## Two clocks, one fast, one slow.

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LSK
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### Two clocks, one fast, one slow.

You have two clocks which you set to the same correct hour and start running at the same time. One clock gains a minute every hour; the other loses a minute every hour. Which clock will be correct more often?

skeptical scientist
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### Re: Two clocks, one fast, one slow.

Spoiler:
Both will be accurate equally often, since both will be accurate again once they have gained/lost 720 minutes, so both will be correct every 720 hours. (On the other hand, a clock which gains/loses times faster will be accurate more frequently.)

In fact, any clock which does not keep perfect time will be correct (to the nearest second/minute/5 minutes/whatever) the same fraction of the time, because the ones which gain/lose time faster will be correct more frequently, but for a shorter period of time.
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kalakuja
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### Re: Two clocks, one fast, one slow.

Huu this one was fun
Spoiler:
I thougt that the clock is closer to corret time the closer the minute pointer is to the correct time's minute pointer. The distance of the pointers in comparison to the correct time is the same for the both clocks: at any moment of time the one ahead is as far from the correct time as the one behind. The hour pointer acts same. Therefore both of the clocks are as correct in comparison to other.

skullturf
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### Re: Two clocks, one fast, one slow.

I decided to assume the clocks are 24-hour clocks, so we can use the word "day" instead of "12-hour period". It doesn't fundamentally change the answer.

(Also, I of course assumed that the hands of the clocks move continuously and uniformly -- we're not talking about clocks that keep perfect time for a while and then suddenly "jump" forward or back.)

Let's say that in addition to the fast clock and the slow clock, we have a third, perfectly accurate, clock in the same room.

Spoiler:
Suppose all three clocks start at time t=0 (say, midnight when December 31st becomes January 1st).

When the true elapsed time is t=60 minutes, the slow clock claims that 59 minutes have passed, and the fast clock claims that 61 minutes have passed.

By uniformity, at time t, the slow clock displays a time of (59/60)t, and the fast clock displays a time of (61/60)t. Informally, the slow clock "thinks" the time is (59/60)t, and the fast clock "thinks" the time is (61/60)t.

In general, any two clocks display the same time as each other if and only if the difference (time that the first clock "thinks" it is) minus (time that the second clock "thinks" it is) is equal to a whole number of days.

The difference (time that the fast clock "thinks") minus (time that the accurate clock "thinks") is (1/60)t. And the difference (time that the accurate clock "thinks") minus (time that the slow clock "thinks") is also (1/60)t.

EDIT: By the way, a brief remark about the psychology or pedagogy behind this puzzle:

Spoiler:
The thoughts that first leaped into my head, for whatever it's worth:

(i) "They've gotta be the same, due to basic symmetry. Being behind by x is "as wrong" as being ahead by x."

(ii) "No wait, maybe they're not the same. Maybe one's right every 59 days and one's right every 61 days, or something. (The very fact that the question is being asked could mean that there's a subtlety.)"

(iii) "No, scratch that, I think they are equivalent after all."

Sabin
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### Re: Two clocks, one fast, one slow.

Heh. I guess the answer truly lies in what is the "right" time. What is the frame of reference and what is its relative velocity expressed as a percentage of the speed of light in a perfect vacuum (for the purpose of calculating time dilation)? Do the clocks gain and lose a minute from the "correct" time, or from one another? Are the clocks analog or digital, and are they reliant on the same power source? Does [imath]\pi = 4[/imath]?