A clock takes 12 seconds to strike 4, how long does it take to strike 12? I have already tried EVERYTHING, but nothing seems to work.
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2This question may be the same as this one:http://math.stackexchange.com/questions/478736/a-clock-takes-6-seconds-to-strike-4-times-at-400-how-many-seconds-will-it-take – NoChance Oct 14 '15 at 01:07
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@NoChance: Not quite the same question, but definitely the same principle. Good catch. – Brian M. Scott Oct 14 '15 at 01:12
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Try Googling "fence post problem"... – DJohnM Oct 14 '15 at 05:51
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This is a bad question. An assumption is needed to solve it with the given info.
A single strike of the clock can be visualised as the sum of a "chime" ($C$) and a "gap" ($G$). The total length of a strike is therefore $C+G$.
$4$ strikes would be depicted as: $CGCGCGC$, the last gap being ignored as the timekeeping would stop once the final note has died off.
We can therefore state: $4C + 3G = 12$.
We are now asked to find how long $12$ strikes would take, and that's $12C + 11G$. The problem is that we have insufficient information to make that determination. If we had information on $8$ strikes (say), we could solve a pair of simultaneous equations and accurately state how long $12$ strikes would take.
However, as things stand, a simplifying assumption of $C \ll G$ needs to be made. This allows us to completely neglect the $C$ terms and say $3G = 12$, hence $G=4$ and get $11G = 44$ ($44$ seconds for $12$ strikes) as the answer.
My issue is: is that assumption justified? In my experience with large clocks (grandfather clocks, clock towers, etc.), the chime definitely takes a finite interval that is non-negligible relative to the gap.
So I conclude: this is a bad, bad problem.
Deepak
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The fact that we're asked to solve it indicates that the second interpretation is intended. It's certainly not a great question, but it really isn't hard to interpret correctly. (But +1.) – Brian M. Scott Oct 14 '15 at 01:02
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1I like the discretion shown by Deepak, who I don't think had any difficulty interpreting the question the way it was intended, but merely had issues with its lack of rigor. – Brian Tung Oct 14 '15 at 02:03
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The striking is "instantaneous". The sounding takes a while to end. – Eric Towers Oct 14 '15 at 02:09
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@Deepak It may be, i use LaTex a lot, but had no exposure to MathJax until coming to this site; it seems extremely similar if not verbatim equivalent. – DrZ214 Oct 14 '15 at 11:44
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@DrZ214 You're quite right, it is MathJax, but the commands are basically LaTeX commands. I believe it's a lot less powerful than "full" LaTeX, though. – Deepak Oct 14 '15 at 12:43
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@Deepak: MathJax is not 'LaTeX based', but rather includes (some of) it capabilities: 'MathJax can display mathematical notation written in LaTeX or MathML markup. Because MathJax is meant only for math display, whereas LaTeX is a document layout language, MathJax only supports the subset of LaTeX used to describe mathematical notation.' (see Wikipedia: MathJax – Features) – CiaPan Oct 15 '15 at 07:34
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@Deepak: What is the meaning of
CG? I mean, does chime mean the duration of the ringing sound itself or the time to produce the chime? And is gap the period till the previous chime dies out? Could you please have a look here too if you have some time? https://math.stackexchange.com/questions/3952831/the-first-thought-for-this-puzzle-is-wrong-but-it-is-also-one-that-seems-most-na – Jim Dec 18 '20 at 22:04 -
@Jim "CG" just means a chime followed by a gap. Pls see 1st answer here: https://math.stackexchange.com/questions/478736/a-clock-takes-6-seconds-to-strike-4-times-at-400-how-many-seconds-will-it-take What I'm calling a "chime" (C) is a "bong" there and what I'm calling a "gap" (G) is an interval (represented as double-dash (--) there). The simplifying assumption (which I disagree with, but seems necessary) is to neglect the times taken for the chimes or bongs and only consider the gaps or intervals. And count carefully to avoid Fencepost Errors: https://en.m.wikipedia.org/wiki/Off-by-one_error – Deepak Dec 19 '20 at 00:13
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@Jim The Chime is the ringing sound. The Gap is the "silent time" before the next ring. I really hope this clarifies, because I can't do better. – Deepak Dec 19 '20 at 00:22
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@Deepak: To me it seems that the "silent time" is the time till the chime died out. So C should be equal to/same as G, i.e. as soon as the chime dies out the next one is produced. But you note (and elsewhere) that the time for a strike is
C + G. This is what is confusing to me. – Jim Dec 19 '20 at 10:45
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HINT: The time taken to strike $4$ is essentially all accounted for by the $3$ intervals between the strokes. When the clock strikes $12$, there a $11$ of those intervals. Assuming that all of the intervals are the same length, how much time will those $11$ intervals require?
Brian M. Scott
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If it takes a clock 12 seconds to strike 4, then it will take another 12 seconds to strike 8 and another 12 seconds to finally strike 12. Adding these together, we get either 36 seconds total or 24 seconds after striking 4.
Brevan Ellefsen
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This is incorrect, I'm afraid; the second $12$ seconds will get you only to the seventh stroke. – Brian M. Scott Oct 14 '15 at 00:44
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@BrianM.Scott I disagree. Let's split this up, starting from 12:00 (which is standard for these questions). We then have 1 tick at 1,2,3, and 4 (so we can cover 4 ticks in 12 seconds). We then have 5,6,7, and 8 (another 12 seconds) and finally 9, 10, 11, and 12 (another 12 seconds). The interval from 12:00 - 4:00 has 4 intervals, not 3. I would say the fundamental problem here is the OP did not say where the clock starts – Brevan Ellefsen Oct 14 '15 at 00:45
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The problem is perfectly clear: the time elapsed during the striking of $4$ is $12$ seconds. That time is essentially all taken up by the intervals between strokes, of which there are three. The starting point is obviously the moment of the first strike. You're thinking of some completely different kind of problem when you introduce that meaningless arbitrary starting point. – Brian M. Scott Oct 14 '15 at 00:52
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@BrianM.Scott So what are you defining as your 3 intervals? I have defined the intervals from 12:00-1:00 (the clock sweeps out 30 degrees), 1:00-2:00 (30 more), 2:00-3:00 (30 degrees) and 3:00-4:00. I disagree that the first strike occurs as the question starts... the OP does not state this, and either interpretation is correct without more criteria. I can see where you are coming from, and I agree that your solution is correct given that the first strike occurs as the question starts, but we need the OP to specify before my answer is "incorrect" – Brevan Ellefsen Oct 14 '15 at 00:56
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What you've just written has nothing to do with the problem. Nothing is being swept out. The three intervals are the intervals of time between consecutive strokes. – Brian M. Scott Oct 14 '15 at 01:00
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@BrianM.Scott That is true, assuming the OP is asking for that. The OP simply states that it "takes 12 seconds to strike 4". Perhaps we have different interpretations here... I take this as "it takes 12 seconds to strike 4:00". Are you interpreting this as "it takes 12 seconds to strike 4 times"? In my interpretation starting point and sweeping out are important. I suppose that if we assume the OP gave us all information needed than your interpretation covers all possible cases while mine requires a starting point, so we could assume yours, but I want the OP to confirm this. – Brevan Ellefsen Oct 14 '15 at 01:03
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I'm giving it its obvious meaning: the length of time required for the clock to sound the hour of $4$ is $12$ seconds. In other words, the duration of the entire striking of $4$ is $12$ seconds. Sounding the hour of $4$ requires striking $4$ times. There is no other reasonable interpretation of the given wording. The only open question is whether we're entitled to ignore as negligible the duration of each impact of the striker. This is, however, both a very plausible assumption and one that is necessary if the problem is to be solved, so we make it. – Brian M. Scott Oct 14 '15 at 01:10