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FIGURE it out ! (Posted on 2004-09-10) Difficulty: 4 of 5
  1. With an unlimited supply of toothpicks of n different colors, how many different triangles can be formed on a flat surface, using three toothpicks for the sides of each triangle?
    (Reflections are considered different, but rotations are not.)

  2. How many different squares?

No Solution Yet Submitted by SilverKnight    
Rating: 3.3333 (3 votes)

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Solution re: Solution for Squares - Correct | Comment 6 of 12 |
(In reply to Solution for Squares by nikki)

First of all, after seeing Charlie’s solution, I realized I was a little too hasty when I "extrapolated" from my triangles solution. For the abcd group, I chose to mimic the "combinations times 2" thought process instead of the "permutations divided by 3 (or 4 for squares)." The two are equivalent for triangles, but not so for squares. I should have been more careful and chosen n!/(n-4)!/4 when moving onto squares. So the answer should actually be:

S = (n^4 + n^2 + 2n)/4

Second, I realized that both of my solutions were a little incomplete. What I actually solved for was n>2 in the triangles, and n>3 in the squares. Here is the complete solution:

Triangles:
For n=1, T= n = 1
For n=2, T= n + n*(n-1) = 4
For n>2, T= (n^3 + 2n)/3

Squares:
For n=1, S= n = 1
For n=2, S= n + 2[n^2 - n] = 6
For n=3, S= n + 2[n^2 - n] + [3/2*(n^3 - 3n^2 + 2n)] = 24
For n>3, S= (n^4 + n^2 + 2n)/4


  Posted by nikki on 2004-09-10 16:45:14
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