We have recently "jumped" to the winter time. This is the time for a riddle regarding the clock face:
Usually, the long hand on the clock indicates the minutes and the short hand indicates the hours. If the minute hand did not exist, we would still be able to tell the time, because the hour hand moves continuously between the whole hours. For instance: at six thirty, the hour hand will be precisely between 6 and 7. What will happen if we swap the long hand with the short hand? In most cases the situation will be meaningless. For instance: when swapping the minute hand with the hour hand at 6:00, the minutes hand will point to 6 (meaning 30 minutes have lapsed from the whole hour) while the hour hand will point to 12 (a whole hour) – a situation which is not possible. However, at 1:46 (approximately) the two hands can be interchanged – the "new" time will be 9:09 (see figure).
How many positions are there on the clock that allow this sort of interchange of the hands?
This is not an easy riddle (and it is "homemade", we don’t think it appears on the internet or in previously printed books). If it is too difficult for you, send us partial solutions and start a discussion.