19 replies to this topic

[TimeSpaceLightForce]

A peculiar Grandfather's clock has both its hour and minute hands 

running in proper rates. Starting at 12 o'clock both runs clockwise.

But the hour hand stops when the minute hand passes over it. 

And the minute hand turn its direction back when it is over the stopped 

hour hand causing the hour hand to run clockwise again.

 

How long will it take for both hands to be on 12 o'clock again?

[BobbyGo]

 

A peculiar Grandfather's clock has both its hour and minute hands 

running in proper rates. Starting at 12 o'clock both runs clockwise.

But the hour hand stops when the minute hand passes over it. 

And the minute hand turn its direction back when it is over the stopped 

hour hand causing the hour hand to run clockwise again.

gfc.jpg

 

How long will it take for both hands to be on 12 o'clock again?

Spoiler for Possibly...

24 hours

[Rob_Gandy]

If I'm reading this right...

Spoiler for I'd say

23 hours. It would normally take 12 hours to get back to 12 o'clock, but the minute hand does one extra revolution each hour 1-11 each taking an hour for 11 more hours.

[CaptainEd]

I see the clock reads 10:12. Do we start now? Or do we start at 12:00?

 

Also, I'm a little unclear what happens when the minute hand and hour hand meet: (When you said the minute hand "passes over" the hour hand, I assume that the minute hand continues in its direction.)

 

* minute hand going clockwise (CW), hits moving hour hand--hour hand stops, minute hand continues CW

* minute hand going CW, hits stopped hour hand--hour hand starts, minute hand goes Counterclockwise (CCW)

* minute hand going CCW, hist moving hour hand--hour hand stops, minute hand continues CCW

* minute hand going CCW, hits stopped hour hand--hour hand starts, minute hand goes CW.

Edited by CaptainEd, 20 February 2013 - 09:14 PM.

[bonanova]

I see the clock reads 10:12. Do we start now? Or do we start at 12:00?

 

Also, I'm a little unclear what happens when the minute hand and hour hand meet: (When you said the minute hand "passes over" the hour hand, I assume that the minute hand continues in its direction.)

 

* minute hand going clockwise (CW), hits moving hour hand--hour hand stops, minute hand continues CW

* minute hand going CW, hits stopped hour hand--hour hand starts, minute hand goes Counterclockwise (CCW)

* minute hand going CCW, hist moving hour hand--hour hand stops, minute hand continues CCW

* minute hand going CCW, hits stopped hour hand--hour hand starts, minute hand goes CW.

 

That is the way I read the OP Cap'n.

 

But it may be the case that a forward-moving minute hand stops the hour hand, while a backward-moving minute hand starts the hour hand again. I think one of these interpretations precludes and answer, and that might make the OP as it stands a double puzzle.

[CaptainEd]

Tricky!

Spoiler for answer and explanation

Spoiler for answer

from the pictured position, it will take 2:38 more before both hands are pointing at 12. However, the minute hand will still be moving CCW. In one hour more, they will once again be pointing at 12, and both hands will be moving CW.

Spoiler for analysis

When both hands are together and moving CW,

it takes 1 hour 5 minutes + epsilon to meet again, when HH stops, MH continues CW.
One hour later, MH meets again and reverses direction (to CCW), while HH starts again.
55 minutes - epsilon later, they meet again, HH stops, MH continues CCW
one hour later, MH meets again and reverses direction to CW, while HH starts again.

 

In other words, in exactly 4 hours, the two hands moved exactly one sixth of the circle (that is, they are both pointing exactly at the number 2 more than last time.

 

So it would take 24 hours for the hands to get together at 12 with both hands moving CW. But only 23 hours for them to get together at 12, because that time, MH is still going CCW, and HH stops.

 

So, from a standing start--both hands at 12, both moving CW, it takes 23 hours to get back to both hands at 12 , 24 hours to do so with both hands moving CW. I agree with Rob's numbers.

BobbyGo may have envisioned the full cycle, but missed that last backward run. 
 

If we are to tell how long from when the clock has HH past 10, MH at 12, we notice several things:
* HH must be moving--it only stops at a little after the odd hours, and exactly on the even hours; it's moving at other times
* MH must be moving CW--same condition
* Therefore it will be (3 hours - 22 minutes) before the hands are together at 12 and 12 (and that will be the 23 hour situation).

Edit: I agree, Bonanova, I'm not totally clear on the mechanism. A third interpretation might be that the MH reverses whenever it hits the HH. I'll take one of Y-san's Excedrins...

Edited by CaptainEd, 20 February 2013 - 10:02 PM.

[TimeSpaceLightForce]

@captainEd, the pic is not the start, both @ 12 o'clock is the start where the hands are moving clockwise, eventually the MH overtakes the HH causing it to stop but the MH keeps going until it reached the stopped HH again where then it reverses its direction (like bumping) while causing the HH to move again(resume running clockwise) . No double interpretations is intended here.

[TimeSpaceLightForce]

If I'm reading this right...

Spoiler for I'd say

23 hours. It would normally take 12 hours to get back to 12 o'clock, but the minute hand does one extra revolution each hour 1-11 each taking an hour for 11 more hours.Both hands should point at 12

[CaptainEd]

@captainEd, the pic is not the start, both @ 12 o'clock is the start where the hands are moving clockwise, eventually the MH overtakes the HH causing it to stop but the MH keeps going until it reached the stopped HH again where then it reverses its direction (like bumping) while causing the HH to move again(resume running clockwise) . No double interpretations is intended here.

OK, TSLF, thanks.

Spoiler for from both up to both up

23 hours.

At that moment, MH is moving counterclockwise. One hour later, MH meets again at (HH=12, MH=12). Analysis as given in my answer above

[TimeSpaceLightForce]

Spoiler for

There is a pattern here..