Triangular Dominoes

In his patent, Richards used a three-digit notation, referring to the pips in clockwise order from the side(s) with the lowest value. Richards illustrated the tiles as two unique sets, with pip values subject to the following restrictions:^[1]

• Pip counts may be repeated
• Starting from the side with the lowest value and proceeding clockwise, the number of pips on a side is equal to or greater than the prior side

In addition to this marking scheme, Richards added the sum of all pips to the center of the tile.^[1]

Richards Triangular Domino sets

Triples	Doubles					Singles
000	001	002	003	004	005
111	011	112	113	114	115		012	013	014	015
222	022	122	223	224	225	023	024	025	123	124	125
333	033	133	233	334	335	034	035	134	135	234	235
444	044	144	244	344	445		045	145	245	345
555	055	155	255	355	455

Percy Alexander MacMahon showed there were 24 possible combinations when each of the three edges of an equilateral triangle are assigned one of four values, and showed the number of unique pieces that can be made in this way is ${\displaystyle {\frac {n}{3}}\cdot (n^{2}+2)}$ for ${\displaystyle n}$ unique values.^[2]: 2 For ${\displaystyle n=5}$, there are 45 unique combinations possible, and for ${\displaystyle n=6}$, there are 76 unique combinations; the reduced set of 35 and 56 in Triangular Dominoes, for 0–4 and 0–5 pips, respectively, result from the additional restriction for increasing values around each side of the tiles when counting clockwise. This can be demonstrated by examination of the "singles" tiles: where 012 is a valid sequence in Triangular Dominoes, 021 is not, and so the mirror image of each "singles" pattern is excluded; there are ten excluded patterns for the set of 0–4 pips and twenty for the set of 0–5 pips. By examination, mirror images of the triples and doubles are identical to the original tiles and so these patterns already adhere to the counting-up restriction.

• Selected included and excluded tiles
• 
  0-1-2, included
• 
  0-2-1, excluded (mirror image of 012)
• 
  0-2-4, included
• 
  0-4-2, excluded (mirror image of 024)
• 
  0-3-5, included
• 
  0-5-3, excluded (mirror image of 035)

These restrictions and resulting tile set of Triangular Dominoes were retained, with markings moved to the corners using Arabic numerals for Triominoes, which was published in 1965.

• Comparison of selected tiles in Triominoes and Triangular Dominoes
• 
  0-1-2 (Triangular Dominoes)
• 
  0-1-2 (Triominoes)
• 
  0-2-4 (Triangular Dominoes)
• 
  0-2-4 (Triominoes)
• 
  0-3-5 (Triangular Dominoes)
• 
  0-3-5 (Triominoes)