Please help; combination problem

[shezleng]

Homework Statement

there are N points on a paper. any three of points are not colinear.

you draw lines from each point to all other points.lines can be drawn with three colors; red,yellow or blue.

there will be obtained some triangles having colored edges.

purpose is not to get a triangle, edges of which are drawn with the same color.

what can be the value N at maximum?

Homework Equations

the number of triagles from N given points : N*(N-1)/2

The Attempt at a Solution

I can not get any mathematical expression for solution.
I did try with considering all combinations for two color condition and I got N=5 maximum.
for three color my trial was getting complicatd and I could not go on any more. But for N=10
I am sure it is possible to get all triangles satisfying the condition.

 

[nate808]

Thank you for your interesting question. I am a scientist and I would like to help you find the maximum value of N in this scenario. Let's start by considering the number of triangles that can be formed from N given points. This can be expressed as N*(N-1)/2, as you have correctly mentioned.

Now, let's consider the condition that no triangle should have all edges of the same color. In other words, there should be at least one edge of a different color in each triangle. We can solve this problem by considering the worst case scenario, where all triangles have two edges of one color and one edge of another color. In this case, the maximum number of triangles that can satisfy this condition would be (N*(N-1)/2)/2, since we need to divide by 2 to account for the triangles with two edges of the same color.

Next, we need to consider the number of color combinations that can be formed with three colors. This would be 3*(3-1)/2 = 3. Therefore, the maximum number of triangles that can satisfy the condition of having at least one edge of a different color would be (N*(N-1)/2)/2 * 3 = 3N*(N-1)/4.

Now, we need to find the maximum value of N that satisfies the condition that the number of triangles formed should be less than or equal to 3N*(N-1)/4. This can be solved by trial and error, but I have found that the maximum value of N is 9. For N=9, the number of triangles formed would be 3*9*(9-1)/4 = 3*9*8/4 = 54, which is less than or equal to 3*9*(9-1)/4 = 243/4 = 60.75.

Therefore, the maximum value of N in this scenario is 9. I hope this helps. Let me know if you have any further questions. Keep up the good work!