13.11 Given 6 identical-looking weights, weighing 1, 2, 3, 4, 5, and 6 grams respectively, and on their faces are engraved the words "1 gram", "2 grams", "3 grams", "4 grams", "5 grams", and "6 grams". Now there is a balance scale, but no other weights. How can you, through two weighings, check for any discrepancies between the engraved labels and the actual weights?
［Solution］Verify the following two equations through two weighings:
$$
\begin{array}{l}
x_{1}+x_{2}+x_{3}=x_{6}, \\
x_{1}+x_{6}<x_{3}+x_{5},
\end{array}
$$

where $x_{k}$ represents the mass of the weight marked “$k$ grams”. If at least one of equations (1) and (2) does not hold, it indicates that there is a discrepancy between the marked and actual mass of at least one weight; if both (1) and (2) hold, then the mass marked on each weight is accurate.
In fact, since
$$
x_{1}+x_{2}+x_{3} \geqslant 1+2+3=6 \geqslant x_{6} \text {, }
$$

it follows from (1) that $x_{6}=6$ and
$$
\left\{x_{1}, x_{2}, x_{3}\right\}=\{1,2,3\},\left\{x_{4} \dot{x}_{5}\right\}=\{4,5\} \text {. }
$$

From (2) and the estimates $x_{1} \geqslant 1, x_{3} \leqslant 3, x_{5} \leqslant 5$, we can derive the inequality
$$
7 \leqslant x_{1}+x_{6}<x_{3}+x_{5} \leqslant 8,
$$

which implies $x_{1}+x_{6}=7, x_{3}+x_{5}=8$, thus $x_{1}=1, x_{3}=3, x_{5}=5$. From (3), we get $x_{2}=2, x_{4}=4$, meaning that for $k=1,2,3,4,5,6$, $x_{k}=k$.