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# Man and Beast 14: Oddly Oblique

MAB 03: From Ungulates Outward covered comprime oblique pieces in the square geometry. MAB 05: Punning by Numbers extended this theme to cubic boards. This article extends it to hex 2d boards and - as MAB 10: The Hybrid Diagonal does with coprime radial pieces - to 3d boards with one or more hex planes.

MAB 05 adds that few cubic pieces form the kind of pairs that square-board ones do, and that m+n:m:n leapers can triangulate on cubic boards just as a radial Ferz - the special case where m=1 and n=0 - can. It identifies the leap of the next m+n:m:n leaper, the Sexton with m=n=1, as two Ferz moves with a 60Â° turn in between - 1:1:0 and 1:0:1. Any m+n:m:n leap can be seen as two diagonal moves with a 60Â° turn in between - m:m:0 and n:0:n. Hex leaps can be seen as two orthogonal moves with a 60Â° turn in between, so m+n:m:n leapers do indeed have duals with half their Square of Leap Length (SOLL), but on the hex board, not their own!

The acknowledgments, overview, and glossary to the series can be found here. Established pieces have a link to their Piececlopedia page. The images used in this series, and indeed in most of my pages, can be found here.

## Pieces

The Ferz' dual is of course the Wazir, and the oblique Sexton's more surprisingly the radial Viceroy, but the Fortnight's dual is, like itself, oblique. Glinski's and McCooey's hexagonal Chess both use this piece, but neither inventor saw fit to devise a new name for it, both reusing Knight. This is unsatisfactory as it can coexist with square-board Knights on 3d hex boards, so in my hex variant Anglojewish Chess I named the hex piece SENNIGHT. In retrospect a name sharing Fortnight's first consonant and vowel might have been better, but the theme entrenched the name Sennight too deeply. As Sennight means the Jewish (and later Christian) 7-day week and Fortnight two such weeks, the names still reflect the pairing. Hex duals of the cubic Arbez, Sustainer, Votary, Endower, Ognimalf, Bandicoot, Pipistrelle, Hemel, and Beggar share their respective first consonant and vowel. The Arbez' dual shares the square-board Zebra's SOLL of 13 (hence Arbez itself) but can coexist with that piece in 3d and so requires its own name. I draw on ungulate names with AUROCHS, a wild ox of ancient Europe. The Sustainer's dual shares the cubic Underscore's SOLL of 19, but as the latter has a definition applicable on xyrixa boards, as featured in Mark Thompson's Tetrahedral Chess, and the pure-hex piece isn't it I name the Sustainer's dual STUDENT, as like many I began my own university studies aged 19. The Votary's dual, SOLL 21, reuses OVERSCORE as the cubic one genuinely cannot coexist with it. The Endower's dual has a prime SOLL with no rectilinear connotations. Such leapers draw on the board's suggestion of a honeycomb - for a SOLL of 31, the longest month length, NEWLYWED after one about to go on their honey-moon. The Ognimalf's dual sharing the Flamingo's SOLL of 37 (again, hence Ognimalf), and the Bandicoot's with its SOLL of 39 still between the Flamingo's 37 and Crane's 40, are the GOOSE (plural GEESE) and BARNOWL. The Pipistrelle's sharing the Fieldmouse's SOLL of 43 is PRIZEMOUSE. The Hemel's, whose SOLL of 49 echoes the square-board Antelope's 25, reuses the cubic name HEPTAGRAM. The Beggar's is BETTONG after a third kind of marsupial alongside the Numbat and Wombat sharing its SOLL of 57. Here are the movement diagrams for the Sennight and Aurochs:

As these duals took values of m up to 7 I decided to fill in the rest. Those of the unnamed 9:5:4 and 9:7:2 leapers share cubic pieces' SOLLs - the Nosrap's 61 and Guineapig's 67, but as I named two cubic pieces as if the duals of these the pure-hex pieces required new names. For the root-61 leaper I resorted to ENTRAP to end like Nosrap, and for the root-67 one GOPHER to maintain the Guineapig's rodent theme. The unnamed 11:6:5 cubic leaper's hex dual, with SOLL 91 - a multiple of 13 whose remainder mod 4 is the 3 impossible on a square-cell board - reuses Zebra's Ze- with ZEMINDAR, a kind of tax collector. For m=7, and n from 3 to 6, the hex leapers have SOLLs 79, 93 (3 times the Newlywed's 31), 109, and 127. The first and last have prime SOLLs with no previous associations and so draw on wax dummies and candles for the first and last with WAXWORKER and CHANDLER. The other two reuse the names of the cubic VACILLATOR and DRIBERYL. Finally as the piece with m=8, n=1 has a leap length less than 9 - in fact root-73, the same as the square-cell Huckster - I include this piece under the name HIGGLER as one similar in rÃ´le as well as name.

It is no accident that all these pieces have odd SOLLs, hence the article's subtitle. As the SOLL of a coprime m+n:m:n leaper is the sum of a multiple of 4 and two odd numbers, and so divides by 4 with remainder 2, half its SOLL is odd. On 2d boards the prime-SOLL pieces and Heptagram are unbound, and those with SOLLs divisible by 3 share the hex Unicorn's binding. All have 12 directions and can triangulate, although the prime-SOLL ones and the Heptagram drop to 8 and become switching when Averse to 1 in 3 cells. Although the prime SOLL and the one 3 times it divide by 4 with different remainders, pieces so paired cannot share a first consonant and vowel as their cubic duals' SOLLs share the remainder 2. These pieces also exist on hex-prism boards, where they are bound to a single hex plane and so weak, and on xyrixa ones, where they tend to be stronger.

Forward-only (FO) pieces I define as having the two most forward moves, flanking an orthogonal in the Glinski/McCooey orientation and a hex diagonal in the Wellisch one. This again reflects breaking oblique moves into radial steps. For the FO Sennight I coin the obvious SENHELM to go with Forthelm. The FO Aurochs, Student, Newlywed, Goose, Barnowl, and Prizemouse are the PAST, PUPIL, RING, GOSLING, GABLE, and PAMPER. The FO Overscore, Heptagram, and Vacillator again reuse ACCENT, OCCULT, and FASHION. The FO Bettong, Entrap, Gopher, Zemindar, and Waxworker are the GNAWER, PRISON, POGOER, TITHE, DUMMY, WICK, and DRAY. The last is a kind of cart as a higgler might use. Such pieces again also exist on xyrixa boards and hex-level - but not hex-ranked - hex-prism ones. Here are the movement diagrams for the Senhelm in the two hex orientations:

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My Nested Chess and offshoots view a square-cell board as alternate squares of a larger square-cell board so that unbound pieces disappear, pieces with the Bishop binding become their unbound duals, and so on. Examples include ...Cobbler-Elephant-Dabbaba-Ferz-Wazir and ...Crane-Charolais-Camel-Knight. Likewise FO sequences include ...Tusk-Dabchick-Cross-Point and ...Canard-Calf-Hump-Helm. Xyrixa boards likewise take alternate cells from a cubic board turning Sexton to Viceroy, Fortnight Sennight, Arbez Aurochs, Sustainer Student, Votary Overscore, Endower Newlywed, Ognimalf Goose, Bandicoot Barnowl, et cetera. The xyrixa board itself has no Fortnight, Arbez, Sustainer, Endower, Ognimalf, Bandicoot, or Pipistrelle as the cubic one has no piece with twice - and so none with half - the SOLL.

Nor does it end there. The cubic Bishop binding means that for any even-SOLL cubic piece, coprime or non-coprime, a xyrixa one has half its SOLL. Pairs of semi-duals expressed as a Rook and Bishop move at right angles remain recognisable. Such pieces correspond to the cubic Eunuch, Nimel, Lecturer, Effarig, Nhirolais, Nurturer, Lemming, Expounder, Loremaster, Wiremaster, Fherolais, Numming, Rootler, Flittermouse, Skimmer, Xemming, Worrier, Tinkerer, Lhorolais, Whirolais, Grumbler, Lumel, Rhorolais, Fimming, Simming, Whorolais, Nhurolais, and Gumming. They recognisably remain the SEXTON, NINJA, ELF, FENCER, NIMEL, UNDERSCORE, LECTURER, EXORCIST, LONGPLAYER, WIDEPLAYER, EFFARIG, NURTURER, ROTCER, FIELDMOUSE, SLIMMER, EXPOUNDER, WOMBAT, NUMBAT, LOREMASTER, WIREMASTER, GUINEAPIG, OPOSSUM, ROOTLER, FLITTERMOUSE, SKIMMER, WORRIER, TINKERER, and GRUMBLER. Likewise pieces corresponding to the unnamed 10:1:1, 10:3:3, 10:7:7, 12:1:1, 12:5:5, 12:7:7, 14:1:1, 14:3:3, and 14:5:5 leapers remain the FEASTER, BROADWAYMAN, LOUNGER, HAMSTER, UPSETTER, ELOPER, WHINGER, HOBBLER, and ROPER. To the ...Cobbler-Elephant-Dabbaba-Ferz-Wazir and ...Crane-Charolais-Camel-Knight chains this adds ...Srene-Eunuch-Sexton-Viceroy, ...Nrine-Nhirolais-Nimel-Ninja, ...Lrene-Lemming-Lecturer-Elf, ...Frene-Fherolais-Effarig-Fencer, ...Nrune-Numming-Nurturer-Underscore, ...Xrene-Xemming-Expounder-Exorcist, et cetera. It follows that this geometry retains the compounds NINTU, LEFANU, FEU, NUNCHAKU, XANADU, LOWENBRAU, WINTHRU, RONDEAU, FICHU, SHIATSU, WOUWOU, TINAMOU, and DEGU. MAB 05 definitions of Hypotenuse pieces also survive the transformation.

Arbitrary pairings can also be preserved. Thus the pieces corresponding to the cubic Dormouse, Vhorolais, Clinger, Onlooker, and Muddler can be the ODDFELLOW, VOTARY, CHIPMUNK, NOSRAP, and MUSKRAT. This adds chains such as ...Vrone-Vhorolais-Votary-Overscore and the compounds VOUDOU, DROPTHRU, CISEAU, NOSFERATU, and MILIEU. As a genuine dual the piece corresponding to the cubic Hherolais remains the HEMEL, and its Hepatgram compound the EHEU. Likewise their FO versions, with CHIPFRYER+GRASP=CHIPPAN still working.

Xyrixa boards have no Rytas, Monk, Dancer, Genome, Epoletna, Propounder, Avower, Viremaster, Octodont, Hogger, Ducker, or Trembler as cubic ones have no - or in some cases no spare - oblique leaper with twice the SOLL. Xyrixa pieces do however have half the SOLLs of the cubic Monk, Genome, Epoletna, Propounder, Rhyrolais, Avower, Viremaster, Octodont, Hogger, Dharolais, Ducker, and Trembler. These require new names preserving the first consonant and vowel. MOUNTIE is short for mounted policeman as a man on a horse fits thrice the Knight's SOLL. GERMINATOR is what pollinators such as bees effectively become indirectly. PENTAGRAM is to leap length 5 what Heptagram is to 7. OPPORTUNIST ends in -ist like Exorcist. RHYMER treats Y as a vowel on the Rhyrolais/Ryrdinal pattern. VAMPIRE completes the Endower/Newlywed/Avower quartet. VIDEOPLAYER fits with other SOLL-33 -player pieces. CORMORANT is a bird in keeping with Flamingo and Goose for SOLL 37. HOUSEMOUSE is part of the "Mousehole Cavalry". DALLIER moves less energetically than a dancer. DRUDGE means a hard worker and that great civiliser of the English language Samuel Johnson modestly used it of himself. TESSELATOR refers to a honeycomb pattern. The FO pieces are MAPLE after the symbol of the nation most noted for its mounties, POLLINATOR, PALMIST after the branch of superstition reading 5-fingered palms, CHANCE, VERSE, FANG completing Rite/Fate/Ring, VIEW, CHOUGH after a smaller bird, HOLE, DELAY, CHORE, and REPEATER.

Many other unnamed colourbound cubic pieces have SOLLs less than 162 - and so have xyrixa duals with leap lengths less than 9. This is a logical point to draw the line, as MAB 03 and 05 cover all rectilinear coprime leapers up to that range. As the 9:2:1, 9:3:2, 9:4:1, and 9:5:2 and 10:3:1 leapers share the leap lengths of the Flittermouse and Pipistrelle, Trembler, Hemel, and Ducker their xyrixa duals share those of the Fieldmouse and Prizemouse, Tesselator, Heptagram, and Drudge. The latter pieces are thus WOODMOUSE as a third kind of mouse, DIVIDER as a synonym, SCEPTIC as one doubting occult symbols such as the Pentagram and Heptagram as well as its SOLL is the Sennight's leap length, and ASSISTANT and INTERN as other hard workers. The FO pieces are DOWELLER, SLICE, DOUBT, and HELP and SUPPORT. The xyrixa duals of the 9:4:3, 9:6:1, 11:4:1, 10:7:1, 9:8:3, and 12:3:1 leapers reuse MIDWIFE, HIGHWAYMAN, GERBIL, PETMOUSE, DUNNART, and SUGARGLIDER respectively after the cubic pieces sharing their own SOLLS, and for their FO versions BIRTH, HEIGHT, REGULAR, TEMPTER, DENUDER, and GUSHER. Those of 10:5:1 and 11:2:1 leapers have the previously unused SOLL of 63, so extend the rodent names: the first VOLE in connection with it being thrice the Overscore's SOLL and the second ZOKOR. Those of the 10:5:3 and 11:3:2, sharing the SOLL of the Guineapig and Gopher, use further rodent names: AGOUTI and MARMOT. The 11:4:1's dual is the BEORNING, after a people in Lord of the Rings noted for their honey cakes. As two other root-73 leapers have human-rÃ´le names the remaining pair, the duals of the 9:7:4 and 11:4:3 leapers, are the AUXILIARY and IRREGULAR. To go with the Petmouse, the 11:5:2 leaper's dual sharing its SOLL of 75 in the ANOMALURE, after a rodent superficially squirrel-like except for its "anomalous" tail. The 12:3:1 leaper's dual puns on the honeyed drink mead with MEDE, a kind of ancient Anatolian. The FO versions of these four are the BANK, BURROW, GATHERER, RAMIFIER, ROBE, ANCILLARY, INSURGENT, MOANER, and DACTYL.

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Hex-prism leapers whose hex coordinate is a hex diagonal I collectively term singly oblique. They have as many directions as hybrid-diagonal pieces: symmetric ones on all boards 12, Glinski-level FO ones 4, Wellisch-level FO ones 2, and hex-ranked FO ones 6. All are bound, to the same proportion of 1 out of their interhex coordinate hex planes as their hex coordinate is of a hex plane. Purely hex leaps can also be exopressed as an orthogonal and a hex diagonal move at right angles. Thus each singly-oblique leaper is a 90Â° rotation of a purely hex piece, and shares its SOLL. The way that orthogonals and NDs alternate between two of the four Dabbaba bindings means that only non-coprime hex pieces have multiple such rotations. An orthogonal and ND at right angles through a cell alternate between the cells's own colour and the same other Dabbaba binding as each other, but a different other Dabbaba binding from any other otrhognal or ND through it. This means that a piece whose leap can be expressed using two different otrhogonals must be to the Dabbaba binding where it started and is therefore non-coprime. Thanks to Ben Reiniger, Joseph DiMuro, and JÃ¶rg Knappen for explaining this to me.

For a Viceroy hex coordinate the piece moving 1 hex plane is MAB 10's Rumbaba. The one moving 2, a Sennight rotation, shares the Fortnight's initials with FOAL. The one moving 3, a Eunuch rotation, is PREFECT, a national government official for a local area, simply as a previously unused title of authority. The one moving 4, a Student rotation, reuses UNDERSCORE as it genuinely cannot coexist with cubic/xyrixa ones. The weak piece moving 5, a rotation of the piece with twice the Sennight's leap, is the SPECTRE. The FO pieces are TRAINER, RUMKEY, CEDILLA, and HAUNT.

When the hex coordinate's SOLL divides exactly by 4, only pieces moving odd numbers of planes are coprime. For a Eunuch hex coordinate the piece moving 1 hex plane is a Aurochs rotation, but I drop the ungulate theme as it is 3d-specific and take 13 as a third of the Barnowl's 39 to give the BAKER whose dozen comprises 12 large loaves and a small one. Lewis Carroll's Hunting of the Snark has a Baker. The one moving 3, an Overscore rotation with thrice the Foal's SOLL, is the OFFSCORE. The one moving 5, a Goose rotation, reuses CORMORANT. For a Corpse hex coordinate the pieces moving 1, 3, and 5 hex planes - respectively Heptagram, Bettong, and Higgler rotations - reuse SCEPTIC, NUMBAT, and HAMSTER. The FO pieces are OVEN, HYPHEN, CHOUGH, DOUBT, MUNCHER, and FRAY.

For a Zombie hex coordinate I start with other monsters, a GOLEM moving 1 hex plane and reusing VAMPIRE for the one moving 2, but break the sequence with the one moving 3, which is non-coprime. The one moving 4, a Prizemouse rotation, reuses FIELDMOUSE. The one moving 5, a rotation of the piece with twice the Aurochs' leap, is the RATIFIER. The FO pieces are SLATE, FANG, MEADOW, and TREATY.

For a Skeleton hex coordinate the piece moving 1 hex plane puns that by terming it SCULLION, meaning a servant working in a scullery. The one moving 2, a Waxworker rotation, reuses MEDE. The one moving 3, a rotation of the piece with twice the Overscore's leap, is the VOCALIST. The one moving 4, whose SOLL of 91 is a multiple of the Aurochs' 13 like the Arbez' 26, is the RAKER. The FO pieces are DISHCLOTH, DACTYL, VOICE, and TINE.

Hybrid-diagonal and/or singly-oblique SOLLs occur in ratios of 3, like pure-hex ones. Pairs include Rumbaba/Prefect, Foal/Offscore, Underscore/Numbat, and Spectre/Vocalist. Those of the Baker, Golem, Cormorant and beyond are off the scale.

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Hex-prism leapers whose hex coordinate is itself oblique I collectively term doubly oblique. They have twice as many directions as pure-hex pieces: symmetric ones on all boards 24, hex-level FO ones 4, and hex-ranked FO ones 12. Those with an unbound hex coordinate are unbound if moving one hex plane, bound to all of alternate hex planes if moving 2, et cetera. Those with a multiple-of-3 hex coordinate are bound to third of whichever hex planes they can reach.

For a Sennight hex coordinate the piece moving 1 hex plane and sharing the Elephant's SOLL of 8 is a HEFFALUMP after an elephant-like creature in the children's fiction of Alan Alexander Milne. The one moving 2 reuses ELF. For the one moving 3 has twice the Dabbaba's leap length I coin SABBABA, a pun on the Jewish week having a Sabbath. The one moving 4 reuses GERMINATOR. For the weak piece moving 5 moves twice as far as the Elephant and Heffalump I coin MINORLUMP. The FO pieces I term RUMTUSK, CLOAK, SABCHICK, SEED, and SABTUSK. Using -LUMP and -BABA for pieces whose SOLLs are odd and even powers of 2, and -TUSK and -CHICK for their FO versions, continues throughout the article.

Pieces with an Aurochs hex coordinate reuse the n:3:2 cubic leaper's name for each piece moving n hex planes: FORTNIGHT, FENCER, LECTURER, RYTAS, SUSTAINER. Likewise the 6:n:1 leaper's name for a Goose hex coordinate: NURTURER, DANCER, GENOME, MIDWIFE, ENDOWER. Reusing the 6:5:n leaper's name for an Entrap hex coordinate is not viable for n=1 as this would duplicate Endower so I substitute AVOWER, but it is for the rest, which reuse MOLERAT, DORMOUSE, DUNNART, and FLITTERMOUSE. FO-piece names are likewise FORTHELM, FOIL, LESSON, ENIV, SUBORDINATE, RUNAROUND, GRACE, CHROMOSOME, BIRTH, FATE, RITE, LOOMER, SLEEP, DENUDER, and ARIA.

For a Student hex coordinate the pieces moving 1 hex plane, with twice the Camel's SOLL, is the CALIBRATOR for one marking a score. Those moving 2, 3, and 5 share the Germinator's, Spectre's, and Lemming's SOLLs respectively and so are the HYBRIDISER, WRAITH, and - to maintain the rodent theme - TUCOTUCO. The one moving 4 reuses ODDFELLOW. The FO pieces are MARKER, GRAFT, SHADE, CUTLER, and QUIRK.

For an Overscore hex coordinate the piece moving 1 hex plane and sharing the Lecturer's SOLL, half the Tucotuco's, is TUTOR. Those moving 2 and 3 reuse PENTAGRAM and MONK. The one moving 4, a fourth with SOLL 37, has another bird name, GULL. As the one moving 5 has SOLL 46 HYLONOME (see notes for pronunciation) completes the Germinator-GenomeHybridiser quartet. The FO pieces are RECAP, SHILLING, FRIAR, GUILLEMOT, and CYLLARUS.

For a hex coordinate of twice the Sennight's leap, the piece moving 1 hex plane shares the Satyr's and Rytas' SOLL of 29, so I pun on satire with TRAVESTY. The one moving 3, a fifth with SOLL 37, has another bird name, BUSTARD. The one moving 5 and sharing the Stork's and, more significantly, Midwife's SOLL of 53 is WETNURSE. The FO pieces are DRAG, BUDGIE, and WETRUN.

For a Newlywed hex coordinate the piece moving 1 hex plane, and sharing the Minorlump's SOLL but much stronger, is the FURTHERLUMP. The one moving 2, and sharing the Oddfellow's SOLL, is the FANATIC. The one moving 3, and sharing the Crane's SOLL of 40, follows Barnowl with SNOWYOWL. The one moving 4 reuses TESSELATOR. For the one moving 5 its SOLL of 56, slightly over the Midwife's 53 and four times the Offscore's 21, inspires the name FOSTERER. The FO pieces are WEDTUSK, RAGE, PALE, REPEATER, and WARD.

For a Barnowl hex coordinate the piece moving 1 hex plane, a third with SOLL 40, has another bird name, TAWNYOWL. The one moving 2 reuses WOODMOUSE. The one moving 3 planes, with twice the Prefect's SOLL, is MONITOR as a synonym for a lesser kind of Prefect. The one moving 4 reuses DRUDGE. As the one moving 5 moves has SOLL 64, the cube of the Dabbaba's, I coin CUBBABA. The FO pieces are STAIN, DOWELLER, SABKEY, CHORE, and CUBCHICK.

For a Heptagram hex coordinate the pieces share square-board 7:n leapers' SOLLs. For those moving 1 and 5 hex planes reusing the reversed names EPOLETNA and OGNIMALF makes sense, and for the FO ones ETSAH and TELLAM. The one moving 2 shares the Midwife's and Wetnurse's SOLL of 53 and so is DRYNURSE. The one moving 3 reuses RHYMER. The one moving 4, and sharing the SOLL of 65 with the Ox/Bittern/Molerat, echoes Molerat with the rodent name HYSTRIX (a kind of porcupine). The FO pieces are DRYRUN, VERSE, and QUILL.

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Where a geometry reuses the names of both a cubic compound's components, reusing the compound's name also makes sense. Thus xyrixa boards inherit Knight+Sexton=CHURCHWARDEN, Knight+Ninja=SAMURAI, Sexton+Ninja=OBERON, Sexton+Camel=TREASURE, Ninja+Camel=MERCHANT, and their FO versions and duals. Both 3d geometries inherit Knight+Elf=ELAND, Camel+Elf=GENIE, and theirs. Many of the pieces in my later article MAB 18: Complex Pieces with Simple Ratios apply to multiple 3d geometries. Hex geometries also generate dual or dual-named compounds. This relegates the rule of Co- compounds having the longest leap or longest coordinate to where both are rectilinear. Otherwise the hex-specific compound takes the prefix. Hence Foal plus Overscore is COROUELLE, Aurochs plus Fencer CONEDRAWEMAG, Knight/Camel/Aurochs/Fencer plus Pentagram COGOUM/COSLEIPNIR/COTUODNES/COELLESIG, Foal plus Heptagram CODACORUM, and Aurochs plus Zoetrope/Zomel COARBORETUM/COABATTOIR. Likewise with FO pieces. The first oblique compound specifically devised for hex use is Knight+Sennight=STALWART in a withdrawn variant. Two more recent relative one-offs, specific to hex-prism boards, are the Sennight+Heffalump and Senhelm+Rumtusk compounds used here. For the former I devised the name SHERIFF as starting with its components' initials followed by the shared first vowel. Fot the latter I use resort to RUSHOFF, from having the same consonants in a different order and starting with the same letters as Rumtusk. Additionally, they both have endings previously unused for compounds of oblique leapers. For this reason I chose an image for the Heffalump that could be marked to indicate compoumnding with the Sennight, and fitted images for the FO pieces around that. Here are my Heffalump, Sheriff, Rumtusk, and Rushoff images:

The cubic Multi-angle leapers named in MAB 07: When Beasts Collide have hex duals. If an angle between diagonal coordinates can be 60Â°, 90Â°, or 120Â° on a cubic board, so can that between orthogonal coordinates on either 3d hex geometry. The Cascade's dual, Knight+Viceroy+Sennight, is CANTERER and the Cataract's, Camel+Sennight+Aurochs, CAPERER to reflect their range of leaping moves. They extrapolate as usual to Zebra+Sennight+Student=ZENTERER, Giraffe+Aurochs+Overscore=GINTERER, Antelope+Aurochs+Goose=NANTERER, Zemel+Overscore+Newlywed=ZEPERER, Satyr+Student+Barnowl=SANTERER, Gimel+Underscore+Heptagram=GIPERER, et cetera. On hex-prism boards these examples, and as far as I can see all such compounds based on a coprime square-board piece, are unbound as they can reach any hex board by their square move and any interhex orthogonal by their hex move. Each piece is the square-board piece's compound with its Wellisch and Glinski/McCooey analogues.

The "Double-strength" subvariant of Honeycomb Chess inspires compounds of two modifications of a square-cell leaper. Substituting the hex diagonal for the hex orthogonal transforms the square-cell leaper into two singly-oblique leapers depending which coordinate is hex and which interhex. These can be seen as rotations analogous to the Unrook pieces of MAB 12: Alternative Fronts, although hex-ranked FO pieces are rotated MAB 12 Superhelms, Superhumps, Superstripes, et cetera. These take the initial of the FO square-cell piece and a single set of endings as no simple hex-prism is a component of more than one. As usual the Knight and Camel, via the H of Helm and Hump, give the real words. The rotated Knight is Foal+Baker=HAYMAKER, after a provider of food to horses as bakers are to humans, and the rotated Camel Prefect+Golem=HANDLER. The rotated Superhelm is Trainer+Oven HILLOCK and the rotated Superhump Rumkey+Slate=HUMMOCK, after kinds of bulges. Zebra extrapolations are Offscore+Vampire=SAYMAKER, Spectre+Scullion=SANDLER, Hyphen+Fang=SILLOCK, and Haunt+Dishcloth=SUMMOCK. Giraffe ones are Underscore+Sceptic=BAYMAKER, Ratifier+Vocalist=BANDLER, Cedilla+Doubt=BILLOCK, and Echo+Voice=BUMMOCK. Antelope ones use further words starting with H: Fieldmouse+Numbat=HEDGER (one creating boundaries between fields), Meadow+Muncher=HEADLOCK. Within the range considered here the Satyr gives Cormorant+Mede=VAYMAKER and Chough+Dactyl=VILLOCK, and the Rector Hamster+Raker=UAYMAKER and Fray+Tine=UILLOCK. Adding the square-cell piece itself gives a triple compound. Initially I tried one-off names for the Knight and Camel so treated but it is simpler to use my original suggestion for the Knight compound - Breeder - as a suffix covering an open-ended number of them. Thus NIGHTBREEDER is Knight+Haymaker, CAMELBREEDER Camel+Handler, ZEBRABREEDER Zebra+Saymaker, GIRAFBREEDER Giraffe+Baymaker, et cetera.

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Long-range pieces usually add the suffix -rider to the first five letters as usual. Examples include SENNIRIDER, AUROCRIDER, NEWLYRIDER, GOOSERIDER, SENHERIDER, PASTRIDER, FOALRIDER, PREFERIDER, GERMIRIDER, TRAINRIDER, MOUNTRIDER, and MAPLERIDER. The Trainrider is unambiguously the rider of the Trainer as the Train listed in MAB 21: Lords High Everything-Else is not a conventional piece. Long-range compounds include STALWRIDER, CANTERIDER, CAPERRIDER, HAYMARIDER, HANDLRIDER, HILLORIDER. Duals prefixed with Co- are allowed five letters between that and the suffix, and long-range versions of breeders keep the -breed- bit so that for example NIGHTBREEDRIDER is distinct from the ordinary Nightrider.

The Haymaker family follow the usual rules of part-symmetric and mixed-range pieces so that Foal+Oven=FOAOCK, Baker+Trainer=BAKOCK, Prefect+Slate=PREOCK, Golem+Rumkey=GOLOCK, Foalrider+Baker=FOALRER, Ovenrider+Trainer=OVENROCK, Preferider+Slate=PREFROCK, et cetera.

Leaping versions of MAB 09's hex SIMURGH and FARRIER are sufficiently long-range not to require riders. These are respectively Sennight+Aurochs+Overscore+Newlywed+... and Sennight+Student+Goose+Entrap+... - different components from the leaping versions of the cubic pieces. Leaping forms of CELEBRANT and NAG have two hex versions. Xyrixa ones match the components of cubic ones but hex-prism ones are Heffalump+Fencer+Monk+Tesselator+... and Heffalump+Calibrator+Nurturer+Avower+... instead.

## Notes

Non-coprime hex-specific pieces have their own article, MAB 17: Hex Heavies. Pieces with a radial and an oblique component, one or both forward-only or Curved or Crooked or hex-specific, are covered in MAB 19: The Vice Squad. The most complex hex-specific pieces have their own article, MAB 20: Far From Square.

Hylonome is a pair of homographs. With the odd syllables stressed and a silent E it is synonymous with hylonomist, a woodland expert. With the even syllables stressed, so as to be a homophone of hylonomy, it is the name of a female mythical Centaur, partner of Cyllarus. I chose the name over the less ambiguous hylonomist partly to echo the Genome sharing its SOLL, and partly to generate the name for the forward-only piece.

I would like to be able to claim Pentagram as a reason for the cubic Heptagram all those years ago but it is sheer chance that this worked out as it did. Xyrixa pieces were not even in my mind when I devised the name Heptagram.

As the mythical character Ganymede is an Anatolian that's probably what the -mede of his name means. On the other hand the -mede of Runnymede, the setting of my early themed variant Magna Carta Chess, is the mead of meadow. On the viscosity or otherwise of the drink mead I cannot comment!

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By Charles Gilman.

Last revised by Ben Reiniger.

Web page created: 2008-05-03. Web page last updated: 2024-01-22﻿