mathschallenge.net

Problem

If you were 35 years old in the year 1225 it would be a very special time mathematically, because 35² = 1225. That is, the square of your age at that moment is equal to the year. This does not happen very often.

Augustus de Morgan, a famous mathematician, was one of those lucky people and in 1864 he wrote:

When was he born?

Solution

42²=1764
If he was 42 in 1764, he would have been born in 1764 42 = 1722. As he wrote it in 1864, he would have been 1864 1722 = 142 years old at the time!!!

43²=1849
If he was 43 in 1849, he would have been born in 1849 42 = 1806, making him 1864 1806 = 58 years old when he wrote the statement.

44²=1936 (not happened by 1864)
So Augustus De Morgan must have been born in 1806.

It is likely that you know somebody with the same special property with their age. When were they born and when is their special year?