Calculating Combinations in a Game of Matching Pieces

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Discussion Overview

The discussion revolves around calculating the number of combinations for placing and orienting pieces on a game board consisting of 9 positions, where each piece has symbols that must match with their neighbors. The scope includes combinatorial reasoning and potential mathematical formulations related to the game mechanics.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant describes the game mechanics, noting that each piece can rotate and must match symbols with neighboring pieces.
  • Another participant questions the initial request for clarification on whether the pieces can be moved or are fixed in place.
  • Some participants propose different mathematical approaches to calculate the combinations, including the use of factorials and powers, such as 9! for arrangements and 4^9 for rotations.
  • One participant suggests starting with a smaller board to manually verify the calculations and understand the problem better.
  • Another participant calculates the total combinations as 9! x 49, considering both the arrangement of pieces and their rotational orientations.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to calculating the combinations, with no consensus reached on a definitive formula. Some calculations and assumptions are challenged or refined throughout the discussion.

Contextual Notes

Participants have not resolved the assumptions regarding the fixed or movable nature of the pieces, nor have they agreed on the correct mathematical formulation for the combinations.

xeon123
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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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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...
 
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?
 
Last edited:
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
 
I think the answer is 49!. Is this right?
 
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?
 
Thanks,
 

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