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When Augustus de Morgan (a mathematician who was born and died in the 19th century) was asked about his age, he replied: “I was x years old in the year x².” What year was he born in?

Could such a strange lot have befallen someone who was born and died in the 20th century?

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    $\begingroup$ I enjoyed solving this problem IRL some years ago, to find the exact day when someone's age would be the square root of the current year. You never did say everything had to be an integer ... $\endgroup$
    – Rand al'Thor
    Nov 19, 2017 at 21:26
  • $\begingroup$ @Randal'Thor some other day $\endgroup$
    – prog_SAHIL
    Nov 20, 2017 at 3:03
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    $\begingroup$ For reference, the 19th century ran from 1 January 1801 (not 1800!) – 31 December 1900. $\endgroup$
    – smci
    Nov 20, 2017 at 3:03
  • $\begingroup$ Check this : puzzling.stackexchange.com/questions/41827/… $\endgroup$
    – Jamal Senjaya
    Nov 20, 2017 at 5:30
  • $\begingroup$ Would you allow 20th Century BC $\endgroup$
    – Tom
    Nov 20, 2017 at 11:15

2 Answers 2

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The only perfect square in the 19th century is:

1849

Which would make him

43 years old on that date, born in 1806

For the 20th century:

It is not possible.
The only perfect square is 1936.
However, that would make him 44 years old and born in 1892, which is in the 19th century.

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  1. When Augustus de Morgan (a mathematician who was born and died in the 19th century) was asked about his age, he replied: “I was x years old in the year x².” What year was he born in?

    This is equivalent to solving $x^2-x-N=0$ where $1800\leq N<N+x\leq1900$ - the notation being that $N$ is his birth year, making his death year at least $N+x=x^2$.

    The solution is $x=\frac{1+\sqrt{1+4N}}{2}$, which is an increasing function of $N$, so we can trivially bound $x$ between $42.929...$ (the solution with N=1800) and $43.588...$ (the square root of 1900).

    If x has to be an integer, then there is only one solution: the one found by Apep. But de Morgan was a mathematician, so I feel obliged to point out that there are actually many solutions. For example, he could be born on 1 January of the year $N=1800$ and be $x=42.918...$ years old on 1 December (the 335th day) of the year $x^2=1842$. So his birth year could actually be anything from

    1800 (the least possible) to 1857 (the year in which $N+x=x^2$ would be 1900).

  2. Could such a strange lot have befallen someone who was born and died in the 20th century?

    Here the problem is the same except with all the bounds increased by 100. As before, we find that

    the solution is $x=\frac{1+\sqrt{1+4N}}{2}$, so we can bound $x$ between $44.092...$ (the solution with N=1900) and $44.721...$ (the square root of 2000).

    This time, x cannot be an integer, but again there are a great many solutions. For example, someone could have been born on 1 January of the year $N=1950$ and be $x=44.654...$ years old on 26 August of the year $x^2=1994$. Their birth year could be anything from

    1900 to 1955, just like before.

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  • $\begingroup$ Next time I will design a puzzle on this ;) $\endgroup$
    – prog_SAHIL
    Nov 20, 2017 at 12:34

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