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**plutoman****Member**- Registered: 2008-01-06
- Posts: 27

Here's a clock-related puzzle I found out about yesterday:

Imagine a two-handed analogue clock (hour and minute hands) where the two hands are indistinguishable, i.e. they are the same size, shape, colour, depth and so on.

During any 12-hour period, at how many different times is the time shown by the clock ambiguous?

NB: 12 o'clock doesn't count.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

Is the answer, sort of guessing a little bit, is it

ooops, times that answer by two because of AM and PM.**igloo** **myrtilles** **fourmis**

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**plutoman****Member**- Registered: 2008-01-06
- Posts: 27

Nice try, but no.

Take 3 o'clock say. On a clock where the two hands are identical, 3 o'clock will look like 12:15, but not quite because at 12:15 the hour hand has moved a bit beyond the vertical. The trick is to find pairs of times where the hands *are* in exactly the same positions.

I hope that helps. Note also that I said a 12-hour period, so we don't need to worry about AM and PM.

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**studymaths****Member**- Registered: 2010-04-11
- Posts: 32

If the hands are indistinguishable isn't the time is always ambiguous?

Except when they line up exactly which will be 11 times (once per hour) if 12 o'clock doesn't count.

So my answer is 3589 out of a possible 3600 seconds are ambiguous.

I'm bound to be missing something, I'm never any good at puzzles!

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**plutoman****Member**- Registered: 2008-01-06
- Posts: 27

studymaths wrote:

If the hands are indistinguishable isn't the time is always ambiguous?

No. At 3 o'clock, the minute hand is pointing vertically (on a bearing of 0° if you like) while the hour hand is pointing due east (bearing of 90°). There is no corresponding time where the hour hand is at 0° and the minute hand is at 90° (12:15 doesn't quite work because the hour hand has moved beyond 0°). So if one hand is pointing due north and the other is due east, you know it must be 3 o'clock.

However there **are** times when you can't tell.

Good luck with the GCSE teaching by the way - can't be an easy task for you. I did my GCSEs in 1996.

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**studymaths****Member**- Registered: 2010-04-11
- Posts: 32

Ah, I knew I was missing something!

I fear the answer is beyond my brain power but will have a think about it. I don't think 11:08pm on a Saturday night is the best time for my brain to be answering maths problems!

Is that an ambiguous time by the way?

*Last edited by studymaths (2011-05-21 10:08:56)*

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**Avon****Member**- Registered: 2007-06-28
- Posts: 80

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**plutoman****Member**- Registered: 2008-01-06
- Posts: 27

Avon, you're pretty close. Would you mind explaining how you got that?

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**studymaths****Member**- Registered: 2010-04-11
- Posts: 32

Plutoman, click on the 132 box to get the explanation. I think I understand it!

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**plutoman****Member**- Registered: 2008-01-06
- Posts: 27

Silly me. For two reasons. Firstly for not clicking on the spoiler box. And

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