jumping. 19b, as a turn, a Bishop may be placed to any empty square in the one(or two) rank(s) exactly three ranks removed from its present rank -- based on method from Altair. 19c, As a turn, only Knight may so move. 19d, Both Bishop and Knight may so move. 19e, This 'horizontal overleap' by Bishop or Knight is to an empty square in a rank four removed. Cumulative Total: 18,310,528,875,000 Chess Variations.

RULE NUMBER 20: Selective drop chess. 20a, none such. 20b, a captured Falcon changes sides and may be dropped thereafter to any empty square as a turn. 20c, Knight only. 20d, Bishop. 20e, Falcon, Bishop, Knight. Cumulative Total: 91,552,734,375,000 Chess Variations.

Between 91 and 92 trillion CV embodiments all specifically defined. For instance, what is notational '1c2a3d4a5e6a7e8a9e10e11e12b13a14d15e16a17e18a19a20b', i.e., number 31,415,151,555,214,515,112 ? In longhand, Falcon Chess 10x10 with fixed castling, normal Knight, Queen, Falcon(that alone captured becomes available for drop later), Half-Duck able to move cylindrically, Reflecting Bishop, Quadra-Pawn that can be moved two at a time, King alternately as 'lame Knight', both King-moving Immobilizer and Promoter, two Warp Points for Rook(Half-Duck), Bishop and Queen only, two (blocking) Black Holes that just the King can use as Warp Points with another one of either side. Eminently playable!

True, 91 trillion is a large number but comprehensible as close to the number of cells in a human body. These over-91 trillion versions are achieved without resorting to shuffling the starting array. Relevantly our comment mid-January 2005 in CVPage computes 226,800 possible mirror-image starting arrays for ten-filed Falcon Chess (It would be the same for Janus Chess and some others.) 'FRNBQKBNRF' certainly might be one popular alternative. (By comparison Fischer Random Chess recommends 960 and Slide-Shuffle 2880 preferred ones on 8x8, and actually 8,294,400 are available on 8x8 there, counting 'ugly' ones like those with same-coloured Bishops and non-mirrors.)

Factoring in the 226,800 possible mirrored initial arrays means quintillions (we pass right by the quadrillions) of CV versions. More importantly, even keeping just the fixed back rank RNBFQKFBNR, there are many other factors that could be brought to bear by groups of 5 or 10 and take the cumulative totals of CVs into quadrillions and quintillions and even ridiculously beyond. To name just a few possibilities not yet integrated: (1) adding piece like an invincible Mamra (Mamra Chess); (2) adding piece like Novo Chess uncapturable General Staff able to trade places with King; (3) different coordination than the 'triangular transference' of RN13, such as pieces separated by two squares or pieces forming a square; (4) requiring captures on only odd-numbered turns; (5) requiring moving a Pawn every other turn; (6) requiring a forward move on even-numbered turns; (7) requiring a move to the left every third turn; (8) specifying a change of power in the enemy type of piece last moved or captured. Most of these stipulations have never been used in any CV as of year 2006. They would likewise be introduced in the five-mode fashion of RNs 3 through 20, steadily mounting the combinations in an endless total.

--George William Duke 2006

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Author: George William Duke.
Web page created: 2006-08-13. Web page last updated: 2006-08-13﻿