To find the sum of all the numbers written in a
Let's first determine how many squares of each size can fit in a
Count of Squares by Size
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squares: There are such squares. -
squares: A square can start anywhere from row 1 to row 9 and column 1 to column 9. Thus, the number of squares is . -
squares: These can start anywhere from row 1 to row 8 and column 1 to column 8, giving . -
squares: These can start in rows 1 to 7 and columns 1 to 7, or . -
squares: These can start in rows 1 to 6 and columns 1 to 6, or . -
squares: These can start in rows 1 to 5 and columns 1 to 5, or . -
squares: These can start in rows 1 to 4 and columns 1 to 4, or . -
squares: These can start in rows 1 to 3 and columns 1 to 3, or . -
squares: These can start in rows 1 to 2 and columns 1 to 2, or . -
square: There is only one such square that covers the entire grid: .
Calculate Coverage of Each Square Size
Now we need to compute how many times each unit square is covered by these different squares.
-
Each
square covers unit square. Therefore, total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares: -
Each
square covers unit squares. Total contribution from squares:
Total Contribution
Now, we sum all these contributions:
Calculating step by step:
Thus, the sum of all sixteen numbers written in the