## Teaser 2932: Triangulation

**From The Sunday Times, 2nd December 2018** [link]

Liam plans to make a set of dominoes. They will be triangular, and one face of each domino will have a number at each corner. The numbers run from 0 up to a maximum digit (9 or less), and the set is to include all possible distinguishable dominoes.

With the maximum digit he has chosen the set would contain a triangular number of dominoes. [A triangular number is one where that number of balls can fit snugly in an equilateral triangle, for example the 15 red balls on a snooker table].

How many dominoes will he need to make?

[teaser2932]

## Jim Randell 11:32 am

on20 March 2019 Permalink |For each maximum digit, this Python program constructs all possible dominoes that are unique by rotation. It runs in 75ms.

Run:[ @repl.it ]Solution:He will need to make 45 dominoes.There is a degenerate solution where only the digit 0 is used. There is only one domino (0, 0, 0), and 1 is a triangular number

T(1) = 1.Running the program we see the number dominoes is given by the following sequence:

which is OEIS A006527 [ link ] the general formula is:

so we don’t have to count the dominoes.

If we consider

d > 9there are larger solutions atd = 12(741 dominoes) andd = 35(15576 dominoes).LikeLike