Better late than never! A small technical SNAFU yesterday interfered with the well-oiled machine that is Problem of the Week. But now we're back on track! The fourth problem has now been posted. The official solution to last week's problem will be up by tomorrow.
For the minute hand there are 6 minute ticks each number. For the hour hand, each of the six ticks represents a 10-minute increment.
We're looking for a time between 3:18 and 3:24. At 3:20, the minute hand will have moved two minute-ticks past the 3, while the hour hand will have moved two 10-minute-ticks past the 3.
We can confirm that 3:20 is the answer another way. The hour hand makes one revolution in 600 minutes, while the minute hand does so in 60 minutes. At 3:00 the hour hand has swept 180 minutes. So we're looking for:
600x = 60x + 180
540x = 180
x = 1/3
1/3*60 = 20, so the answer is 3:20.
The minute hand moves ten times as fast as the hour hand. So it is straightforward to set up and sum the infinite series: the hands cross at 3.333... hours after "10:00", which is 3:20:00.
Using this method, it is straightforward to find the other eight times the hands cross: 10:00:00, 1:06:40, 2:13:20, 4:26:40, 5:33:20, 6:40:00, 7:46:40, and 8:53:20. These are the integer multiples of 1.111... hours.
I got the same answer, using Another Matt's logic. Socially, I'm kinda glad we didn't end up with a decimal clock. With our human desire for simplicity and business' desire to squeeze every moment it can out of a person's life, if we had a decimal system I have no doubt the standard work schedule would be working seven new-hour days and getting two days off out of each ten. On the plus side, there'd be enough days in the week to add a few more Norse gods to the list. Using the US system of starting the week on the last day of the weekend, I propose day 10 be called Lokisday. I often get into trouble on Lokisday nights.
We really need some sort of spoiler tags here; I have to carefully scroll the previous commenters off the page first so I don't inadvertently cheat. :)
Assume there are 60 minute divisions between each number. Between 10 and 1 we have 0-60, then 60-120, and so on. At exactly 3:00, the minute hand is at 0, the hour hand is at 180. After t minutes, the minute hand is at 0 + 10t, the hour hand is at 180 + t. They cross at 10t = 180 + t, or t = 20. Therefore, they coincide at exactly 03:20.