A Special Squarish Age

Source

I got this problem from Rustan Leino, who says Chuck Chan created it and Greg Nelson suggested introducing the name "squarish" to simplify the problem statement.

I solved it and wrote up my solution.

Problem    

Let's say that a number is squarish if it is the product of two consecutive numbers. For example, 6 is squarish, because it is $2 \times 3.$

A friend of mine at Microsoft recently had a birthday. He said his age is now squarish. Moreover, since the previous time his age was a squarish number, a squarish number of years have passed. How many years will he have to wait until his age will have this property again?

Solution     Reveal