Can you draw 3 overlapping circles (Venn diagram) such that all 7 of the formed regions have the same area?
For the case of two circles and 3 equal regions I found this answer:
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asked Sep 18, 2019 at 5:18
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1$\begingroup$ My intuition tells me that this isn't possible, but I dunno how to prove it this late at night. Great concept, regardless! $\endgroup$– Brandon_JSep 18, 2019 at 5:46
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2$\begingroup$ Well, if you remove the restriction that the 3 areas be circles, it's certainly possible. There's no requirement that a Venn diagram be made only of circles, that's just the easiest and classic example. $\endgroup$– Darrel HoffmanSep 18, 2019 at 13:57
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$\begingroup$ @DarrelHoffman While there may be thousands of solutions, it might still be interesting to find a shape that can do that. $\endgroup$– StrawberrySep 18, 2019 at 16:34
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1$\begingroup$ @Strawberry (Irregular) pentagons will do this easily, for example. E.g. let all intersected regions (for 2 or all 3 areas) be equilateral triangles with side 1 (and height $\frac{\sqrt3}{2}$), and the remaining areas (belonging to only 1 area) be (isosceles) triangles with base 2 and height $\frac{\sqrt3}{4}$. $\endgroup$– trolley813Sep 18, 2019 at 18:34
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$\begingroup$ @trolley813 Oh, I see what you mean now. $\endgroup$– StrawberrySep 19, 2019 at 10:34
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1 Answer
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Well, that's impossible.
Proof:
Let $A$ be an area of any of the 7 regions. Now, let us consider only 2 of the overlapping circles (removing 3rd circle for a while) and notice that their overlapping area is $2A$, and each of the non-overlapping parts (belonging to one of the circles, but not both) should also have an area of $2A$. So, the only way to arrange this circles (it will be true for any pair of the circles) is as in the linked question (the distance between their centers must be $2x$, where $x\approx0.403972$). So, the centers of the circles must form a right triangle with side $2x$. Plotting this gives the following graph:
Now it's clear that the regions have unequal area, even without any calculations (for example, that's because the bottom side of the grey curved triangle, where all 3 circles overlap, lies well below the x-axis, but the intersection points of red and green circles have the same y-coordinates, since their centers lie on the x-axis, so the grey area must be definitely greater then the brown one).
Python code for plotting: Try it online! (unfortunately it will not run there, since tio.run does not support external packages, likematplotlib).
answered Sep 18, 2019 at 6:20
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1$\begingroup$ Great work! I wasn't sure if this had a solution. $\endgroup$ Sep 18, 2019 at 6:32
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1$\begingroup$ @DmitryKamenetsky Thanks! I've added a link to Python code used for plotting. $\endgroup$ Sep 18, 2019 at 6:39
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1$\begingroup$ Got to be honest @DmitryKamenetsky - when I see the OP say "I wasn't sure if this had a solution" it makes me very wary about attempting to solve their future puzzles... $\endgroup$– StivSep 18, 2019 at 6:59
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1$\begingroup$ @Stiv Hmmm... ok I will remember this for my future puzzles and try to make sure they have a solution. For me proving that a puzzle doesn't have a solution is also just as interesting. $\endgroup$ Sep 18, 2019 at 7:33
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3$\begingroup$ @DmitryKamenetsky I totally agree with you - as a researcher myself a negative finding is often just as useful as (if not more important than) a positive result :) However, in my opinion being 'not sure' if there is a solution to a puzzle is different to already knowing that there isn't one. When that's the case it seems to become more of an exercise in mathematical proof and I wonder whether maybe this type of question would be more appropiate on the Math SE rather than Puzzling... Just my opinion! :) $\endgroup$– StivSep 18, 2019 at 8:05