Genady Posted September 9, 2023 Share Posted September 9, 2023 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? Link to comment Share on other sites More sharing options...
Sensei Posted September 9, 2023 Share Posted September 9, 2023 7 hours ago, Genady said: What is the difference between the orange and the blue areas? Wavelength of light of orange and blue photons? 😛 Link to comment Share on other sites More sharing options...
Genady Posted September 9, 2023 Author Share Posted September 9, 2023 51 minutes ago, Sensei said: Wavelength of light of orange and blue photons? 😛 Well, that too. Link to comment Share on other sites More sharing options...
TheVat Posted September 9, 2023 Share Posted September 9, 2023 Spoiler Perhaps one can imagine a single circle that travels all around the string loop. The relative color areas would tend towards equality. Link to comment Share on other sites More sharing options...
Sensei Posted September 9, 2023 Share Posted September 9, 2023 2 hours ago, Genady said: Well, that too. A good quiz question cannot have non-ambiguous answers. Link to comment Share on other sites More sharing options...
joigus Posted September 9, 2023 Share Posted September 9, 2023 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. 1 Link to comment Share on other sites More sharing options...
Genady Posted September 9, 2023 Author Share Posted September 9, 2023 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. Link to comment Share on other sites More sharing options...
Genady Posted September 9, 2023 Author Share Posted September 9, 2023 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: Link to comment Share on other sites More sharing options...
Genady Posted September 9, 2023 Author Share Posted September 9, 2023 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. Link to comment Share on other sites More sharing options...
Genady Posted September 10, 2023 Author Share Posted September 10, 2023 Hint: Spoiler Formula for sum of interior angles of polygon. Link to comment Share on other sites More sharing options...
sethoflagos Posted September 10, 2023 Share Posted September 10, 2023 (edited) 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 1 Link to comment Share on other sites More sharing options...
Genady Posted September 10, 2023 Author Share Posted September 10, 2023 (edited) 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 Link to comment Share on other sites More sharing options...
sethoflagos Posted September 10, 2023 Share Posted September 10, 2023 (edited) 1 hour ago, Genady said: Hide contents @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 Link to comment Share on other sites More sharing options...
Genady Posted September 10, 2023 Author Share Posted September 10, 2023 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 Link to comment Share on other sites More sharing options...
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