A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one $1 imes 1$ square, two $2 imes 2$, two $3 imes 3$, and the one $4 imes 4$.

Question

A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one $1 	imes 1$ square, two $2 	imes 2$, two $3 	imes 3$, and the one $4 	imes 4$.

What is the sum of all sixteen numbers written in this grid?
Solution:
Here is the completed grid:

[asy]
unitsize(0.6 cm);

for(int i = 0; i <= 4; ++i) {
  draw((i,0)--(i,4));
  draw((0,i)--(4,i));
}

label("$4$", (0.5,3.5));
label("$6$", (1.5,3.5));
label("$6$", (2.5,3.5));
label("$4$", (3.5,3.5));
label("$6$", (0.5,2.5));
label("$10$", (1.5,2.5));
label("$10$", (2.5,2.5));
label("$6$", (3.5,2.5));
label("$6$", (0.5,1.5));
label("$10$", (1.5,1.5));
label("$10$", (2.5,1.5));
label("$6$", (3.5,1.5));
label("$4$", (0.5,0.5));
label("$6$", (1.5,0.5));
label("$6$", (2.5,0.5));
label("$4$", (3.5,0.5));
[/asy]

$ 104
Vt Problem 12 Part (b)

What about the same problem with a $10 	imes 10$ grid?

GPT-4o mini · Answer

To find the sum of all the numbers written in a $10 \times 10$ grid, where each number represents the count of all squares of various sizes that cover a particular unit square, we can approach it systematically.

Let's first determine how many squares of each size can fit in a $10 \times 10$ grid.

Count of Squares by Size

$1 \times 1$ squares: There are $10 \times 10 = 100$ such squares.
$2 \times 2$ squares: A $2 \times 2$ square can start anywhere from row 1 to row 9 and column 1 to column 9. Thus, the number of $2 \times 2$ squares is $9 \times 9 = 81$.
$3 \times 3$ squares: These can start anywhere from row 1 to row 8 and column 1 to column 8, giving $8 \times 8 = 64$.
$4 \times 4$ squares: These can start in rows 1 to 7 and columns 1 to 7, or $7 \times 7 = 49$.
$5 \times 5$ squares: These can start in rows 1 to 6 and columns 1 to 6, or $6 \times 6 = 36$.
$6 \times 6$ squares: These can start in rows 1 to 5 and columns 1 to 5, or $5 \times 5 = 25$.
$7 \times 7$ squares: These can start in rows 1 to 4 and columns 1 to 4, or $4 \times 4 = 16$.
$8 \times 8$ squares: These can start in rows 1 to 3 and columns 1 to 3, or $3 \times 3 = 9$.
$9 \times 9$ squares: These can start in rows 1 to 2 and columns 1 to 2, or $2 \times 2 = 4$.
$10 \times 10$ square: There is only one such square that covers the entire grid: $1 \times 1 = 1$.

Calculate Coverage of Each Square Size

Now we need to compute how many times each unit square is covered by these different squares.

Each $1 \times 1$ square covers $1$ unit square. Therefore, total contribution from $1 \times 1$ squares: \[ 1 \times 100 = 100 \]
Each $2 \times 2$ square covers $4$ unit squares. Total contribution from $2 \times 2$ squares: \[ 4 \times 81 = 324 \]
Each $3 \times 3$ square covers $9$ unit squares. Total contribution from $3 \times 3$ squares: \[ 9 \times 64 = 576 \]
Each $4 \times 4$ square covers $16$ unit squares. Total contribution from $4 \times 4$ squares: \[ 16 \times 49 = 784 \]
Each $5 \times 5$ square covers $25$ unit squares. Total contribution from $5 \times 5$ squares: \[ 25 \times 36 = 900 \]
Each $6 \times 6$ square covers $36$ unit squares. Total contribution from $6 \times 6$ squares: \[ 36 \times 25 = 900 \]
Each $7 \times 7$ square covers $49$ unit squares. Total contribution from $7 \times 7$ squares: \[ 49 \times 16 = 784 \]
Each $8 \times 8$ square covers $64$ unit squares. Total contribution from $8 \times 8$ squares: \[ 64 \times 9 = 576 \]
Each $9 \times 9$ square covers $81$ unit squares. Total contribution from $9 \times 9$ squares: \[ 81 \times 4 = 324 \]
Each $10 \times 10$ square covers $100$ unit squares. Total contribution from $10 \times 10$ squares: \[ 100 \times 1 = 100 \]

Total Contribution

Now, we sum all these contributions: \[ 100 + 324 + 576 + 784 + 900 + 900 + 784 + 576 + 324 + 100 \]

Calculating step by step: \[ 100 + 324 = 424 \] \[ 424 + 576 = 1000 \] \[ 1000 + 784 = 1784 \] \[ 1784 + 900 = 2684 \] \[ 2684 + 900 = 3584 \] \[ 3584 + 784 = 4368 \] \[ 4368 + 576 = 4944 \] \[ 4944 + 324 = 5268 \] \[ 5268 + 100 = 5368 \]

Thus, the sum of all sixteen numbers written in the $10 \times 10$ grid is $\boxed{5368}$.