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#1 Posted 7:36pm 07-02-09 Open with smile!There are 6 points in a plane such that no 3 are collinear. Each point is joined to another by a coloured line which is either blue or red. Find the minimum no. of monochromatic triangles.
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#2 Posted 7:37pm 07-02-09 Re: Open with smile!plz post ur solution. i dont hav d answer
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#3 Posted 11:01pm 07-02-09 Re: Open with smile!no one????????????????????
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#4 Posted 11:09pm 07-02-09 Re: Open with smile!a wild guess 19....more chances of being wrong....
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#5 Posted 11:11pm 07-02-09 Re: Open with smile!tell me i'm right or wrong...
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#6 Posted 11:11pm 07-02-09 Re: Open with smile!19 or 38.....
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#7 Posted 11:14pm 07-02-09 Re: Open with smile!let's wait for othersl....
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#8 Posted 11:37pm 07-02-09 Re: Open with smile!sry prateek... i dunno the ans....i wildly guessed 20..but not sure abt it
Plan ur work and work out ur plan! Edited on 11:38pm 07-02-09 |
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#9 Posted 09:00am 08-02-09 Re: Open with smile!maybe 35 my method minimum monochromaticity wil be when out of [p]6[/p]C[ss]2[/ss] total lines 8are of one colour and 7 are of another colour and from the 7 same colour lines [p]7[/p]C[ss]3[/ss] triangles can be made please tell me ur views on this soln
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#10 Posted 0:24pm 08-02-09 Re: Open with smile![image]9576076.jpg[/image] yup there are [b]two monochromatic triangles[/b]
http://abhirupsarkar.webs.com/ Edited on 0:05pm 09-02-09 |
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#11 Posted 0:55pm 08-02-09 Re: Open with smile!wat u mean by monochromatic triangles |
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#12 Posted 1:01pm 08-02-09 Re: Open with smile!triangles having three sides of same colour
http://abhirupsarkar.webs.com/ |
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#13 Posted 1:06pm 08-02-09 Re: Open with smile!i agree wid abhirup "1" |
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#14 Posted 2:45pm 08-02-09 Re: Open with smile!According to me the answer should be two. I would argue this way: Start with a situation where all the lines are coloured red. WLOG we can consider a hexagon. Call a monochromatic triangle [i]good[/i], and [i]bad[/i] otherwise. Then, to begin with, we have 20 good triangles. Now we change the colours of the lines one by one and see how many triangles turn bad . First if we take two opposite bases, we turn for each of the bases, 4 triangles bad. So, now we have only 12 good triangles. This way as each of the bases turn colour, a total of 18 triangles turn bad with one side in each such triangle of the wrong colour. Now only two triangles are left and changing the colour of one of the lines does not change the number of good triangles. This means we reached the case of minimum number of good triangles. Hence the minimum is [b]2[/b] (In abhirup's case we have one red and one blue triangle making that two good triangles) [The assumption here is that we are considering only triangles formed by the six vertices]
Edited on 2:47pm 08-02-09 |
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#15 Posted 6:19pm 08-02-09 Re: Open with smile!in abhirups figure i can see more than 2 same coloured triangle infact many of them in the central blue region
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#16 Posted 6:27pm 08-02-09 Re: Open with smile!no subhash you dont know which of those lines do really intersect in space
The map is not the territory. Edited on 6:30pm 08-02-09 |
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#17 Posted 6:30pm 08-02-09 Re: Open with smile!hey is answer 1....
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#18 Posted 6:30pm 08-02-09 Re: Open with smile!no phil question was There are 6 points[b] in a plane [/b]such that no 3 are collinear. Each point is joined to another by a coloured line which is either blue or red. Find the minimum no. of monochromatic triangles. so it is one plane so no need of meeting in space it is one plane this from prophet sir's solution [b]The assumption here is that we are considering only triangles formed by the six vertices[/b] but is this given in the question
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#19 Posted 6:31pm 08-02-09 Re: Open with smile!but in abhirups post there are only 2 monochromatic Δs one red and one blue maybe i am missing some of your Δs
The map is not the territory. Edited on 6:32pm 08-02-09 |
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#20 Posted 6:58pm 08-02-09 Re: Open with smile!how can u miss so many triangles because there are somany blue lines there are many blue triangles within the outer hexagon
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#21 Posted 02:03am 09-02-09 Re: Open with smile!yes so there is only one blue Δ that can be formed using the points on hexagon i don't know how you are seeing many Δ's
The map is not the territory. |
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#22 Posted 5:25pm 09-02-09 Re: Open with smile![image]9680496.jpg[/image]
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#23 Posted 5:25pm 09-02-09 Re: Open with smile!checkk thetriangle that ive shaded that is also monochromatic isnt it
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#24 Posted 5:31pm 09-02-09 Re: Open with smile![11] but we have to take Δs formed by the points on the hexagon only !! see prophet sir's post wat he's written in the end i mean the minimum will be found using that assumption
The map is not the territory. Edited on 5:32pm 09-02-09 |
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#25 Posted 5:35pm 09-02-09 Re: Open with smile!ya got it now philip but the figure and the lines are 2D there is no question of 3D "There are 6 points [b]in a plane[/b]" this is from the question so it is 2d am i wrong here
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#26 Posted 6:56pm 09-02-09 Re: Open with smile!I made the assumption that the triangles are those formed by the vertices as otherwise the question becomes too complicated to handle. If, instead (as subash indicates) you want to consider triangles formed from all possible intersections, the first hurdle is : just how many triangles are formed. If the points form a regular hexagon you have a number of pairs of lines that dont interect. In the general case, too its difficult too predict how many triangles are there Thats why if you limit it to just the triangles with these six points as vertices, you are assured of 20 triangles. What is more, no matter what their orientation, I can shuffle the points around to form a hexagon without altering the number of triangles (hence the [b]WLOG[/b]). This simplification is what made me use that assumption. Otherwise you will be on a wild goose chase.
Edited on 6:56pm 09-02-09 |
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#27 Posted 11:00pm 09-02-09 Re: Open with smile!yup.....we have to make the assumption that the triangles are those formed by the points as vertices
http://abhirupsarkar.webs.com/ |
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#28 Posted 00:35am 11-02-09 Re: Open with smile!thanx evry1!
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#29 Posted 00:37am 11-02-09 Re: Open with smile!so finally wats the ans....
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#30 Posted 00:41am 11-02-09 Re: Open with smile!it is 2...as explained by prophet sir
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#31 Posted 00:44am 11-02-09 Re: Open with smile!2 or 20....
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#32 Posted 00:48am 11-02-09 Re: Open with smile!2 coz we r considering only those which formed from vertices of the hexagon
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#33 Posted 09:29am 11-02-09 Re: Open with smile!Ppl are awaiting one clarification from you, aditya. The triangles in question are just the ones formed from the 6 points as vertices isnt it? |
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#34 Posted 09:35am 11-02-09 Re: Open with smile!yes.......initially i thought all the possible triangles, but then it wud b much difficult 2 find the ans.....so, only the triangles which r formed by vertices.
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