Circle Division Solution
An explanation of a neat circle puzzle involving combinatorics, graphs, Euler's characteristic formula and pascal's triangle. Music: Wyoming 307 by Time for ...
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In my last video, I posed the following question: If you take n points on a circle, then connect every pair of them with a line, how many sections to these lines cut the circle into? What was strange is that when n is less than 6, and when n is 10 for some reason, the answer is always a power of 2, but for all other values of n, the answer seems completely unrelated powers of 2. What love about this problem is that it brings together many disparate concept: counting functions, graphs, one of Euler's famous equations, and Pascal's triangle. You might be wondering if changing the placement of points change the number of sections It actually can! For instance, watch how this small region in the middle disappears if we adjust things so that three lines go through the same point. B...
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In my last video, I posed the following question: If you take n points on a circle, then connect every pair of them with a line, how many sections to these lines cut the circle into? What was strange is that when n is less than 6, and when n is 10 for some reason, the answer is always a power of 2, but for all other values of n, the answer seems completely unrelated powers of 2. What love about this problem is that it brings together many disparate concept: counting functions, graphs, one of Euler's famous equations, and Pascal's triangle. You might be wondering if changing the placement of points change the number of sections It actually can! For instance, watch how this small region in the middle disappears if we adjust things so that three lines go through the same point. B...
