| 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 | ||
| ||
StarCraft 2 StarCraft: Brood War Britney Dota 2  62550Calm  6450Sea  3249Jaedong 2267Horang2 1502Mini 482Hyuk 477Soma 377Stork 368BeSt 362[ Show more ] Counter-Strike Super Smash Bros Other Games singsing1819 B2W.Neo710 hiko296 Mlord201 DeMusliM198 Pyrionflax172 XaKoH  171KnowMe130 Trikslyr130 RotterdaM53 QueenE52 Liquid`VortiX41 Liquid`LucifroN19 ZerO(Twitch)11 NotJumperer2 |
|
Replay Cast
The PondCast
KCM Race Survival
WardiTV Map Contest Tou…
Gerald vs herO
Clem vs Cure
ByuN vs Solar
Rogue vs MaxPax
ShoWTimE vs TBD
OSC
CranKy Ducklings
Escore
RSL Revival
Replay Cast
WardiTV Map Contest Tou…
[ Show More ] |
|
|
EPT
