Domino numerology


A great deal has been written by mathematicians and puzzlers on domino puzzles and the mathematics concerning dominoes. I will not try te reproduce all, merely to summarize some of the results.


Number of tiles
The number of tiles depends of course on te size of the domino set. The general formula is : (n+1)(n+2)/2, where n is the highest number on your set.

Thus in an ordinary double 6 set are 7*8/2 = 28 tiles.


Number of spots
The number of spots in a set is given by the formula : n(n+1)(n+2)/2.

Thus in your double 6 set there are 6*7*8/2 = 168 spots.


Average spots per tile

It is easy to derive from the above that the average number of spots on a tile is n. So the average number of spots on a tile in our double 6 set is 6.

Summary for other size sets
Set

size

 Number

of tiles

 Number

of spots

 Average

number

of spots

/ tile
6 28 168 6
9 55 495 9
12 91 1092 12
15 136 2040 15
18 190 3420 18

Interested visitors might find further information in:

1) Games and Puzzles issue 44, jan. 1976, Article 'The cohesive structure of dominoes' by Jaap Creutzberg. Published by Edu-Games.

2) Dominospelen and Dominopuzzels (in dutch), by K.W.H. Leeflang, Kosmos, Amsterdam, 1972.

Triominoes
Goliath enterprises has been publishing triominoes, a superform of dominoes, consisting of triangles with values 0 through 4 in each of the 3 corners. I suppose the name triominoes has been trademaerked by them.

Number of tiles
The number of tiles with highest value n is equal to 1+n(n+1). So the number of possible tiles with value n<=1 is 1+0*1 + 1+1*2 = 4.

n Number of tiles

with highest value

n = 1+n(n+1)

 Number of tiles

in set n

(all values)

 Number of spots

on tiles with highest

value n

 Total spots in set Average

spots / tile
0 1 1 0 0 0
1 3 4 6 6 1.5
2 7 11 27 33 3
3 13 24 76 109 4.58
4 21 45 162 271 6.02
5 31 76 ? ? ?
6 43 119 ? ? ?

A lot of unanswered questions here:

  1. What is the mathematical formula for the number of triominoes in a set of size n?
  2. What is the formula for the number of spots in such a set?
  3. What is the formula for the average number of spots per tile?

Mail me and I will publish your results (with your name) at this page.

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