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You want to draw a circle with a 4 inch radius. A trivial task for you and your trusty compass. When you go to grab your compass which has not had much love for a while you find it is rusted shut; stuck at 5 inches. Is it still possible to complete the task and draw a perfect circle with a 4-inch radius using the compass that can only draw circles with a 5-inch radius?

You may use other things as well to solve the problem such as a straightedge.

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you can if you have infinite time you can construct a segment of 4 inch distance, but not draw a circle of 4 inch so draw a circle of 5 inch and then in every direction draw a segment of four –  Willemien Feb 5 at 23:49
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No, you already answered the question for yourself. If it could draw a circle with a 4-inch radius, then it would not be a compass "that can only draw circles with a 5-inch radius", would it? –  Karl Kronenfeld Feb 6 at 0:21

1 Answer 1

up vote 17 down vote accepted

My first approach

Yes, just place the center 3 inches above the paper. If that is a possibility?

Different rendering

To put it differently, if you like, you could draw a 5-inch-circle, use scissors to cut a radius, then form a cone by overlapping at the cutting line until the proportion of the height to the radius of the base is 3 to 4...

That will happen when the overlap covers $\frac{1}{4}$ of the surface of the cone...

Illustrations

To fully earn your votes, here is an illustration:

enter image description here

To see the situation from arbitrary angles, consult the following dynamic 3D-graph:

GeoGebra-illustration

In particular, see what it looks like from above - the $1:4$ proportion of the overlap becomes evident!

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Yes that is one solution. You can use a 3 inch block to raise the center. –  David Caliri Feb 5 at 23:54
    
I think the initial one-line solution was perfect because it was also practical. IMHO, no need to add anything... –  Vadim Feb 6 at 0:00
    
@Vadim: You are quite right, although my last edit suggests a way to measure the exact result by eye, if I am not mistaken :) –  String Feb 6 at 0:02
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Very nice. Yes I agree the first answer wins in practicality, but the second is more imaginative. I asked this question to open a talk I gave and the only answer I got was WD-40. –  David Caliri Feb 6 at 0:14
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@DavidCaliri: I added an illustration of the cone-solution. Be sure to visit the link adress to see the dynamic version of it! –  String Feb 6 at 1:11

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