ISI2018-MMA-2

Question

The number of squares in the following figure is

$$\begin{array}{|c|c|c|c|}\hline \text{} & & & \\\hline \hline \text{} & & & \\\hline \hline \text{} & & & \\\hline \hline \text{} & & & \\\hline  \end{array}$$

• A. $25$
• B. $26$
• C. $29$
• D. $30$

Comments

Here ans will be 30

Square of length 1 is 16

"          "     "        2. " 9

"          "     "       3  is  4

"          "     "       4.  " 1

Answer 1

There is a general formula for finding out the number of squares in a square = $\frac{n(n+1)(2n+1)}{6}$ where n represents the side of a $n*n$ square.

Derivation of the formula :
Square of length 1 is 16 (i.e. $4^2$ )

Square of length 2 is  9 (i.e. $3^2$)

Square of length 3 is  4 (i.e. $2^2$)

Square of length 4 is  1 (i.e.$1^2$).

So in general number of squares in a square of side n is $1^2$ + $2^2$ + $3^2$ +...... $n^2$ = $\frac{n(n+1)(2n+1)}{6}$
Hence by the formula we have $\frac{4.5.9}{6}$ = 30

Comments

is it necessary to  given figure must be square or whatever.

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If the figure is not a square, this formula will not be applicable since rectangle will have 2 sides of different sizes i.e. length and breadth