In the top row of a 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?

s u m o f t h e v a l u es in t h e t o p ro w : S = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 1 − 2 1 − 2 8 ​ = 2 8 − 1 = 255 t h e s u m o f a ll t h e n u mb ers o n t h e c h ess b o a r d : S ⋅ 1 + S ⋅ 3 + S ⋅ 9 + S ⋅ 27 + S ⋅ 81 + S ⋅ 243 + S ⋅ 729 + S ⋅ 2187 = = S ⋅ ( 1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187 ) = 255 ⋅ 1 − 3 1 − 3 8 ​ = = 255 ⋅ − 2 − 6560 ​ = 255 ⋅ 3280 = 836 , 400 A n s . t h e s u m o f a ll t h e n u mb ers o n t h e c h ess b o a r d i s 836 , 400

Answered by kate200468 | 2024-06-10

The sum of all the numbers on the chessboard is calculated to be 836400, by summing the products of the top row values (powers of 2) and left column values (powers of 3).
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Answered by kate200468 | 2024-12-24