How Many Ways to Arrange Marbles in a 6x6 Grid Without Color Repetition?

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The problem of arranging 36 marbles in a 6x6 grid without color repetition involves using 6 colors with 6 marbles each. The key to solving this combinatorial challenge lies in recognizing that while switching two marbles of different colors disrupts the solution, permuting rows, columns, or colors maintains valid arrangements. The discussion highlights the complexity of finding unique solutions and suggests that multiple approaches exist, though some may be more tedious than others.

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gonzo
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Okay, say you have a 6x6 grid. You have 36 marbles, in 6 colors, with 6 of each color. You want to arrange the marbles on the grid so that no row or column contains two marbles of the same color.

How many ways are there to do this?
 
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I guess there are many approaches to this problem. Some more tedious than other.
I think the following will work:
First consider a particular solution. (Draw the grid with numbers 1 to 6 for example).

To what extend is this solution unique?
Clearly, switching any 2 marbles of different colors will no longer give a solution.
However, switching columns or rows will give different solutions, as will any permutation of the colors on the balls.
Are these the only possibilities?

(I`m not claiming this is the best way to do the problem, but it will work).
 

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