Question

We number both the rows and the columns of an $8$ $\times$ $8$ chess-board with the numbers $1$ to $8$. A number of grains are placed onto each square, in such a way that the number of grains on a certain square equals the product of its row and column numbers. How many grains are there on the entire chessboard?

• A. $1296$
• B. $1096$
• C. $2490$
• D. $1156$

Answer 1

The correct option is A $1296$

In the first column, we have, successively,
$1\times 1,1\times 2,1\times 3,.....,1\times 8$ grains.
So, in the first column : $1\times (1+2+3+4+5+6+7+8)$
In the second column : $2\times (1+2+3+4+5+6+7+8)$
In the third column : $3\times (1+2+3+4+5+6+7+8).$
Finally, in the eighth column : $8\times (1+2+3+4+5+6+7+8)$
Since $1+2+3+4+5+6+7+8=\left(1/2\right)\times 8\times (1+8)=36$, there are ${36}^2=1296$ grains on the board.