Question

Musketeering swordplay. At the royal fencing competition in France, the first four places were taken by Athos, Porthos, Aramis, and D'Artagnan. The sum of the places taken by Athos, Porthos, and D'Artagnan was 6; the sum of the places taken by Porthos and Aramis was also 6. What was the place taken by each of the musketeers if Porthos ranked higher than Athos?

Answer 1

Let:
\(p_1\)=position of Athos
\(p_2\)=position of Porthos
\(p_3\)=position of Aramis
\(p_4\)=position of D'Artagnan

Then we know:\[
\begin{align}
p_1+p_2+p_3+p_4&=10\tag{1}\\
p_1+p_2+p_4&=6\tag{2}\\
p_2+p_3&=6\tag{3}\\
p_2>p_1\tag{4}\\
\\
\text{(1)-(3) gives}\quad p_1+p_4&=4\tag{5}\\
\text{(2)-(5) gives}\quad p_2&=2\\
\text{(4) then gives}\quad p_1&=1\\
\text{(3) then gives}\quad p_3&=4\\
\text{(5) then gives}\quad p_4&=3\\
\end{align}\]
So final positions were:
1st: Athos
2nd: Porthos
3rd: D'Artagnan
4th: Aramis

Answer 2

Nice try, but incorrect.
" The sum of the places taken by Athos, Porthos, and D'Artagnan was 6"
this would not be possible if anyone of these people placed 4th, so that means Aramis must have gotten 4th.

" the sum of the places taken by Porthos and Aramis was also 6"
Knowing that Aramis placed 4th, this easily yields that Porthos took 2nd

"Porthos ranked higher than Athos"
From what we already know this can only mean Athos took third, leaving first place to D'Artagnan.

Your only mistake:
"Porthos ranked higher than Athos"
must be \[p_2<p_1\]

Answer 3

D'oh - of course - I misinterpreted "ranked higher"!