Circles on a string

[Genady]

Here are n circles of radius r arranged in a closed string. The centers of the adjacent circles are connected with straight lines. What is the difference between the orange and the blue areas?

 

[Sensei]

7 hours ago, Genady said:

What is the difference between the orange and the blue areas?

Wavelength of light of orange and blue photons? 😛

 

[Genady]

51 minutes ago, Sensei said:

Wavelength of light of orange and blue photons? 😛

 

Well, that too.

[TheVat]

Spoiler

Perhaps one can imagine a single circle that travels all around the string loop.  The relative color areas would tend towards equality.

 

[Sensei]

2 hours ago, Genady said:

Well, that too.

A good quiz question cannot have non-ambiguous answers.

 

[joigus]

Spoiler

If your problem is well defined (that is, it does not depend on particular angles, whether it's a regular or irregular polygon, etc, it should be solved by exploration with some simple cases.

I've tried square and rectangle with n=4,6,8 plus general n-gon and I get that the difference of areas is always 2A, where A is the area of one circle.

Proving rigorously that it should be so for more general configurations (as well as independent of n) could be trickier.

 

[Genady]

13 minutes ago, joigus said:

  Reveal hidden contents

If your problem is well defined (that is, it does not depend on particular angles, whether it's a regular or irregular polygon, etc, it should be solved by exploration with some simple cases.

I've tried square and rectangle with n=4,6,8 plus general n-gon and I get that the difference of areas is always 2A, where A is the area of one circle.

Proving rigorously that it should be so for more general configurations (as well as independent of n) could be trickier.

 

Spoiler

The answer is correct. The general proof is possible using school level geometry (my school anyway.) Look at n=3 and you will see how it comes out.

 

38 minutes ago, TheVat said:

  Reveal hidden contents

Perhaps one can imagine a single circle that travels all around the string loop.  The relative color areas would tend towards equality.

 

Spoiler

Try it with 3 circles.

 

[Genady]

When you guys solve the OP, consider this modification. The circles do not necessarily fill the string, but may have gaps, like this, for example:

[Genady]

PS. A thought just crossed my mind, that the question and the answer work perfectly well even for a "string" with only two circles on it.

[Genady]

Hint:

Spoiler

Formula for sum of interior angles of polygon.

 

[sethoflagos]

Spoiler

OP

The straight lines form an n-sided polygon for which the sum of external angles must equal 2pi radians (axiomatic)

If the external angle formed at the centre of a circle is A radians, the excess area of orange over blue is (2A/2pi)*(pi.r^2) = Ar^2 (NB A can be negative)

By first axiom, sum of all n values of Ar^2 = 2pi.r*2

 

Subsidiary....

I see no difference from the OP. The lengths of the sides of the polygon are irrelevant. The proof is the same (isn't it?)

 

Edited September 10, 2023 by sethoflagos

[Genady]

Spoiler

@sethoflagos: "... n-sided polygon for which the sum of external angles must equal 2pi radians ..."

But sum of exterior angles of a square, for example, is 6pi ...

 

Edited September 10, 2023 by Genady

[sethoflagos]

1 hour ago, Genady said:

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@sethoflagos: "... n-sided polygon for which the sum of external angles must equal 2pi radians ..."

But sum of exterior angles of a square, for example, is 6pi ...

 

Spoiler

Maybe this is a language thing.

For me 'external angle' is defined as in https://www.varsitytutors.com/hotmath/hotmath_help/topics/polygon-exterior-angle-sum-theorem

The exterior angles (by this understanding) of a square sum to 2pi radians, not 6.

 

Is there some fine distinction between 'exterior' and 'external'? I'm 64. Tell me I'm fundamentally mistaken 🤪.

 

Edited September 10, 2023 by sethoflagos
balance

[Genady]

6 minutes ago, sethoflagos said:

  Reveal hidden contents

Maybe this is a language thing.

For me 'external angle' is defined as in https://www.varsitytutors.com/hotmath/hotmath_help/topics/polygon-exterior-angle-sum-theorem

The exterior angles (by this understanding) of a square sum to 2pi radians, not 6.

 

I see. Yes, it is not what I thought it is. Sorry for that.

Now I understand your answer. It is correct! +1