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?
Details and assumptions
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 to fill the squares of a 3x3 grid with non-negative integers such that the sum of each row and the sum of each column is 7, we can use a systematic approach:
Step 1: Start by considering the 3 rows and 3 columns separately.
Step 2: Since the sum of each row needs to be 7, and there are 3 rows in total, we can distribute the value of 7 among the rows. Since the numbers in each grid square can be non-negative, the possibilities are:
Row 1: 7, 0, 0
Row 2: 0, 7, 0
Row 3: 0, 0, 7
Row 1: 6, 1, 0
Row 2: 1, 6, 0
Row 3: 0, 0, 7
Row 1: 6, 0, 1
Row 2: 0, 6, 1
Row 3: 1, 0, 6
Row 1: 5, 2, 0
Row 2: 2, 5, 0
Row 3: 0, 0, 7
Row 1: 5, 1, 1
Row 2: 1, 5, 1
Row 3: 1, 1, 5
Row 1: 5, 0, 2
Row 2: 0, 5, 2
Row 3: 2, 0, 5
Row 1: 4, 3, 0
Row 2: 3, 4, 0
Row 3: 0, 0, 7
Row 1: 4, 2, 1
Row 2: 2, 4, 1
Row 3: 1, 1, 5
Row 1: 4, 1, 2
Row 2: 1, 4, 2
Row 3: 2, 1, 4
Row 1: 4, 0, 3
Row 2: 0, 4, 3
Row 3: 3, 0, 4
Step 3: Repeat the same process for the columns. At each step, distribute the remaining values of 7 among the columns in different ways.
Step 4: Once we have exhausted all possible distributions for both the rows and columns, count the distinct arrangements.
From the above list of possibilities, you can see that there are a total of 10 different ways to fill the squares of the 3x3 grid such that the sum of each row and the sum of each column is 7.
Note: Rotations and reflections are distinct arrangements, so be careful not to count them as duplicates.