## Friday, October 14, 2011

### Telling time without the glasses

This new lamp that you bought for your bedroom - it projects the magical lace pattern on the walls and makes you want to fall asleep in the fairyland of your bedroom any time of the day.  Yesterday you fell under this spell and took a nap. When you woke up, you quickly glanced at the clock. Without your glasses on both clock handles looked exactly the same size.  You guesstimated the time and went back asleep. When you finally put yourself up and managed to get to your appointment you were almost an hour late.

What time might have been on the clock that confused you?
Want something harder - how many times like these are throughout the day when without telling which clock handle is which you can't guesstimate the time?

Answer ideas accepted any time until midnight on Saturday October 15th (EST), on our Family Puzzle Marathon. They will be hidden till then and everyone who submitted something reasonable will get a puzzle point.

Jerome said...

If what is happening is what I think is happening, the clock hands are one on top of the other. I'm old enough that two things are true:

1. I remember learning to tell time on a 12 hour clock, and the minute and hour digits never coincide on a digital clock(that is the red hour,minute digits are not on top of each other ever).

2. I get up often enough at night that I could be confused frequently by the hands overlapping on a mechanical clock. I would be willing to bet it has happened to me at least once.

Let us assumem that it happened once at midnight which is the most likely time for someone to wake up.

Call that 0 hour.

About 1 hour later, it will happen again a little after 1. But what time is that exactly?

It turns hout thatthe minute hand will go around 12 times throgh a 360o Rotation for every 1 that the hour hand goes around.

Therefore the times will coincide 12/11 times the number of hours after midniht.

1 hour on a clock = 360/12 = 30o

For example, at 1 oclock
12/11 * 30 hour = 32.7272727272 degrees.

ie, the clock's hand's will coincide at an angle of 32.7o from midnight, going clockwise of course.

1 hour = 60 minutes = 30o
x = 2.72727262 o

x = 5 minutes 27 seconds later than 1 oclock.

Each hour after that will add a little more to the amount of time that must pass before the two hands coinside.

For example (again).

At 7 oclock when you do the calculation you get
229.09090909 o which is roughly 38 minutes and 11 seconds after 7 oclock. Roughly translated that means 20 to 8.

Tom said...

I’m not familiar with these lamps. “Nap” usually will imply afternoon, though not always. This photo appears that the wall slants “away to our right.” If the shorter hand is pointing pretty much toward the napper and the longer hand points 180 degrees away, for instance 7:06 approx, then the slant (parallax?) might make it appear to be about 1:35 to foggy eyes.

Anytime the hands are nearly 180 degrees opposed, I can misread them. That will happen I think 11 times per cycle on a 12-hour clock like this, or 22 times daily. Plus, when the two hands overlap, again I think 22 times daily. So I’m betting 44 times a day I might get confused.

I can get confused even oftener if there’s a hand showing seconds. I can get confused even more if it is quite dark!

Maria said...

Tom - I will get a picture of this lamp for you. It is quite dangerous.

What I meant was not two handles coinciding but seeing two handles that appear to be the same size and mistaking hour handle for the minute handle. This is not always possible. For most of the times throughout the day one can tell which handle is which even when they appear to be the same size. Take 6:00 - hour handle is at 6, minute at 12. The other way around is impossible: when minute handle is at 6, hour handle can't be precisely on 12. Handles are ambiguous but their positions dictate the only possible scenario for what handle represents hours and what minutes.

But take 12:05 and 13:00 - in both cases there are two handles, one around 12 and another around 1. One can assume it is 12:05 and continue napping arriving approx. an hour later to the meeting.

Same with 11:55 and 11:00
Given two handles that appear to be of the same size, one on 11 and another on 12, we can confuse the time and be late for approx. an hour.

Also 5:33 and 6:27 - two handles of the same size, one between 5 and 6 and another between 6 and 7.

It seems that handles should be around 5 mins apart from each other to allow to visually separate between them yet allow realistic time in both configurations. And it seems that this happens at least once an hour.

Any other insights?