De Morgan's saying: "I was x years old in the year x^2"

Question

Augustus de Morgan, when asked about his age, remarked:

I was $x$ years old in the year $x^2$.

He died in $1871$. Examining possible squares, we find $41^2 = 1681$, $42^2 = 1764$, $43^2 = 1849$, $44^2 = 1936$. Clearly, the only one which makes sense in this problem would be $43^2 = 1849$. So de Morgan was $43$ in $1849$. Therefore, he was born in $1806$.

Is there a systematic way to answer such a question without trial and error?

isn't this absurd, He said this when obviously he was alive so you shouldn't know when he will die and use that to find his age !? or I misunderstood something ? — Davood Karimi, Sep 29 '22 at 16:33
@DavoodKarimi A contemporary of de Morgan would only go up to the year he said this. But that information is not (readily) available to us. — okzoomer, Oct 03 '22 at 20:41

score 3 · Accepted Answer · answered Feb 10 '22 at 16:27

3

The year is $x^2$ and he died in 1871. Assuming he lived less than 100 years:

$1871-100<x^2<1871$

Taking the square root of both sides $42\le x\le44$ .

Still need trial and error, but there are only 3 candidates.

answered Feb 10 '22 at 16:27

TurlocTheRed

5,683

1

Thank you very much. Clever solution:-)! – Dimitris Feb 10 '22 at 16:29
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When you take the square root, you have to round the result towards the right direction: if he was born in 1771, then there's no way his mystery age could be 42, because $42^2 = 1764 < 1771$. Same thing for 44. So only 43 remains. – Charlie Vanaret Feb 10 '22 at 16:44

De Morgan's saying: "I was x years old in the year x^2"

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