Time Lines
Logic puzzles require you to think. You will have to be logical in your reasoning.
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.
HintEach 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 )
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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
coolkcoolk
Oct 27, 2009
 had no idea...got me thinking though! good one! 
Stack1607
Oct 28, 2009
 Didn't get it 
precious1026
Oct 31, 2009
 Its one of those things were it can never make logical sense to some minds. 
didida
Nov 04, 2009
 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? 
didida
Nov 04, 2009
 Nevermind, I just got it, thank you! 
DarkMessiah
Nov 26, 2009
 Cool puzzle... and I got it right 
DarkMessiah
Nov 26, 2009
 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... 
rainbirdbrain
Nov 30, 2009
 That was a good one! I got it, but I had to think about it for a while. 
dalfamnest
Nov 30, 2009
 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. 
patiencewithaP
Nov 30, 2009
 I hate it when I go in the wrong direction. Great teaser with a great explanation! 
craniac
Nov 30, 2009
 Sorry  not my thing. 
auntiesis
Nov 30, 2009
 I couldn't get it. 
UptheHill
Nov 30, 2009
 
VitalStatistic
Dec 01, 2009
 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 
alexcunn
Dec 01, 2009
 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. 
MarcM1098
Dec 01, 2009
 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. 
xdbtcp
May 12, 2011
 Wouldn't it be 2:11 for it to be exactly on the NEXT line? I don't get it... 
JasonD
Jan 31, 2012
 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. 
JasonD
Jan 31, 2012
 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) 
MarcM1098
Feb 01, 2012
 Thanks for your thorough explanation JasonD 
Babe
Nov 30, 2012
 And so TIME MARCHES ON!!!! 
HABS2933
Nov 30, 2012
 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. 
cutebug
Nov 30, 2012
 It only takes a little TIME to figure this one out. 
freakshow07
Nov 30, 2012
 Woohoo I got it!!!! Wasn't that hard actually. You just have to think proportionally. 
isis523
Nov 30, 2012
 tough 1 finallyy got it. The hint helped. apparently the hour hand is doesn't always land directly on a linenumber. How 'bout that. Its what is not mentioned. great teaser. 
ftlink
Nov 30, 2012
 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 
iggy39
Nov 30, 2012
 OK whatever, I'm just stupid. 
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