A special squarish age


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*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 has passed.  How many years would he have to wait until his age would have this property again?

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    Assuming that “this property” in the question refers to the ‘squarishness’ of age only (and not the number of years since last squarish age being also squarish), we have to wait 14 years, when he will be 56 (7 * 8). His current age is 42 (6 * 7), and the last time his age was squarish was when he was 30. (5 * 6), the number of years passed since then being 42 – 30= 12 which is again squarish (3 * 4). 

    The next time BOTH these conditions (i.e., age being squarish AND the number of years since last squarish age also being squarish) becomes true will be when he’s 110 yrs (if he lives till that age). 110 = 10 * 11, and the previous squarish age would be 90 (9 * 10), and the difference, 20 yrs (which also is squarish, being 4 *5)

    Viji_Pinarayi Expert Answered on 17th November 2015.

    As an ‘afterthought’, I’d add: The previous time both the conditions had come true was when he was 12, the previous time his age was squarish was when he was 6, and the number of years since then was again 6.

    (There are only 3 possible times this can happen in a human lifespan, and all the 3 are covered in the solution and this comment.)

    on 18th November 2015.
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