Puzzle: find the weight of each penguin 5 penguins are weighed on a scale in pairs. The zookeeper records the weight of each pair of penguins. They are 16, 18, 19, 20, 21, 22, 23, 24, 26, and 27 pounds. What is the weight of each penguin?
Since they are weighed in pairs, each penguin is weighed four times.
So the sum of the weights of the five penguins is
$$(16+18+19+20+21+22+23+24+26+27)/4=54$$
However, I don't know how to continue from there to find the weight of each penguin.
 A: Let the penguins' weights in pounds, in increasing order, be $w_1$, $w_2$, $w_3$, $w_4$ and $w_5$. You've already shown that
$$w_1 + w_2 + w_3 + w_4 + w_5 = 54 \tag{1}\label{eq1A}$$
The combined weight of the $2$ lightest penguins is the smallest recorded value, while the combined weight of the $2$ heaviest penguins is the largest recorded value, i.e.,
$$w_1 + w_2 = \color{red}{16}, \; \; w_4 + w_5 = \color{blue}{27} \tag{2}\label{eq2A}$$
Next, \eqref{eq1A} minus the sum of the $2$ expressions in \eqref{eq2A} gives
$$w_3 = 54 - (\color{red}{16} + \color{blue}{27}) = \color{green}{11} \tag{3}\label{eq3A}$$
The second smallest recorded weight must be of the lightest penguin and the third lightest one so, by also using \eqref{eq3A} and the LHS of \eqref{eq2A}, we have
$$w_1 + w_3 = 18, \; \; w_1 = 18 - \color{green}{11} = 7 \; \; \to \; \; w_2 = \color{red}{16} - 7 = 9 \tag{4}\label{eq4A}$$
Similarly, as user2661923's comment basically suggests, the second largest recorded weight must be of the third heaviest penguin and the heaviest one so, by also using \eqref{eq3A} and the RHS of \eqref{eq2A}, we then get
$$w_3 + w_5 = 26, \; \; w_5 = 26 - \color{green}{11} = 15 \; \; \to \; \; w_4 = \color{blue}{27} - 15 = 12 \tag{5}\label{eq5A}$$
Thus, $w_1 = 7$, $w_2 = 9$, $w_3 = \color{green}{11}$, $w_4 = 12$ and $w_5 = 15$. We've already shown how the $2$ smallest and $2$ largest recorded weights come from the penguins' weights, with the remaining ones being $19 = 7 + 12 = w_1 + w_4$, $20 = 9 + \color{green}{11} = w_2 + w_3$, $21 = 9 + 12 = w_2 + w_4$, $22 = 7 + 15 = w_1 + w_5$, $23 = \color{green}{11} + 12 = w_3 + w_4$, and $24 = 9 + 15 = w_2 + w_5$.
