Multipath Chess Pieces
By George Duke
What is a chess piece? Not so much a tangible artifact as a method of movement from one square, or cell, to another. Classically there are two types of chess pieces, Leapers and Riders. Knight is Leaper; Rook and Bishop are Riders, and Queen compound Rider. A better classification adds a third grouping, Multipath Pieces, which move to a given square by more than the one pathway of simple Rider, but not automatically like a Leaper. FALCON of Falcon Chess is the quintessential multipath piece having as it does three(3) pathways to squares three away obliquely from a departure square.
Rook and Bishop have straight-line paths to their squares, but sometimes Chess Variants have rules of movement for pieces' changing direction 36 degrees (after a Knight leap), 45, 54, 90, or 135 degrees. That opens the possibility of multiple paths. Simply Ralph Betza's Bent Riders are not ipso facto multipath, but most multiple-path pieces are bent riders with change(s) of direction. Described in H. J. R. Murray's 'History of Chess' 1912 page 348, Gryphon dates from year 1211, "A move compounded of one step diagonally followed by any number straight." Once this Bent Rider is permitted the reverse order too, that is, any number straight followed by one step diagonally, that becomes multipath, in this case two-path, piece.
How can a multipath Piece not be a bent rider? 8x8 Cylinder Chess board connects files a and h. So, a Rook (or Queen) on e4, for example, can reach arrival square g4 by way of e4-d4-c4-b4-a4-h4-g4 or else e4-f4-g4. Therefore, in Cylinder Chess R and Q are multipath with two routes to each attainable square not in same file or diagonally separated from departure square; and there is not the angled change of direction of a bent rider.
G. Balbo's Elbow Chess (ECV) creates pieces falling within this new classification by requiring a right-angled turn midway through any move. Double-Move Chesses (ECV) give rise to situations where orthodox pieces have what is equivalent to double paths. Billiards Chess (ECV) Queens (but not Bishops) have double paths to some of their squares. There a Queen on d2 may move to d8 Rook-like d2-d3-d4-d5-d6-d7-d8 or reflecting Bishop-like d2-c3-b4-a5-b6-c7-d8. One or other or both pathways may be impeded by intervening pieces, but either pathway validates the move from d2 to d8. Not too far a field, Co-Chess Variants bring forth double capturing options. Within CV Page Novo Chess' Motor Unit, having four of six Falcon movements, is two-fold in its motion from and to certain squares of its intricate game board.
In at least two cases, ROSE and Halfling Rose, Betza creates multipath movers out of Bent Riders. CVP Piececlopedia page on Rose shows the Rose's paths having continual 36- and 54-degree turns. In the diagram 'Half-Rose' on b4 moves b4-d5-f4 or b4-d3-f4, two possibilities to the same square. Or Half-Rose at b4 goes b4-c6-e7-g6-h4, or b4-c2-e1-g2-h4, again two ways. Rose is two-path to all its squares and HALF-ROSE is two-path to its (0,5) and (0,7) squares.
Another multipath piece, Carlos Cetina's SISSA moves any number as Bishop followed by the same number as Rook, or vice versa. Therefore, Sissa has at least two pathways to each destination square on any rectangular board; any or all may be blocked. More specifically, Sissa moves to squares of Rook and also those of Nightrider. However, Sissa does not leap like NN or slide like Rook. It does not move like they do whilst having the same arrival squares. Sissa moves to each 'N-Rider square' by two unique paths and 'Rook square' by four(4) routes. Four pathways may be cut to two by proximity to edge. Moreover, intervening pieces may reduce any four-fold way to 3-, 2-, or one, or prohibit move altogether; and any two-fold way to 1 or 0. In second drawing (both Sissa and Coherent Chess) Sissa at c3 captures Queen at e7 by its 'modified Nightrider route' c3-d4-e5-e6-e7. The other pathway c3-c4-c5-d6-e7 is blocked by Rook at c5, but as two-fold pathway square, one path is sufficient to complete move (and capture). For emphasis, Sissa goes to N-Rider squares by two-fold way, and Rook squares by four-fold way. Also making the same distinction is the angled change of direction. When it turns 45 degrees, Sissa goes to N-Rider square; when 135 degrees, Rook square. In same drawing, Sissa on c3 may capture Rook on c5 by four-fold pathways c3-d3-e3-d4-c5, or c3-d4-c5-d5-c5, or c3-b3-a3-b4-c5, or c3-b4-a5-b5-c5. No pieces intervene, but if they did, any one pathway would be sufficient to move and capture c5 from c3.
Eric Greenwood's Renaissance Chess here and in Pritchard's ECV has two relevant pieces. CAVALIER moves "one square diagonally and any number straight" or vice versa--the Gryphon from 13th Century extended to become multipath. DUKE goes "one straight and any number diagonally" or vice versa, the logical counterpart to Cavalier. Both Cavalier and Duke then have a two-fold way to attainable squares. When blocked by either colour, a move may yet be valid by the other pathway.
Tim Stiles invented some doubly bent riders that qualify. Wolf's squares not on Rook's path are more or less conventionally one-way. Wolf's squares along Rook's path are fittingly two-way. Insofar as not blocked, Wolf reaches any possible Rook square beyond Dabbabah locations by two alternatives whilst not following Rook's pathway itself. Although Cetina's Sissa unimpeded moves (four-way) to any Rook square, the three pieces (Rook, Wolf, Sissa) all utilize entirely different pathways to the same 'Rook squares', Wolf and Sissa belonging to our new species following plural trips.
Edgar Rice Burroughs' Jetan has many pieces with multiple pathways. Jetan suffers from different interpretations of rules and discrepancy between text and appendix. Primarily using Larry Smith's descriptions in Jetan article, six of eight Jetan piece-types have multiple paths in both sets of rules in 'Chessmen of Mars', text and appendix, giving latter precedence as Burroughs' intended revision. In appendix WARRIOR moves two squares orthogonally with 90-degree change of direction allowed. So, Warrior's two-path squares are the diagonally adjacent ones, 'Ferz squares'. A second piece, PADWAR moves two diagonal in any combination of directions. Therefore, when Padwar 'turns' 90 degrees, it is going to a two-path square. Padwar multiple-, that is two-, path squares are 'Dabbabah squares'. Notice that Warrior's one-path squares and Padwar's two-way squares are the very same Dabbabah squares. Both types of arrival squares, one-way and two-way, can be blocked, but it takes at least two pieces (of either colour) to block a two-way one. Warrior and Padwar are two of the five multi-pathers in Jetan.
The rule Burroughs lays down for DWAR, moving three squares straight (orthogonal) in any direction or combination, brings forth three types of squares, the cases alternately one-way, two-way and three-way. Wazir squares are two-way, Knight squares three-way and (0,3) one-way. Next, THOAT moves one straight and one diagonal either order and any direction. Besides its two-way Wazir squares, of interest Thoat goes to 'Knight squares' two different ways, following the move definition.
Fifth, FLYER moves three squares diagonally in any combination of directions. If interpreted as non-jumping and not allowing doubling back, Flyer moves to Camel and Ferz squares by three-way and two-way alternatives respectively. However, Smith's and others' interpretation, following Burroughs' words to the letter and ignoring the superior implementation, is that Flyer cannot be impeded. That way Flyer is a jumper, or leaper, and does not belong in this listing.
Jetan's CHIEFTAIN, Smith's 'Chained Wild Chieftain', moves three squares in any combination and direction of both straight and diagonal steps. It reaches all the forty-nine squares within a seven-by-seven array of squares centered about a starting square. This three square movement deserves elaboration because there are multiple pathways. Each corner square can be reached only one way, by three diagonal steps in same direction. A square adjacent to the Chieftain's starting square can be reached twelve different ways, by various permitted combinations of straight and diagonal steps. Further (describing all the available squares in one sentence), Chieftain reaches from its starting square the various squares within surrounding 7x7 array by permitted combinations of its three-square move, by one way for corner squares, seven ways for the four outside squares in the straight directions, six ways for those at the opposite corner of a two by four and of a three by three rectangle of squares, three ways for the squares at the opposite corner of a three by four rectangle, twelve ways for the twenty remaining squares, and (if permitted) an additional twenty-four ways for the starting square itself. By inadvertence, one extreme form of multiple routes is Burroughs' Chieftain!
CV Page article 'Passed Pawns, Scorpions and Dragon' explains FALCON, SCORPION and DRAGON as respectively three-way, four-way and five-way multipath Movers. Because of overlap in the pathways, two pieces intervening may block 3-way Falcon, but it depends on their positioning. Three pieces are always sufficient to block F. Scorpion too may be blocked from moving to a particular square by just two intervening pieces, if properly positioned, but even three pieces (of either colour) improperly situated may be insufficient. Likewise, Dragon can be 'stopped' from making a particular move by as few as two pieces, but even four pieces do not necessarily forestall it: Dragon may still have one of its pathways interweaving among as many as four potential blockers.
Suggesting more than one path, Double Chess problems go back one millennium. Jetan's five or six piece-types multipath originate by 1920s. Novo Chess' Motor Unit comes out of 1930s, Billiards dates from the next decade, Elbows Chess a 1970's invention. With roots in 800-year-old Gryphon, Renaissance's Cavalier and Duke appear about 1980. Falcon is documented from 1992, followed by Scorpion and Dragon in year 2000. Sissa, the half-Rook half-Bishop, hails from 1998, about the time Betza experiments with Rose and Half-Rose. Beyond chronology, any rule of movement writ large (having real-world counterparts allegorically) describes something multiform and Multipath, whereof reduction to mere 'Leaper' or 'Rider' is actually the special case.