, 2021

We systematically work through pairs of the given integers to see which pairs add up to a third given integer. Starting with the smallest possible pairs, we have: $$4 + 27 = 31 \qquad 12 + 15 = 27 \qquad 12 + 27 = 39$$ There are no other pairs that add up to a third given integer.

This means that each of the three sums in the problem must be one of the three sums above.

In the sums above, the only integer that appears three times is 27.

In the sums in the problem, the only variable that appears three times is $c$.

Therefore, $c=27$.

This also means that the sum $a+b=c$ must be the sum $12+15=27$.

Since 12 appears in two sums and 15 does not, then $a=15$ and $b=12$.

Matching the values that we know already with the equations that we have, we obtain $$\begin{aligned} a + b & = c & & & 15 + 12 & = 27 \\ b + c & = d & & & 12 + 27 & = 39 \\ c + e & = f & & & 27 + 4 & = 31\end{aligned}$$ Therefore, $a+c+f=15+27+31 = 73$.