This is the day 4 post of the challenge quiz week by 2IIM Online CAT Coaching :

Challenge Quiz Puzzle – Day 4:

Consider positive integer x less than 201. What is the probability that [√x] = [√(x+5)] ? [x] is the greatest integer less than or equal to x.

**Solution**

This one is much simpler than the others that we have seen. Let us start by doing some trial and error.

When x = 1 this does not work. As [√x] = 1 and [√(x+5)] = 2

When x = 2 this does not work. As [√x] = 1 and [√(x+5)] = 2

When x = 3 this does not work. As [√x] = 1 and [√(x+5)] = 2

When x = 4 this does not work. As [√x] = 2 and [√(x+5)] = 3

We can sense that this is going to have something to do with perfect squares. Between x and x + 5, we should not have a perfect square in between. That’s it. Beyond this, this question becomes very easy to solve.

When x = 5, 6, 7 or 8, this does not work. As [√x] = 2 and [√(x+5)] = 3

When x = 9, we have a breakthrough. [√x] = [√(x+5)] = 3

When x = 10, we have a breakthrough. [√x] = [√(x+5)] = 3

It does not work for 11, but now we can see a pattern. Pick a perfect square, say n^{2} subtract 6 from it. We have n^{2} – 6. When x is any value from (n-1)^{2} to n^{2} – 6, this should work.

So, n = 9, 10 work; n = 16, 17, 18, 19 will work. N = 25, 26, 27, 28, 29, 30 will work and so on.

So, number of numbers that will work will go as 2 + 4 + 6 + 8 +… and so on.

Let us think about the last few numbers. 196, 197, 198, 199 and 200 will work.

Before this, the numbers from 169 onwards will work till 190. This is a set of 22 numbers.

So, the total number of numbers that will work = 2 + 4 + 6 + 8 +……+ 22 + 5.

= 2( 1+ 2 + 3 + …11) + 5 = 132 + 5 = 137.

So, the probability is 137/200.