Question

You need to take $n$ arbitrary points on or inside a square of side $2cm$ that there will always be a pair of points at a distance of not more than $\sqrt{2}cm$. What is the minimum value of $n$?

• A. $2$
• B. $4$
• C. $5$
• D. $8$

Answer 1

Answer: (C)

$5$

Draw a square of side 2 cm.

Divide it into 4 parts and assign 4 points to each part such that the distance between each is less than $\sqrt{2}$ cm

There cant be more than a point in the same region as the longest separation would be

along the diagonal which is only $\sqrt{2}$ cm long.

Hence when we insert the 5th point we can not put it in any of the regions such that its distance wouldn't exceed $\sqrt{2}$ cm.

It is necessary for each region to contain a point after taking 4 points as explained above. The border case is only reached with four points at the corners and the 5th one at the centre of the square.