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wu :: forums    riddles    medium (Moderators: ThudnBlunder, SMQ, Icarus, Eigenray, Grimbal, william wu, towr)    Missing Vertex « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Missing Vertex  (Read 575 times) ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Missing Vertex   « on: Mar 23rd, 2009, 4:24pm » Quote Modify The sum of the face angles at all but one of the vertices of a given simple polyhedron is 5160o. What is the sum of the face angles at the missing vertex? IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: Missing Vertex   « Reply #1 on: Mar 24th, 2009, 2:50am » Quote Modify polyhedral formula, V + F - E = 2 number of angles, #A = 2E   sum(180 - An) = F*360 #A*180 - sum(An) =  F*360 E*360 - sum(An) =  F*360   E*360 -  F*360 = sum(An) V*360 = sum(An) + 720 sum(An) = 0 mod 360 5160 = -240 mod 360   So the remaining angles sum to 240 degrees (as it has to be between 0 and 360). IP Logged Wikipedia, Google, Mathworld, Integer sequence DB JohanC Senior Riddler     Posts: 460 Re: Missing Vertex   « Reply #2 on: Mar 24th, 2009, 4:28am » Quote Modify on Mar 23rd, 2009, 4:24pm, ThudanBlunder wrote: .... of a given simple polyhedron ...   What's your definition of a simple polyhedron? If the polyhedron isn't convex, Towr's conclusion is only valid modulo 360, I think. IP Logged ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Re: Missing Vertex   « Reply #3 on: Mar 24th, 2009, 6:44am » Quote Modify on Mar 24th, 2009, 4:28am, JohanC wrote:   What's your definition of a simple polyhedron? If the polyhedron isn't convex, Towr's conclusion is only valid modulo 360, I think. 'Simple' means topologically equivalent to a sphere.   IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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