# Burnside's Lemma and Dominoes

A domino is a thin rectangular piece of wood with two adjacent squares on one side (the other side is black). Each square is either blank or has 1, 2, 3, 4, 5 or 6 dots.

Using Burnside's Lemma, show how many dominoes are there.

Workings:

I know that the formula for determining dominoes is $\frac{(n+1)(n+2)}{2}$.

So I would just plug $6$ into that to get $28$.

But I don't know how to use Burnside's Lemma to show this.

Any help will be appreciated.

Think about the following problem : on the set $X$ of pairs of integers between 0 and 6, you can consider a symmetry given by

$$s: (i,j)\mapsto (j,i)$$

It is clear that a domino is an orbit of $X$ for this action. So that, the number of dominoes $n$ is given (using Burnside) by :

$$n=\frac{1}{2}(|X|+|Fix(s)|)$$ $$n=\frac{1}{2}(7^2+7)$$

$$n=28$$

I used Burnside with the group generated by $s$ (only two elements) acting on X.

• Alright I see thanks – hockeynl Mar 22 '15 at 19:35