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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

Inspired by http://www.mathisfunforum.com/viewtopic.php?id=18730.

**Q1 (easy):** How many times in a 12-hour period do the hour and minute hands point in opposite directions?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Hi;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

**Q2 (trickier):** Let *θ* be an arbitrary angle greater than 0° and smaller than 180°. How many times in a 12-hour period do the two hands of a clock make an angle of *θ* between themselves?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Hi scientia;

I will work on that one as soon as I get done with the easy stuff first.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

Take your time.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Hi;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,347

Hi;

Happy New Year!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**OzMark****Member**- Registered: 2013-06-10
- Posts: 9

Q1, refer also also

http://www.mathsisfun.com/puzzles/a-que … -time.html

The question asks for the distance from the bottom of the clock face at 6 o'clock to the tip of the clock hands, what they mean is when do both the hands make an angle to a vertical line on the clock face through 6 o'clock.

The answers for when the hands are at either side of the clock face are ;

12:55 and 23sec, 1:50 46s, 2:46 9s, 3:41 32s, 4:36 55s, 5:32 18s, 6:27 42s, 7:23 5s, 8:18 28s, 9:13 51 s, 10:09 14s & 11:04 37s

There are 11 more solutions when the minute and hour hand are directly on top of each other ;

12:00 00s, 1:05 27s, 2:10 55s, 3:16 22s, 4:21 49s, 5:27 16s, 6:32 44s, 7:38 11s, 8:43 38s, 9:49 5s & 10:54 33s

Also we need to assume the length of the minute and hour hands is the same, or else we cant solve the question at all.

The solution given only cites one of these solutions 8:18 28s, I suppose that is made with reference to the picture.

I can make an explanation of the equations I used if I get a reply to this post, I wasnt going to be bothered if noone reads these.

Have Fun !

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,336

hi OzMark

Welcome to the forum.

I've not tried this puzzle but it looks like you've discovered a bug; maybe the wording should be changed ... perhaps indicating that equal angles are wanted rather than distances. I'll raise it with the page creator, MathsIsFun, himself.

Hope you enjoy your membership. Looking forward to your contributions.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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