Folklore

In Italy, they produce clocks where the hour hand makes one revolution in a day, and the minute hand makes 24 revolutions, with the minute hand being longer than the hour hand (in regular clocks, the hour hand makes two revolutions in a day, and the minute hand makes 24). Consider all positions of the two hands and the zero mark on the Italian clocks that also occur on regular clocks. How many such positions exist on the Italian clocks in a day? (The zero mark indicates 24 hours on Italian clocks and 12 hours on regular clocks.)

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Let some position of the hour hand of ordinary clocks before noon be taken at time $t$ (counting from the beginning of the day; $0 \leq t<12$ : we measure time in hours). Since the hour hand of Italian clocks moves twice as slowly as the hour hand of ordinary clocks, it will coincide with the considered position of the hour hand of ordinary clocks at the moment $2 t$. The minute hands (of ordinary and Italian clocks) occupy the same position if the difference in times is an integer number of hours. Thus, the condition for "coincidence of positions" is that the number $2 t-t=t$ is an integer. In the specified interval, there are exactly 12 integer values - from 0 to 11 (which correspond to the moments $0,1, \ldots, 11$ o'clock exactly on ordinary clocks).

## Answer

12 positions.

## Problem