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The only digits which will appear the same when reflected in the glass table-top are $0, 1, 3,$ and $8$. So it it necessary to find the number of times in a $24$-hour period that the display on the clock is made up only of these digits.

There are two possibilities for the first digit: $0$ or $1$.
There are four possibilities for the second digit: $0$, $1$, $3$ or $8$.
There are three possibilities for the third digit: $0$, $1$ or $3$.
There are four possibilities for the fourth digit: $0$, $1$, $3$ or $8$.

To find the total number of possible times, we can multiply together the number of possibilities for each digit.

Therefore the display and its reflection give the same time on
$2 \times 4 \times 3 \times 4 =96$ occasions

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.