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Charles Gilman wrote on Sat, May 29, 2010 07:07 AM UTC:

When I named the 2:2:1 leaper after a Japanese stealth warrior (or 'Nearly Invisible Nutter Jumping Around', as one Terry Pratchett fan calls them) all I had in mind was a pun on its SOLL of 9, extrapolating from the existing 2:1:1 Sexton's 6 and continued with my 3:1:1 Elf's 11. Little did I realise at the time how strong it was not just as a piece but as a concept. Where for a 2d oblique piece a pair of directions has one further pair of its directions at right angles, like standard (Bishop) diagonals, and for a typical cubic one it has none, like the nonstandard (Unicorn) diagonals, a pair of Ninja directions has two pairs, like orthogonals. Huge numbers of pieces with SOLLs divisible by 9 can therefore be expressed as two or three moves in Ninja directions, as all moves can from Wazir steps. Here is a table comparing cubic moves comprising Wazir steps, Trebuchet leaps, and Ninja leaps. Note how many pieces can be formed of the same combinations of Trebuchet and Ninja leaps. Note that cubic SOLLs include neither 7 nor 63. Note also that the Cassowary and Zhemois moves cannot be formed of Ninja ones at right angles because the zero coordinate cannot be achieved from a 6:6:3 leap and a 2:2:1 or 4:4:2 one as it can from the latter two.

1	1:0:0 Wazir	3:0:0 Trebuchet	2:2:1 Ninja
2	1:1:0 Ferz	3:3:0 Tripper	[2:1:2]+[1:2:-2] Tripper [2:2:-1]+[2:-1:2] Nimel
3	1:1:1 Viceroy	3:3:3 Zombie	[2:2:-1]+[2:-1:2]+[-1:2:2] Zombie [2:1:-2]+[2:-2:1]+[1:2:2] Exorcist
5	2:1:0 Knight	6:3:0 Chamois	2[2:2:1]+[2:-1:-2] Chamois 2[2:1:2]+[1:2:-2] Chipmunk
6	2:1:1 Sexton	6:3:3 Sessowary	2[1:2:2]+[2:1:-2]+[2:-2:1] Sessowary 2[2:2:-1]+[2:-1:2]+[-1:2:2] Expounder 2[2:1:2]+[1:2:-2]+[2:-2:-1] Propounder
9	3:0:0 Trebuchet 2:2:1 Ninja	9:0:0 Tritrebuchet 6:6:3 Nhimois	2[2:2:-1]+2[2:-1:2]+[1:-2:-2] Tritrebuchet 2[1:2:2]+[2:1:-2]+[2:-2:1] Nhimois 2[2:2:-1]+2[2:-1:2]+[-1:2:2] Opossum 2[2:1:2]+2[1:2:-2]+[2:-2:1] Ultimatum
10	3:1:0 Camel	9:3:0 Cassowary	3[2:1:2]+[1:2:-2] Clinger 3[2:2:1]+[2:-1:-2] Hogger
11	3:1:1 Elf	9:3:3 Letdown	3[2:2:1]+[2:-1:-1]+[1:-2:2] Letdown 3[2:2:-1]+[2:-1:2]+[-1:2:2] Loner 3[1:2:2]+[2:-2:1]+[2:1:-2] Whiner
13	3:2:0 Zebra	9:6:0 Zhemois	3[2:1:2]+2[1:2:-2] Bystander
14	3:2:1 Fortnight	9:6:3 Fossowary	3[2:2:-1]+2[2:-1:2]+[-1:2:2] Fossowary

Charles Gilman wrote on Mon, Jun 14, 2010 06:27 AM UTC:

Since stating that no cubic piece has a SOLL of either 7 or its multiple by
9, 63, I have noticed something further: no cubic piece has a SOLL one
short of a multiple of 8 full stop. Geometries with a hex component have a
monopoly over the Sennight and Foal (7), Mountie (15), Germinator and
Hybridiser (23), Newlywed and Vampire (31, perhaps the oddest juxtaposition
of same-SOLL piece names), Barnowl (39), Tesselator (47), Drudge (55), and
Mede (79). As orthogonal leapers' SOLLs always divide by 8 with remainder
0, 1, or 4 it follows that square-cell leapers' SOLLs generally always
divide by 8 whose remainder is a sum of two of those modulo 8: 0, 1, 2, 4,
or 5 - and likewise cubic ones a sum of three of them modulo 8: 0, 1, 2, 3,
4, 5, or 6.

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