### Brain Teasers

# Time Lines

A clock has 60 lines on it; one at each minute interval. Everyone knows that the hour and minute hands point to the same line at 12:00. Can you figure out what time it is for these situations?

1. The hour hand is exactly on one line, and the minute hand is exactly on the NEXT line.

2. The hour hand is exactly on one line, and the minute hand is exactly on the PREVIOUS line.

1. The hour hand is exactly on one line, and the minute hand is exactly on the NEXT line.

2. The hour hand is exactly on one line, and the minute hand is exactly on the PREVIOUS line.

### Hint

Each hour, the minute hand moves 60 intervals, while the hour hand moves 5 intervals. The hour hand is exactly on a line every 12 minutes. ( 60 / 5 = 12 )### Answer

1. The time is 2:12.2. The time is 9:48.

The hour hand is exactly on a line every 12 minutes, so we only need to look at times ending in 00, 12, 24, 36, and 48.

The hands exactly overlap at 12:00. They are 5 lines apart at 11:00 and 1:00, and farther apart at other hours.

At times ending in 12, the hands are closest at 2:12, where they are 1 line apart.

At times ending in 24, the hands are closest at 4:24, where they are 2 lines apart.

At times ending in 36, the hands are closest at 7:36, where they are 2 lines apart.

At times ending in 48, the hands are closest at 9:48, where they are 1 line apart.

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## Comments

had no idea...got me thinking though! good one!

Didn't get it

Its one of those things were it can never make logical sense to some minds.

I was just wondering, it doesn't have to be 12 minutes after or before the hour does it? Can it be, for instance, 121 for the first one?

Nevermind, I just got it, thank you!

Cool puzzle... and I got it right

didida, no it can't be 121 for the first one if that is what your saying.. however, if your considering the minute and our hand not to be moving simultaneously and proportionally then your right...

That was a good one! I got it, but I had to think about it for a while.

Good one - and just a little different from the other clock problems i know. I managed to get it right, too. Your explanation was clear. Thanks.

I hate it when I go in the wrong direction. Great teaser with a great explanation!

Sorry -- not my thing.

I couldn't get it.

My head hurts

Seriously, I'm normally pretty good at these sorts of things, but my brain just isn't with it right now. I even made up a spreadsheet to find the answer, and I can see you are right, but I still can't follow why

Great teaser anyway. I think I'll recheck it in a day or two

Seriously, I'm normally pretty good at these sorts of things, but my brain just isn't with it right now. I even made up a spreadsheet to find the answer, and I can see you are right, but I still can't follow why

Great teaser anyway. I think I'll recheck it in a day or two

the hour hand moves exactly the same as the minute hand just slower....so why would the hour hand only be exactly on a line every 12 minutes. it would be exactly on a line every 1 minute, just like the minute hand. so the answer could be anything.

I explained that in the hint. The lines mark out intervals for the *minute* hand; the hour hand only travels 5 lines between one hour and the next. Thus it takes 12 minutes for the hour hand to move from one line to the next one. It that still doesn't make sense, then grab an analog clock and move the hands.

Wouldn't it be 2:11 for it to be exactly on the NEXT line? I don't get it...

xdbtcb, I started to write that down when I worked out the answer, so I think I know what you're thinking.

At 2:12, the HOUR hand would be at the "eleven minute mark", but the minute hand would be (of course) at the twelve minute mark.

At 2:12, the HOUR hand would be at the "eleven minute mark", but the minute hand would be (of course) at the twelve minute mark.

This is how I figured it out:

The hour hand hits an exact line at 0,12,24,36, and 48 minutes past each hour.

If we number the marks from 1 to 60, the equivalent minute mark will be given by 5n+1 at twelve minutes past every hour. E.g.:

1:12 .... n=1... equivalent mark = 5+1 = 6 (the hour hand is one tick past the 5)

2:12 .... n=2... equivalent mark = 10+1 = 11 (the hour hand is one tick past the 10)

3:12 .... n=3... equivalent mark = 15+1 = 16

etc.

Generalizing, the minute & hour hands will line up at all integral solutions for the following:

0 = 5n + 0

12 = 5n + 1

24 = 5n + 2

36 = 5n + 3

48 = 5n + 4

Ergo, the only exact lineup is noon/midnight (n=0). We already knew that.

The riddle asks for the minute hand & hour hand to be one "tick" away, so we need integral solutions to any of the following:

0 +/- 1 = 5n + 0

12 +/- 1 = 5n + 1

24 +/- 1 = 5n + 2

36 +/- 1 = 5n + 3

48 +/- 1 = 5n + 4

A.k.a.,

59 or 1 = 5n + 0

11 or 13 = 5n + 1

23 or 25 = 5n + 2

35 or 37 = 5n + 3

47 or 49 = 5n + 4

The solutions are n = 2, which corresponds to 2:12 (hour hand at the 11th tick)

and n=9, which corresponds to 9:48 (hour hand at the 49th tick)

The hour hand hits an exact line at 0,12,24,36, and 48 minutes past each hour.

If we number the marks from 1 to 60, the equivalent minute mark will be given by 5n+1 at twelve minutes past every hour. E.g.:

1:12 .... n=1... equivalent mark = 5+1 = 6 (the hour hand is one tick past the 5)

2:12 .... n=2... equivalent mark = 10+1 = 11 (the hour hand is one tick past the 10)

3:12 .... n=3... equivalent mark = 15+1 = 16

etc.

Generalizing, the minute & hour hands will line up at all integral solutions for the following:

0 = 5n + 0

12 = 5n + 1

24 = 5n + 2

36 = 5n + 3

48 = 5n + 4

Ergo, the only exact lineup is noon/midnight (n=0). We already knew that.

The riddle asks for the minute hand & hour hand to be one "tick" away, so we need integral solutions to any of the following:

0 +/- 1 = 5n + 0

12 +/- 1 = 5n + 1

24 +/- 1 = 5n + 2

36 +/- 1 = 5n + 3

48 +/- 1 = 5n + 4

A.k.a.,

59 or 1 = 5n + 0

11 or 13 = 5n + 1

23 or 25 = 5n + 2

35 or 37 = 5n + 3

47 or 49 = 5n + 4

The solutions are n = 2, which corresponds to 2:12 (hour hand at the 11th tick)

and n=9, which corresponds to 9:48 (hour hand at the 49th tick)

Thanks for your thorough explanation JasonD

And so TIME MARCHES ON!!!!

It took a bit of work to figure it out, but wasn't overly difficult which was nice. Especailly considering the dream I had last night in which I was forced to take a series of advanced level Trig exams (I haven't been in school for 20 years so not sure where that came from). I Really like your explanation too Jason, not sure I would have been able to explain it that well.

It only takes a little TIME to figure this one out.

Woohoo I got it!!!! Wasn't that hard actually. You just have to think proportionally.

tough 1 finallyy got it. The hint helped. apparently the hour hand is doesn't always land directly on a line-number. How 'bout that. Its what is not mentioned. great teaser.

very good. I usually do OK with these type of problems, but didn't think outside the box. Your original explanation to the answer was also a puzzle in itself. It takes twelve minutes for the hour hand to move exactly to a minute marker, hence 12 x 5 minutes equal 60. If the minute hand is not exactly on an increment of 12, The hour hand is between markers (even if it is just by a hair). At 2:12, the hour hand is one marker past 20 and the minute hand is 2 markers past 20.

GIVE US ANOTHER ONE ADDING THE SECOND HAND......HA HA

GIVE US ANOTHER ONE ADDING THE SECOND HAND......HA HA

OK whatever, I'm just stupid.

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