I’ve just posted the new Problem of the Week, along with an official solution to last week’s problem. But this one will have to hold you for a while, since I’m taking next week off.
POTW 5 is weirdly phrased. There is only one time “between 4 and 5” in which the hands coincide, regardless of which way the hands are turning. Rounded to the nearest minute, I believe that time shown on the clock is 4:20. However, if the question is asking how much time has passed between noon and when that backwards-clock shows “4:20” on its dial, the answer would be the mirror image of 4:20, which is 7hrs and 40 minutes (so, the true time is 7:40pm, since we started at noon).
So the backwards answer is 7:40.
The minute hand and the hour hand will coincide 13 times every 12 hours, which means the interval between such coincidences is a bit more than 55 minutes. (55.3846… minutes, if you do the calculation.) The instance between 4:00 and 5:00 will be the fifth such coincidence after 12:00, which is (rounding to the nearest minute) 277 minutes after 12:00, or 4:37.
eric@1: After seeing your solution (which posted while I was typing mine) I went back and re-read the problem. You seem to have assumed that both hands are rotating anticlockwise, and that the watch is showing a time between 4:00 and 5:00 (even under those assumptions, your solution is incorrect; it should be 7:38). I interpreted the problem as having one hand rotate clockwise and the other anticlockwise; for purposes of this problem it doesn’t matter which is which, since the correct time is stated to be between 4:00 and 5:00.
I’ll let Jason decide whether your solution is reasonable, but if I were the judge, I would say no. It’s clear that Jason is looking for a time between 4:00 and 5:00 by a correctly operating clock.
Eric Lund @3: I certainly did not think of the hands going in different directions, I only considered the case of them both running backwards or forwards. But I also just SWAGged the minutes. Your 7:38 would thus be the exact answer to which I was approximating.
Eric Lund’s interpretation is correct. We are told that the two hands rotate in opposite directions. So let’s assume that the hour hand moves clockwise while the minute hand moves anti-clockwise. When the correct time is 4:05, a normal clock would have its minute hand on the 1 and its hour hand 1/12 of the way between the 4 and the 5. On this clock, by contrast, the hour hand will be in the normal place, but the minute hand will actually point to the 11.
Interesting – since the problem doesn’t ask for two answers, I will assume that both cases (ie minute hand goes anticlockwise, or hour hand does) give the same answer. For simplicity I will therefore assume that the hour hand correctly goes clockwise, and therefore is located somewhere between 4 and 5.
Divide the clock into 60 minutes, going clockwise, starting at 12 = 0 degrees. Assume they coincide at t minutes after 4 o’clock. The hour hand is at 20 + t/12; the minute hand is at 60 – t. As these are the same minute, this works out to approximately 4:37
Alternately, assume that the hour hand goes anticlockwise instead. Then the hour hand would be somewhere between 7 and 8; using the same 60 minute positioning, it is at 40 – t/12. The minute hand (now going clockwise) is at t. This gives us the same answer of about 4:37, as expected.
jrosenhouse wrote (October 3, 2016):
> I’ve just posted the new Problem of the Week [ http://educ.jmu.edu/~rosenhjd/POTW/Fall16/POTW5F16.pdf ] …
With the two dial clock hands rotating in opposite senses, they will meet eleven times, after having started in coincidence from some particular mark, until again coinciding at/with the same mark on the clock face.
(That’s a somewhat more general and hopefully more correct statement of the idea which may be gathered from Eric Lund’s description, #2).
By the usual assumption of uniform motion wrt. the clock face markings, the angular separation between successive coincidence rays is 1/13 of a full circle (=~= 27° 41’32.3’’); corresponding to a duration of 12/13 hours.
According to the problem statement,
– “ the two hands started together at noon”,
i.e. having coincided at/with mark “12”; and
– the “correct time” of the specific coincidence of interest “was between 4:00 and 5:00”,
i.e. in particular: the coincidence of interest was observed by the hour hand between having passed mark “4” and having passed mark “5”.
Two cases may be distinguished:
(1): The hour hand rotates as usual (clockwise), moving from hour marks “12” to “1” to “2” to “3” … to “12”,
and the minute hand rotates in the opposite sense.
The coincidence of interest was then the fifth coincidence after noon (the coincidence at “12”);
and it was observed by the big hand after having passed mark “4” and before passing mark “5” (on its one clockwise roundtrip from “12” back to “12”).
It occurred consequently 5 * 12/13 = 60/13 =~= 4 hours 37 minutes after the noon coincident.
(2): The minute hand rotates as usual (clockwise), and the hour hand rotates in the opposite sense:
“12” to “11” to “10” to “9” to “8” to “7” to “6” to “5” to “4” … to “12”.
The coincidence of interest was then the eigth coincidence after noon (the coincidence at “12”);
and it was observed by the big hand after having passed mark “5” and before passing mark “4” (on its one counter-clockwise roundtrip from “12” back to “12”).
It occurred consequently 8 * 12/13 = 96/13 =~= 7 hours 23 minutes hours after the noon coincident.
For both cases, the “correct time”, i.e. the coincident “position” of both hands between marks “4” and “5” is the same:
=~= 138° 27’ 41.5’’ (clockwise) from the “12” mark.
Only 2 realms after our lifelong demise… and 1 of em aint too cool.
You are taking off when there are two big chess stories to cover?
1. What did you think of “Queen of Katwe”?
2. Any comments about Trump’s “grand chess masters” remark?
POTW 6 is up! You can use Jason’s link to get to it.
Though I admit I think I must be reading it wrong. Hour hands only point “directly to” a minute mark when the minute hand is on the 12, leaving to only two exact answers: 11:59:00* and 12:01:00. Am I interpreting the question wrong?
*Because the problem also states “…after 12…” this is the 11:59:00 that occurs 11 hours and 59 minutes after the 12 o’clock start of the exercise, not the 11:59:00 that occurs one minute before the start of the exercise.
The way I interpret the problem is that both the continuously moving hour hand is pointing directly at one of the minute marks, which happens once every 12 minutes. Thus the hour hand and the minute hands are both pointing at minute marks at 12:00 (both pointing to minute mark 0 at the 12 o’clock position). 12:12 (the hour hand is pointing to minute mark 1 and the minute hand to minute mark 12, etc). So both hands are pointing directly at a minute mark at 0, 12, 24, 36, and 48 minutes after each hour.
The problem also requires that the minute marks be consecutive. I get two answers. At 2:12 the hour hand is pointing at the 11 minute mark, and the minute hand at the 12 minute mark. And at 9:48 the hour hand is pointing at the 49 minute mark and the minute hand is pointing at the 48 minute mark.
Editing fail in #11: the first sentence should not include the “both”.
Ah, you’re right. I was confusing the action of the second hand with the action of the minute hand.
So the hour hand will be exactly on a tick mark at the 12, 24, 36, and 48 minute mark of each hour. 2:12 is a solution, 4:24 is not (two tick separation), 6:36 is not (three tick separation), and 9:48 is.
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Folks, I’m done.
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