## March 2004

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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 wasxyears old in the yearx^{ 2}. Which year was he born in? Can anyone reading this blog say that he will bexyears old in the yearx^{ 2}? (Whole numbers only here, please.) How about other powers? Can anyone alive say: I shall bexyears old in the yearx^{ N}, for someNgreater than 2?

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*Solution*

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.