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 $1 \times 1$ square, two $2 \times 2$, two $3 \times 3$, and the one $4 \times 4$.

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 $10 \times 10$ grid?

1 answer

To find the sum of all the numbers written in a 10×10 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 10×10 grid.

Count of Squares by Size

  1. 1×1 squares: There are 10×10=100 such squares.

  2. 2×2 squares: A 2×2 square can start anywhere from row 1 to row 9 and column 1 to column 9. Thus, the number of 2×2 squares is 9×9=81.

  3. 3×3 squares: These can start anywhere from row 1 to row 8 and column 1 to column 8, giving 8×8=64.

  4. 4×4 squares: These can start in rows 1 to 7 and columns 1 to 7, or 7×7=49.

  5. 5×5 squares: These can start in rows 1 to 6 and columns 1 to 6, or 6×6=36.

  6. 6×6 squares: These can start in rows 1 to 5 and columns 1 to 5, or 5×5=25.

  7. 7×7 squares: These can start in rows 1 to 4 and columns 1 to 4, or 4×4=16.

  8. 8×8 squares: These can start in rows 1 to 3 and columns 1 to 3, or 3×3=9.

  9. 9×9 squares: These can start in rows 1 to 2 and columns 1 to 2, or 2×2=4.

  10. 10×10 square: There is only one such square that covers the entire grid: 1×1=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 1×1 square covers 1 unit square. Therefore, total contribution from 1×1 squares: 1×100=100

  • Each 2×2 square covers 4 unit squares. Total contribution from 2×2 squares: 4×81=324

  • Each 3×3 square covers 9 unit squares. Total contribution from 3×3 squares: 9×64=576

  • Each 4×4 square covers 16 unit squares. Total contribution from 4×4 squares: 16×49=784

  • Each 5×5 square covers 25 unit squares. Total contribution from 5×5 squares: 25×36=900

  • Each 6×6 square covers 36 unit squares. Total contribution from 6×6 squares: 36×25=900

  • Each 7×7 square covers 49 unit squares. Total contribution from 7×7 squares: 49×16=784

  • Each 8×8 square covers 64 unit squares. Total contribution from 8×8 squares: 64×9=576

  • Each 9×9 square covers 81 unit squares. Total contribution from 9×9 squares: 81×4=324

  • Each 10×10 square covers 100 unit squares. Total contribution from 10×10 squares: 100×1=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 10×10 grid is 5368.

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