In my March diary I posed the following brain-teaser.
Augustus De Morgan, the 19th-century English mathematician (whose name, as Martin Gardner pointed out, is an anagram of "O Gus, tug a mean surd!"), noticed that he was x years old in the year x 2. Which year was he born in? Can anyone reading this blog say that he will be x years old in the year x 2? (Whole numbers only here, please.) How about other powers? Can anyone alive say: I shall be x years old in the year x N, for some N greater than 2?
To avoid quibbles, I'm going to restrict the meaning of "is x years old in
year y" to
"celebrates x-th birthday in year y."
Then, if you were born in year N, you will be x years old in year N + x. We require this to be x 2, and so N, the year of your birth, is x 2 − x, for some whole number x in the range of normal human lifespans. Putting x equal to 42, 43, 44, 45, 46 gives birth years 1764, 1806, 1892, 1980, 2070. If you don't have one of those birth years, you're out of luck.
De Morgan was born in 1806, so he was 43 years old in 1849, which is the square of 43. Readers born in 1980 will enjoy the thrill of being 45 years old in 2025, which is the square of 45.
There is nothing much to be said about higher powers; though a person born in 2184 will be doubly blessed. He will attain age 3 in 2187, the seventh power of 3; and he will be 13 in 2197, the cube of 13.