In the top row of an $8 \times 8$ chessboard, Tom writes the values 1, 2, 4, 8, 16, 32, 64, 128. In the leftmost column, Tom writes the values 1, 3, 9, 27, 81, 243, 729, 2187. In each of the remaining empty squares, Tom writes the product of the leftmost number in that square's row and the topmost number in that square's column. What is the sum of all the numbers on the chessboard

The top row contains all powers of 2 from up to 0 to up to 7 in ascending order, whereas the leftmost column contains all powers of 3 from up to 0 to up to 7 in descending order.

Based on all the information given in statement, we determine that the product in each square of the chessboard ([tex]s(i, j)[/tex]) is represented by the following expression:

[tex]s(i, j) = 2^{i-1}\cdot 3^{j-1}[/tex] (1)

Where:

[tex]i[/tex] - Row index.
[tex]j[/tex] - Column index.

And the sum of all the numbers on the chessboard is determined by the following formula:

[tex]p = \Sigma\limits_{i=1}^{8}\Sigma\limits_{j=1}^{8} s(i,j)[/tex] (2)

Which can be found rapidly by using the following algorithm in Python:

sum = 0

for i in range(1, 9):

for j in range(1,9):

sum += ((2** (i-1))* (3** (j-1)))

print(sum)

After performing this algorithm, we found that the sum of all the numbers on the chessboard is 836400. [tex]\blacksquare[/tex]

To learn more on sum functions, we kindly invite to check this verified question: https://brainly.com/question/13013054

To learn more on algorithms, we kindly invite to check this verified question: https://brainly.com/question/22952967