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Each unit square of the 4x4 grid below is to be filled in with the number 1, 2, 3 or 4, so that:

Each row contains the numbers 1, 2, 3, and 4 in some order,

Each column contains the numbers 1, 2, 3, and 4 in some order, and

Each  square (outlined in bold) contains the numbers 1, 2, 3, and 4.




Some of the squares have already been filled in. Find the number of ways of filling in the rest of the grid.

 May 23, 2023
 #1
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We can solve this problem by considering the different cases.

Case 1: The number 1 is in the top left corner.

In this case, the number 4 must be in the bottom left corner. The number 2 must be in the top right corner, and the number 3 must be in the bottom right corner. The remaining squares can be filled in in any order, so there are 3! = 6 ways to fill in the grid.

Case 2: The number 2 is in the top left corner.

In this case, the number 1 must be in the bottom left corner. The number 4 must be in the top right corner, and the number 3 must be in the bottom right corner. The remaining squares can be filled in in any order, so there are 3! = 6 ways to fill in the grid.

Case 3: The number 3 is in the top left corner.

In this case, the number 1 must be in the bottom left corner. The number 2 must be in the top right corner, and the number 4 must be in the bottom right corner. The remaining squares can be filled in in any order, so there are 3! = 6 ways to fill in the grid.

Case 4: The number 4 is in the top left corner.

In this case, the number 2 must be in the bottom left corner. The number 1 must be in the top right corner, and the number 3 must be in the bottom right corner. The remaining squares can be filled in in any order, so there are 3! = 6 ways to fill in the grid.

Therefore, the total number of ways to fill in the grid is 6 + 6 + 6 + 6 = 24.

 May 24, 2023

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