## Task 10/66

"Proof" that the weight of a locomotive is equal to the weight of a brick:

Let the weight of the locomotive be $m$, and the weight of the brick be $z$.

Then, let $m+z=e s$ be the sum of both. Therefore, $m-2 s=-z$ and $m=-z+2 s$. Multiplying these two equations together, we get:

$$
m^{2}-2 m s=z^{2}-2 z s
$$

or, after adding $s^{2}$ to both sides

$$
m^{2}-2 m s+s^{2}=z^{2}-2 z s+s^{2} \rightarrow(m-s)^{2}=(z-s)^{2} \rightarrow \text { thus } m=z
$$

Where is the error?
The error lies in the last conclusion. From

$$
(m-s)^{2}=(z-s)^{2}
$$

one cannot infer that $m-s=z-s$ and thus $m=z$, but only $|m-s|=|z-s|$. This results in either $m-s=z-s$ or $m-s=-(z-s)$. In the present case, the second of these relations actually holds. Therefore, we again obtain the initial equation $m+z=2s$.