The squares of a 3×3 grid are filled with non-negative integers such that the sum of each row and the sum of each column is 7. How many different ways can the squares be filled? The numbers in each grid square does not need to be distinct. Rotations and reflections are distinct arrangements.
To find the number of different ways the squares can be filled, we can break this problem into smaller steps:
Step 1: Determine the possible range of values for the numbers in the grid squares.
Since the sum of each row and each column is 7, and there are 9 squares in total, the sum of all the numbers in the grid must be 63 (7 * 9). This means that the sum of the values in each square must also add up to 63.
Step 2: Determine the possible combinations of numbers that add up to 7.
Since the numbers in each row and column must add up to 7, we can list down all possible combinations of non-negative integers that add up to 7.
Here are the possible combinations:
- 0 + 0 + 7 = 7
- 0 + 1 + 6 = 7
- 0 + 2 + 5 = 7
- 0 + 3 + 4 = 7
- 1 + 1 + 5 = 7
- 1 + 2 + 4 = 7
- 1 + 3 + 3 = 7
- 2 + 2 + 3 = 7
Step 3: Determine the number of ways to fill the grid squares.
Since each square can have any non-negative integer, we can multiply the number of combinations from step 2 by the number of squares in the grid. In this case, since we have a 3x3 grid (9 squares), we need to consider the number of combinations raised to the power of 9.
In total, there are 8 possible combinations of numbers that add up to 7. Therefore, the answer to the question is 8^9 = 134,456 different ways the squares can be filled.