| Blogs > blankspace |
|
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.  | ||
|
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  | ||
|
Av4st
Canada92 Posts
| ||
|
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 | ||
|
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. | ||
|
Marradron
Netherlands1586 Posts
n(n-1) /2 + n Gotta check if its right  nvm these are edges or something not correct. | ||
|
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 | ||
|
Complete
United States1864 Posts
| ||
|
Complete
United States1864 Posts
(x^4-6x^3+23x^2-18x+24)/24 | ||
|
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? | ||
|
n.DieJokes
United States3443 Posts
| ||
|
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.... | ||
|
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? | ||
|
JodoYodo
Canada1772 Posts
Every time a line intersects another line, you bisect two regions into four regions | ||
| ||
|
|
Replay Cast
CrankTV Team League
WardiTV Qualifier
Epic.LAN
Big Brain Bouts
SHIN vs Elazer
Percival vs Nicoract
Reynor vs Lambo
Replay Cast
RSL Revival
Clem vs Lambo
Scarlett vs Cure
CranKy Ducklings
Epic.LAN
IPSL
Dragon vs Hawk
[ Show More ] |
|
|
EPT
