A Poorly Designed Clock
The hour and minute hands of a clock are indistinguishable. How many moments are there in a day when it is not possible to tell from this clock what time it is?
Labels: logic, mathschallenge
posted by Zaux at
3:54 PM
18 Comments:
22 times.
11 times when the hands coincide, and 11 more times when they are exactly opposed.
I am the anon above - I forgot to enter my avatar name.
Although not exactly wrong, times like 11:43 will look like 7:57 (or something like that) - i.e. when they're about 90 degrees apart. But I'm not going to count them.
Hmmm, not sure if I got that quite right. Never mind!
It can only be at Noon and Midnight. If it is anything else you would be able to tell because the hour hand only slightly changes vs. the minute hand. For ex. at 6:15 the hour hand would already be 1/4 of the way to the 7 and the same can be said for any other time.
Hi Wiz ... the 22 times you referenced are actually the moments when the time is determinable (except for AM/PM) ... right?
Hi Bev ... good thought ... obviously if one sits and watrches the clock, movement will define the minute hand ... for this problem, assume the clock viewing is only a glance and not a time study. Since AM/PM is not important in our pursuit, the 12 o'clock position would be obvious...
Wiz... actually there are 22 times a day when the hands coincide and the time is obvious ... time is not so obvious when the hands are opposed.
I'm beginning to wonder if any times can be misinterpreted assuming exact ambiguity. This one's a bit trickier than I had supposed. I'm assuming a 12 hour clock and that am/pm ambiguity is irrelevant.
Hi Chris ... the problem states "in a day" ... thus a 24 hour time period
I'm going to post this solution to see if you guys get it:
Let us first note that for the problem to make sense, we must assume that the hands move continuously, and that we are not tasked with deciding whether a time is AM or PM. Note that we can tell what time it is when the two hands coincide, even though we can't tell which hand is which; this happens 22 times a day, since the minute hand goes around 24 times while the hour hand goes around twice, in the same direction.
This reasoning turns out to be good practice for the proof. Imagine that we add to our clock a third "fast hand", which starts at 12 at midnight and runs exactly 12 times as fast as the minute hand.
Now we claim that whenever the hour hand and the fast hand coincide, the hour and minute hands are in an ambiguous position. Why? Because later, when the minute hand has traveled 12 times as far, it will be where the fast hand (and thus also the hour hand) is now, while the hour hand is where the minute hand is now. Conversely, by the same reasoning,all ambiguous positions occur when the hour hand and the fast hand coincide.
So, we need only to compute the number of times a day this coincidence occurs. The fast hand goes around 12^2 x 2 = 288 times a day, while the hour hand goes around just twice, so this happens 286 times.
Of these, 22 are times when the hour and and minute hand (thus all 3 hands) are coincident, leaving 264 ambiguous moments.
Answer --> 264 moments of ambiguity
I see your logic, but the question would have been impossible to answer without knowing what you define as a "moment".
There is no such chronological period as a moment but you have defined it as the ticks of an imaginary fast hand. I had no way of knowing that!
Maniac, surely you're kidding. A moment is a well known concept. A quick Google reveals:
A particular point in time.
An indefinitely short time.
Here and now: at this time.
.. also, Zaux has been careful to eliminate "ticks", the hands move continuously (that means smoothly) - they don't tick.
I haven't fully absorbed the solution (still on first coffee). But not sure why you knock off the 22 completely coincident hand moments. I certainly like the technique used.
Hi Chris ... when the hands are coincident the time is obvious. The clock has 60 minute marks ... thus if the hands coincide on the 18th mark, it is 3:18 ... on the
28th mark, it 5:28 ... etc. Those moments happen 11 times per 12 hour period, or twice per day ... thus the 22 times where the odd clocks time is obvious.
So it is, doh!
look evryone if both the hands are coinciding at any point (2.10,3.15,4.20,5.25,6.30..........and so on) u can easily tell whats the time like if both the hands are on 10 then it must definetly be 10.50........... so other than 12 out of 12 hours we can judge the exact time and rest other moments u cant be sure...thats it
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