Calculating Combinations in a Game of Matching Pieces

[xeon123]

Hi,

Before ask my question, let me explain how this game works.

It exist a board with 9 positions and 9 pieces. Each position of the board is for a piece. A piece is composed by four symbols on each side. Each piece must match their neighbors, and each one can rotate. The goal of this game is to put all the pieces in some order, in a way that each piece matches with their neighbor.

An example of a final is in attachment.


I would like to calculate how many combinations of the pieces exist for a possible the solution?

 

[gsal]

I am not entirely sure what you are asking.

The board itself allows for 4 unique position for every one of the 9 pieces; so, I think there are a total of 4⁹ positions...now, how many of those are solution? Well, it depends on how the symbols have been drawn in the pieces...

 

[xeon123]

I was thinking differently.

I don't know how many solutions exists. But I would like to know how many combinations of pieces in the board exist? Is it 4^9^'? Since I've 9 places and 9 pieces, I've 9⁹ possible combinations. And each piece can rotate 4 times, I've 4^9^9. Am I correct?

 

[gsal]

Hhhmmm...maybe I misunderstood the problem...I thought the 9 pieces where fixed in place and their position could not be changed...that they could only be rotated...

If the 9 pieces can also be moved around, then, that a different story...

I got to work, now, so I won't look into it...but, if you are not sure about the solution, why don't you start an exercise with a smaller board? A board where you can actually manually count and know the solution and then see what the formula matches? Anyway, just a suggestion.

gotta go

 

[xeon123]

I think the answer is 4^9!. Is this right?

 

[gsal]

ok, I am back...have a few more minutes.

Here is the thing...the 9 pieces can be placed in the board in 9! different ways...do you agree with this? If you start filling up the board, you have the choice of 9 pieces for the first position, 8 choices for the second, 7 for the third and so on...and so, you can position the 9 pieces on the grid in 9! different ways...

The above accounts for the position of the pieces on the board...now, we need to consider rotation...and so, for every one of the 9! positions, we can have 4⁹ rotational orientations...

So, I think it might just be 9! x 4⁹

what do you think?

Like I said, how about considering a 2x2 board and doing the exercise by hand? Did you do that?

 

[xeon123]

Thanks,