Mathematics Question Involving Watches

[Shore Guy]

There is a question whose solution eluded me for years. I occasionally revisited it over time only to be frustrated by not succeeding to solve it by brute force and giving up after a few minutes. Then one night after retiring to bed and revisiting it again, I solved it in 10 or so seconds, without paper and pencil which I never take to bed with me (I'm not that crazy.)

The question is:

Determine between 12 noon and 12 midnight, the EXACT times that the minute and hour hands coincide. Express the answers as hours and fractions of hours beyond 12 noon.

I will give the solution and proof in this thread in a few weeks after allowing members to ponder this. This can be done in two or three lines. When you see the solution, you will say to yourself, why didn't I think of that!

If any member comes up with this elegant (and really beautiful proof), I will recognize it. I will not comment or even look at a number crunching or brute force attempt, valid or invalid.

[Thatguy]

Read op wrong....deleted

[Floorguy]

[Avionics Guy]

Originally Posted by Shore Guy

The question is:

Determine between 12 noon and 12 midnight, the EXACT times that the minute and hour hands coincide. Express the answers as hours and fractions of hours beyond 12 noon.

Seems pretty straightforward to me. Anyway, the position of a watch hand can be thought of as the phase of a sine wave. Create two equations: 1) one that gives you the phase of a 1/60Hz (i.e., revolutions per second) sine wave as a function of time, and 2) one that gives you the phase of a 1/3600Hz (i.e., revolutions per hour) sine wave as a function of time. Now find when the two equations are equal. I'm probably over-thinking this and there probably exists a more elegant solution.

[MonBK]

Is this a trick question?

[Shore Guy]

No

[MonBK]

Originally Posted by Shore Guy

No

Then the first time it happens would be 1.05

[Shore Guy]

At 1:05, the hour hand is no longer pointed at 1. I will no longer comment on incorrect replies.

[MonBK]

Originally Posted by shore guy

at 1:05, the hour hand is no longer pointed at 1. I will no longer comment on incorrect replies.

1.06

[kilyung]

Originally Posted by Shore Guy

The question is:

Determine between 12 noon and 12 midnight, the EXACT times that the minute and hour hands coincide. Express the answers as hours and fractions of hours beyond 12 noon.

12/11 * hour {where hour is 0 to 11}

[Shore Guy]

Kilyung,

You solved it! That was fast.

For those that need an explanation, there are 11 times that the hands coincide in the 12 hour period, not counting the initial position but counting 12 midnight. Moreover the duration between successive coinciding is constant and therefore is 12/11.


Kilyung, you rule!

[AK797]

Originally Posted by Shore Guy

Kilyung,

You solved it! That was fast.

For those that need an explanation, there are 11 times that the hands coincide in the 12 hour period, not counting the initial position but counting 12 midnight. Moreover the duration between successive coinciding is constant and therefore is 12/11.


Kilyung, you rule!

This is why he has so many gold watches!

[ocwatching]

I knew that Caveman is a smart dude!

Originally Posted by Shore Guy

Kilyung,

You solved it! That was fast.

For those that need an explanation, there are 11 times that the hands coincide in the 12 hour period, not counting the initial position but counting 12 midnight. Moreover the duration between successive coinciding is constant and therefore is 12/11.


Kilyung, you rule!

[brandrea]

My head hurts

[MonBK]

Originally Posted by brandrea

My head hurts

Mine as well, I'm going back to the "let's insult NJ people" thread.

[REFZ]

In 24 hour scale:
12:00:00
13:05:27
14:10:54
15:16:21
16:21:49
17:27:16
18:32:43
19:38:10
20:43:38
21:49:05
22:54:32
24:00:00

[Brian Page]

12 times

[omx5o]

Every 3600 + (3600 / 11) = 3927.272727273 seconds.

[Brian Page]

Originally Posted by omx5o

Every 3600 / 11 = 327.272727273 seconds.

I stand corrected 11 -

[omx5o]

Originally Posted by Brian Page

I stand corrected 11 -

I just edited it to:

Every 3600 + (3600 / 11) = 3927.272727273 seconds, or 1:05:27.27272727 recurring.

Probably still wrong, but I'm late for bed.

[Shore Guy]

REFZ,

Appreciate the visual representation. However as I am sure you are aware, the tabulated numbers in your table are rounded and not EXACT.

[watchwatcher]

Yeah...I was going to say that... Nice job, Mike.

[Shore Guy]

omx5o,

Answer was to be in hours and fractions thereof. You did more work to put it into hours, minutes, and seconds.

Your answer however was correct.