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20 votes
20 votes

A $1 \times 1$ chessboard has one square, a $2 \times 2$ chessboard has five squares. Continuing along this fashion, what is the number of squares on the regular $8 \times 8$ chessboard?

  1. $64$
  2. $65$
  3. $204$
  4. $144$
  5. $256$
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8 Answers

Best answer
29 votes
29 votes

Now lets see how many $7 \times 7$ squares are possible

 

These two patterns can shift to right as well as follows:

So, $7\times 7$ squares possible is $4$

Now lets see how many $6 \times 6$ squares are possible

 

So, $6\times 6$ squares possible is $9$

Now lets see how many $5 \times 5$ squares are possible:

$4$ vertical moves $\times  4$ horizontal moves $=4^2$ possibilities.

Proceeding like this,

  • $8\times 8$  squares possible $: 1\times 1=1$
  • $7\times 7$  squares possible $:2\times 2=4$
  • $6\times 6$  squares possible $:3\times 3=9$
  • $5\times 5$  squares possible $:4\times4=16$
  • $4\times 4$  squares possible $: 5\times5=25$
  • $3\times 3$  squares possible $: 6\times6 =36$
  • $2\times 2$  squares possible $: 7\times7 =49$
  • $1\times 1$  squares possible $:8\times8=64$

Total squares $\quad: 204$

Now we can generalize like with $n \times n$ chess board total squares $=1^2+2^2+3^2+\ldots +n^2 = \frac{n(n+1)(2n+1)}{6}$

Correct Answer: $C$

edited by
17 votes
17 votes
No. of squares on chessboard of $n\times n$ is equal to sum of squares of $n$ terms for $8\times 8$ chessboard,

$\begin{align} &=\frac{n\left(n+1\right)\left(2n+1\right)}{6} \\&=\frac{8\times 9\times 17}{6}\\&=204.\end{align}$
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4 votes
4 votes

Let T(n,n) be the number of squares in a n*n chess board.

T(1,1) = 1 

T(2,2) (i.e) number of squares in a 2*2 chessboard = number of squares in a 1*1 chessboard + 22

Similarly T(n,n) = T(n-1,n-1) + n2  if n>2 with base conditions T(1,1) = 1 and T(2,2) = 5

Solving, we will get T(8,8) = 204.

                             

edited by
3 votes
3 votes
By simple observation it will be-

1^2 + 2^2+ 3^2+ ...................+ n^2.

where n is the size given, in the question its 8.

sum of squares of numbers.
Answer:

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