In the top row of an 8*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 every other square that doesn't have a number yet, 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?

So here is the 8x8 chessboard with the given info filled in

1 2 4 8 16 32 64 128
3
9
27
81
243
729
2187

Add up the first row, you get 255. (1+2+4+...+128)
If you multiply out the second row and add it up, it equals 765. (3+6+12+...+384)
Notice that 255*3=768
This pattern continues on for the other rows, and you can probably solve it from here

EDIT: 255*3=765

sorry typo :P

To find the sum of all the numbers on the chessboard, we need to calculate the value of each square and then add them all together.

First, let's calculate the numbers in the first row (top row). Tom writes the values 1, 2, 4, 8, 16, 32, 64, 128. These are already given and do not require any calculations.

Next, let's calculate the numbers in the first column (leftmost column). Tom writes the values 1, 3, 9, 27, 81, 243, 729, 2187. These are also given and do not require any calculations.

Now, we need to calculate the numbers in the other squares. Each square is the product of the number in its row (from the first row) and the number in its column (from the first column).

For example, the number in the second row and second column is the product of 2 (from the first row) and 3 (from the first column), which is 6. Similarly, the number in the third row and third column is the product of 4 and 9, which is 36. We need to perform this calculation for each square.

Here is the complete chessboard with the calculated values:

1 2 4 8 16 32 64 128
3 6 12 24 48 96 192 384
9 18 36 72 144 288 576 1152
27 54 108 216 432 864 1728 3456
81 162 324 648 1296 2592 5184 10368
243 486 972 1944 3888 7776 15552 31104
729 1458 2916 5832 11664 23328 46656 93312
2187 4374 8748 17496 34992 69984 139968 279936

Now, let's sum all the numbers on the chessboard:

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 + 9 + 18 + 36 + 72 + 144 + 288 + 576 + 1152 + 27 + 54 + 108 + 216 + 432 + 864 + 1728 + 3456 + 81 + 162 + 324 + 648 + 1296 + 2592 + 5184 + 10368 + 243 + 486 + 972 + 1944 + 3888 + 7776 + 15552 + 31104 + 729 + 1458 + 2916 + 5832 + 11664 + 23328 + 46656 + 93312 + 2187 + 4374 + 8748 + 17496 + 34992 + 69984 + 139968 + 279936

Calculating this sum will give us the final answer, the sum of all the numbers on the chessboard.

Do it yourself