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 $x^2$. Jasper Jenkins [possibly not a real person], wishing to improve on this, told me in 1925 that he was $a^2 + b^2$ in $a^4 + b^4$; that he was $2m$ in the year $2m^2$; and that he was $3n$ years old in the year $3n^4$. Can you give the years in which De Morgan and Jenkins were respectively born?

(Source: 536 Curious Problems and Puzzles by Henry Ernest Dudeney)

Squaring away the year of birth for De Morgan is pretty easy. The maximum value possible for $x$ is the floor of the square root of 1871, which is 43. 43 squared is 1849, which seems like a reasonable value. This would mean that De Morgan was born in 1806, which is correct.

The second part seems harder because it contains information that is extraneous to getting the solution (though I suppose one might consider the extra statements "bonus rounds"). But really all that one need focus on is the last statement, that he was $3n$ years old in the year $3n^4$. In other words:

\[ y + 3n = 3n^4 \]

Or:

\[ y = 3n(n^3 - 1) \]

Where $y$ is the year of his birth. This one is the easiest to work with because there is only one variable and the cubic term blows up quickly, meaning only a few values of $n$ need to be tried. Reformulating a little, one has to try different values of the function:

\[ y(n) = 3n(n^3 - 1) \]

Starting at zero, these are:

\begin{align*}
y(0) &= 0 \\
y(1) &= 0 \\
y(2) &= 42 \\
y(3) &= 234 \\
y(4) &= 756 \\
y(5) &= 1860 \\
y(6) &= 3870
\end{align*}

Like I said, the cubic term has let us dash through Antiquity, jump over the Early Middle Ages and land right in the Victorian era before going almost two millennia into the future. (By comparison, the naive guess-and-check method used here, but with the second statement that involves only a squared term, requires thirty-two iterations, starting from zero, to get the right answer.) 1860 is indeed the right answer and it can be verified by noting that, in this instance, the third statement means that he was 15 ($3n$) in 1875 ($3n^4$).

Final answer: De Morgan was born in 1806; Jenkins, in 1860.

The Logic Question Almost Everyone Gets Wrong

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person? [Possible answers: yes, no, cannot be determined]

(Source: an article in The Guardian by Alex Bellos)

This is the sort of question like the one about the cost of a bat and a ball where a reflexive answer will tend to be wrong. Because Anne's marital status has not been given upfront, as it has with Jack and George, one might immediately suspect that it cannot be determined. But consider that "being married" is an obviously binary predicate. So look at both possibilities: if Anne is married, then a married person (Anne) is looking at an unmarried one (George) and if Anne is unmarried then a married person (Jack) is still nonetheless looking at an unmarried person (Anne). Anne's marital status is irrelevant, so the answer is yes.