What is the least number of children that can be arranged in rows of 12 16 or 18 in each now?

The question states that the boys can also be arranged in a solid square, meaning that 180 cannot be the answer, as its square root is not a whole number.

You need a number that contains the factors 12, 15, 18, and is also a perfect square of a whole number.

That would be 900, which is a multiple of the three factors and is the square of 30.

The LCM (least common multiple) is an important number/concept. It is easily found by finding the union of the sets of the prime factors of the numbers.

   12       {2, 2, 3}   15       {3, 5}

   18       {2, 3, 3}

Now find the smallest (least) superset that contains all three of these sets.      It is {2, 2, 3, 3, 5}

So, the LCM = 2 * 2 * 3 * 3 * 5 = 180

p.s., I don't know what Square Root has to do with this.