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blankspace
United States292 Posts
Example: For the case n=3, you have a triangle inscribed in the center of the circle. The number of regions is thus 4. For the case n=4, you have a square with diagonals drawn, so that gets you 8. I will tell you though that the answer is not 2^(n-1), but it is not complicated.  | ||
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Oracle
Canada411 Posts
I'll get back to you once i figure out the first part EDIT: I dont think this is possible with my knowledge of graph theory, since # faces is ill-defined for non-planar graphs, and everything K_5 and above is nonplanar... and im stuck  | ||
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Av4st
Canada92 Posts
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Oracle
Canada411 Posts
for n = 5 you have ![[image loading]](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/4-simplex_graph.svg/188px-4-simplex_graph.svg.png) which has 11 faces. Plus the 5 imaginary ones on the outside, so 16 total. 2^4 + 1 = 17 | ||
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blankspace
United States292 Posts
Also note that the number of regions for n=7 is 57 so looking for some sort of 2^something formula is not going to work. | ||
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Marradron
Netherlands1586 Posts
n(n-1) /2 + n Gotta check if its right  nvm these are edges or something not correct. | ||
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Oracle
Canada411 Posts
3 | 3 | 1 + 3 = 4 4 | 6 | 4 + 4 = 8 5 | 10 | 11 + 5 = 16 6 | 15 | 24 + 6 = 30 7 | 21 | 50 + 7 = 57 For anyone else attempting a graph-theory approach | ||
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Complete
United States1864 Posts
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Complete
United States1864 Posts
(x^4-6x^3+23x^2-18x+24)/24 | ||
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blankspace
United States292 Posts
+ Show Spoiler + I think that is the correct answer, though you should realize that you can express it in a more elegant form (namely as the sum of binomial coefficients. What is your reasoning/how did you get the answer? Did you assume it was a polynomial and interpolated? | ||
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n.DieJokes
United States3443 Posts
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LaLuSh
Sweden2358 Posts
C(n, 4) + C(n, 2) + 1 Number of lines drawn between n points can be expressed as a series (n-1)+(n-2)+...(n-(n-1)) Or more practical as binomial coefficient: C(n,2). Expressing number of possibilities to connect 2 points in an n point circle by a line. C(n,4) gives the number of possible intersects between any 4 of the n points in the circle. Adding them together and the extra region in the center (+1), gives you the total. + Show Spoiler + Watched the spoilers for help.... | ||
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blankspace
United States292 Posts
+ Show Spoiler + That is the correct answer  Can you justify why your answer is correct though? As in why should counting these intersections give you the number of regions? | ||
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JodoYodo
Canada1772 Posts
Every time a line intersects another line, you bisect two regions into four regions | ||
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