## Find the length of the string

HansV
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Location: Wageningen, The Netherlands

### Find the length of the string

In today's The Guardian: Can you solve it? Are you in the smartest 10 per cent?
A string is wound symmetrically around a circular rod. The string goes exactly four times around the rod. The circumference of the rod is 4cm and its length is 12cm. Find the length of the string.
String.jpg
Note: only basic mathematics is needed.
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Regards,
Hans

Rudi
gamma jay
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Location: Cape Town

### Re: Find the length of the string

Tip: If you go to the loo and unwind the inner cylinder of a standard toilet paper roll, and add 4 cm to its length, you'll almost be there.
(Now when the lounge becomes quiet, we'll all know where the members are!!)
Regards,
Rudi

HansV
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Joined: 16 Jan 2010, 00:14
Status: Microsoft MVP
Location: Wageningen, The Netherlands

### Re: Find the length of the string

And now for the real answer...
Regards,
Hans

StuartR
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Joined: 16 Jan 2010, 15:49
Location: London, Europe

### Re: Find the length of the string

Spoiler
If you cut the roll lengthwise then you will see the thread making lines across a rectangle. There are 4 lines, each of which is the hypotenuse of a 4x3x5 right angle triangle, so the total length is 5cm * 4 = 20cm
Roll.png
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StuartR

HansV
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Joined: 16 Jan 2010, 00:14
Status: Microsoft MVP
Location: Wageningen, The Netherlands

### Re: Find the length of the string

That was my solution too.
Regards,
Hans

StuartR
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Joined: 16 Jan 2010, 15:49
Location: London, Europe

### Re: Find the length of the string

I added a diagram to make it more clear
StuartR

Rudi
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Location: Cape Town

### Re: Find the length of the string

You guys and your math(s)!! Way to clever for me!
Regards,
Rudi

PaulB
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Location: Ottawa ON

### Re: Find the length of the string

Very clever!
Regards,
Paul

Today's earworm

Rudi
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Joined: 17 Mar 2010, 17:33
Location: Cape Town

### Re: Find the length of the string

Question:
Spoiler
Out of interest, if you could not cut the roll to make it flat, would there be a clever way to work it out using pi or circumference or something like that, even though it might be a bit more complex than the request for a basic solution?

Some interesting formulas here:
- Lenth of line around an object
- Calculating Total Length of Wraps Around A Cylinder
Regards,
Rudi

HansV
Posts: 75279
Joined: 16 Jan 2010, 00:14
Status: Microsoft MVP
Location: Wageningen, The Netherlands

### Re: Find the length of the string

Spoiler
I don't think a more complicated calculation would help.

You don't even have to cut the cylinder open. In your imagination, apply some ink to the string, then roll the cylinder over a sheet of paper. You'll basically get Stuart's diagram. After four complete revolutions, you'll have a straight line representing the entire string.
S1681.png
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Regards,
Hans

AlanMiller
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Location: Melbourne, Australia

### Re: Find the length of the string

Spoiler
As soon as I saw the 4 (circumference) and the 12cm/4 turns = 3, I knew a 3-4-5 triangle would appear somewhere. Sure enough it did.
Alan