Enigma 1694

Spoiler for New Scientist Enigma 1694:

Our town clock has a standard 12-hour face, divided into 60 minute marks. The hands do not move smoothly: the minute hand moves sharply on to the next minute mark at the end of each minute, and the hour hand moves at the end of each 12 minutes.

The other day, a prankster swapped the hands (an operation that only took a few seconds between movements of the mechanism). Immediately after the swap, the clock showed the wrong time, but it was a valid time; in other words, the relative positions of the hands made sense. Within the next 10 minutes, the clock had shown the right time for exactly 2 minutes.

What time did the clock show immediately before the hands were swapped?

Let’s think about what valid clock positions are. If the hour hand is on the “zeroth” mark (i.e. pointing directly at 12) then it must be less than 12 minutes since the start of the hour; the minute hand must be on one of marks 0 through 11 (from pointing to 12 to one mark past 2). The same is true if the hour hand is on marks 5, 10, 15, …, 55 — that is, if it’s pointing directly to any number. If it’s on marks 1, 6, 11, …, 56 — pointing to the first mark after any number — then it must be from 12 to 23 minutes past the hour, so the minute hand must be pointing to one of marks 12 to 23. And so on:

 Hour hand mark Minute hand mark 0, 5, 10, …, 55 0–11 1, 6, 11, …, 56 12–23 2, 7, 12, …, 57 24–35 3, 8, 13, …, 58 36–47 4, 9, 14, …, 59 48–59

Now consider the times when the swapped hand clock is reading the correct time: In other words, if the hands were at that moment swapped back, the clock would read the same. This means both hands are pointing to the same mark. 12:00 is such a time; so are 1:05 and 2:10; the next time is 3:16 (see the above table; hour hand mark = minute hand mark = 16 is valid); then 4:21, 5:27, 6:32, 7:38, 8:43, 9:49, 10:54, and 11:59. Of these the only ones that occur within 10 minutes of each other (in fact within less than 1 hour 5 minutes of each other) are 11:59 and 12:00. So the starting time must have been between 11:51 and 11:58.

At the starting moment, the hour hand was at mark 59 and the minute hand was at a mark from 51 to 58. After swapping the minute hand was at mark 59 and the time was wrong but valid, so the hour hand was at mark 4, 9, 14, …, or 59. Only one of these is in the range from 51 to 58, namely 54. The time was 11:54 (marks 59, 54); after the hands were swapped the clock read 10:59 (marks 54, 59), then read invalid times until 11:59 and 12:00.