Before the water, the ticket punching machines can punch some - or even all - of the nine numbered fields on the ticket. Depending on the setting of the machine, the inspectors ask that the machine does not punch the same fields if someone places their ticket not as prescribed, but upside down. How many such settings are possible for the machine?

$$
\begin{array}{|l|l|l|}
\hline 1 & 2 & 3 \\
\hline 4 & 5 & 6 \\
\hline 7 & 8 & 9 \\
\hline
\end{array}
$$
Solution. If we do not consider any of the conditions of the problem, then the ticket can be punched in $2^{9}$ ways (allowing for the possibility that there is no hole on it). From this, we need to subtract the number of punchings that are symmetric about the 2, 5, 8 axis (among these is also the case where no number has been punched out). A pattern of such punching can be obtained by arbitrarily punching some of the $2,5,8$, and the $1,4,7$, fields. After that, we reflect the holes about the 2, 5, 8 axis. Therefore, the number of symmetric patterns is $2^{3} \cdot 2^{3}=2^{6}$, so the number of punchings in question is: $2^{9}-2^{6}=64 \cdot 7=448$.