Count the number of unique permutations of this puzzle

Question

I have a puzzle that has 9 square slots. It looks like a rubik's cube face. In each square (except the middle one) you can put a coin in the middle.

The goal is to count the number of unique puzzles that are possible. A puzzle is unique by symmetrizing it with respect to the horizontal or vertical axis or by rotating it in the centre.

Here is an example of the same puzzle :

The second is the symmetric of the first one by the horizontal axis. The third is the symmetric of the first one by the vertical axis. The fourth is the first by rotating it 90° from the centre.

I'm a bit stuck. Do you have any ideas ?

Answer 1

Tell me if you think I am wrong.

Let $r$ correspond to a rotation of $90$° and $s$ be a reflection with respect to the vertical axis.
Let $G = (I,r,r^2,r^3,s,rs,r^2s,r^3s)$ the group of all actions (no more) to a puzzle ($I$ corresponds to the identity) and $X$ the set of all the possible puzzle.
Finally, let $F_{g} = \{x\in X| g\cdot x = x\}$ where $g\in G$.

According to Burnside's theorem, we have : $$|X/G| = \frac{1}{|G|}\sum_{g\in G} |F_g|$$

Now we have to calculate all the $F_g$'s. We have :

• $|F_I| = 2^8$
• $|F_r| = 2^2$
• $|F_{r^2}| = 2^4$
• $|F_{r^3}| = 2^2$
• $|F_s| = 2^5$
• $|F_{rs}| = 2^5$
• $|F_{r^2s}| = 2^5$
• $|F_{r^3s}| = 2^5$

So we get : $|X/G| = 8^{-1}*(2^8+2*2^2+2^4+4*2^5) = 51$

There is 51 puzzles possible.

Answer 2

ANSWER FOR 2 X 2 (See comments)

Note first that there are $2^4=16$ patterns since each square can contain a coin or not.

The square has 8 symmetries: 4 reflections and 4 rotations. The Orbit-Counting theorem says that you should first find how many patterns are fixed by each symmetry.

The identity symmetry This fixes all 16 patterns.

Reflection in a vertical line This fixes 4 patterns

And so on.

For the 8 symmetries I obtain 16,4,4,8,8,2,2,4.

You then find the average of these numbers - in this case $48/8=6$.

The theorem says this is the number you require.

Please ask if this brief explanation is not clear.