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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.

By Another Matt (not verified) on 27 Sep 2016 #permalink

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.

By Eric Lund (not verified) on 27 Sep 2016 #permalink

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.

Socially, I'm unhappy we didn't end up with a duodecimal number system.

By Another Matt (not verified) on 27 Sep 2016 #permalink

In reply to by eric (not verified)

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.