Henry Ernest Dudeney/Modern Puzzles/23 - De Morgan and Another/Solution

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Modern Puzzles by Henry Ernest Dudeney: $23$

De Morgan and Another
Augustus de Morgan, the mathematician, who died in $1871$, used to boast that he was $x$ years old in the year $x^2$.
My living friend, Jasper Jenkins, wishing to improve on this, tells me he was $a^2 + b^2$ in $a^4 + b^4$;
that he was $2 m$ in the year $2 m^2$;
and that he was $3 n$ years old in the year $3 n^4$.
Can you give the years in which De Morgan and Jenkins were respectively born?


Solution

Augustus De Morgan was born in $1806$.

Jasper Jenkins was born in $1860$.


Proof

Note that Dudeney was writing this in the $1920$s.

The square numbers around the $18$th and $19$th century are:

$42^2 = 1764$
$43^2 = 1849$
$44^2 = 1936$

of which only $1849 = 43^2$ can plausibly fit the parameters for Augustus De Morgan.

Hence it is deduced that Augustus De Morgan was born on $1849 - 43 = 1806$.

When we inspect his page, we see that indeed he was born on $18$th June $1806$.


As for Jasper, we need to inspect square numbers around the $900$ to $1000$ region:

$2 \times 30^2 = 2 \times 900 = 1800$
$2 \times 31^2 = 2 \times 961 = 1922$
$2 \times 32^2 = 2 \times 1024 = 2048$

Clearly Jasper was $2 \times 31 = 62$ in $1922$.

Hence it appears Jasper was born in $1860$.

We check the $4$th powers over the range $600$ to $700$ and find:

$3 \times 5^3 = 3 \times 625 = 1875$

which corroborates the above: Jasper was $3 \times 5 = 15$ in $1875$.


Continuing to explore the $4$th powers, we have this list:

$1^4 = 1$
$2^4 = 16$
$3^4 = 81$
$4^4 = 256$
$5^4 = 625$
$6^4 = 1296$
$7^4 = 2401$

and we have gone high enough.

Inspecting these numbers, we have that:

$625 + 1296 = 1921$

at which time Jasper was $5^2 + 6^2 = 25 + 36 = 61$.

$\blacksquare$


Sources

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