$7 \cdot 77$ There are 900 three-digit numbers $(100,101, \cdots, 999)$, and these three-digit numbers are printed on cards, with one three-digit number printed on each card. However, some cards, when flipped, still show a three-digit number, such as 198, which when flipped shows 861; some cards do not, such as 531, which when flipped makes no sense. Therefore, some cards can be used in two ways, so fewer cards can be printed. How many fewer cards can be printed at the minimum?
[Solution] There are 5 digits that still make sense when inverted: $0,1,6,8$, 9. A three-digit number that still makes sense when inverted must remain a three-digit number, so the tens place has 5 choices. However, since the hundreds and units places cannot be 0, there are only 4 choices for these positions. Therefore, the total number of such numbers is $5 \times$ $4 \times 4=80$.

Among all the three-digit numbers that still make sense when inverted, some numbers invert to another three-digit number, while others invert to themselves. Clearly, only the former can be used for double-sided cards, the latter cannot.

The digits that remain the same when inverted are only 3: $0,1,8$, so the tens place of the latter type of number has only 3 choices. The hundreds place can be chosen from $1,6,8,9$. The units place then corresponds to $1,9,8,6$. It is easy to see that there are 12 such numbers.

In summary, the number of three-digit numbers that still make sense when inverted but are not equal to themselves is 68. When printing cards, these can be used double-sided, saving half, i.e., a maximum of 34 fewer cards can be printed.