Class 11

Math

Co-ordinate Geometry

Straight Lines

We number both the rows and the columns of an $8$ $×$ $8$ 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?

- $1296$
- $1096$
- $2490$
- $1156$

$1×1,1×2,1×3,.....,1×8$ grains.

So, in the first column : $1×(1+2+3+4+5+6+7+8)$

In the second column : $2×(1+2+3+4+5+6+7+8)$

In the third column : $3×(1+2+3+4+5+6+7+8).$

Finally, in the eighth column : $8×(1+2+3+4+5+6+7+8)$

Since $1+2+3+4+5+6+7+8=(1/2)×8×(1+8)=36$, there are $36_{2}=1296$ grains on the board.