Calculating Combinations in a Game of Matching Pieces

In summary: I think it would be more helpful if you could summarize the conversation for me.In summary, the conversation revolves around a game with a board and 9 pieces that can be rotated and placed in different positions. The goal is to match the pieces with their neighbors. The question is how many combinations of pieces exist for a possible solution. The answer is 9! x 49, taking into account both the position and rotation of the pieces. It is suggested to try a smaller board to better understand the concept.
  • #1
xeon123
90
0
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?
 

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  • #2
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...
 
  • #3
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?
 
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  • #4
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
 
  • #5
I think the answer is 49!. Is this right?
 
  • #6
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?
 
  • #7
Thanks,
 

FAQ: Calculating Combinations in a Game of Matching Pieces

How can I generate all possible combinations for a game?

To generate all possible combinations for a game, you can use a mathematical concept called permutations and combinations. You can also use a computer program or spreadsheet to help you generate the combinations. Simply list all the possible choices for each round of the game, and then use the formula nCr (n choose r) to calculate the total number of combinations.

What is the difference between permutations and combinations?

Permutations and combinations are both methods used to calculate the number of possible outcomes in a given situation. The main difference is that permutations take into account the order of the elements, while combinations do not. In other words, permutations are concerned with arranging a set of objects in a specific order, whereas combinations are concerned with selecting a subset of objects without considering the order.

Can I use a formula to calculate the number of combinations for a game?

Yes, you can use the formula nCr (n choose r) to calculate the number of combinations for a game. This formula is used when you want to select a subset of r objects from a total of n objects, without considering the order. The formula is nCr = n! / (r! * (n-r)!), where n! (n factorial) represents the number of ways to arrange n objects in a specific order.

Is there a limit to the number of combinations a game can have?

The number of combinations a game can have depends on the total number of choices available and the number of choices you are allowed to make in each round. Generally, the number of combinations can be very large, especially for games with a large number of choices and rounds. However, it is possible to limit the number of combinations by setting constraints or rules for the game.

How can I use the combinations to improve my chances of winning the game?

The combinations in a game represent all the possible outcomes, so there is no guaranteed way to use them to improve your chances of winning. However, understanding the total number of combinations and the probability of each combination occurring can help you make strategic decisions in the game. You can also use the combinations to analyze past games and identify patterns or strategies that may increase your chances of winning.

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