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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 Solution for three toothpicks to a side Comment 12 of 12 |
The solution, if each side of the polygon were made of three colored toothpicks in simply to substitute n^3 for n where n is the number of colors where each side is composed of a single toothpick.

... for triangles, the equation is transformed from
(n^3 + 2n)/3 to (n^9 + 2n^3)/3.
... for squares, the equation is transformed from
(n^4 + n^3 + 2*n)/4 to (n^12 + n^9 + 2n^3)/4.

[One can reference The On-Line Encylopedia of Integer Sequences to find the equations for the "number of ways to color vertices of a triangle using <= n colors, allowing only rotations" (Sloan A006527) and the "number of ways to color vertices of a square using <= n colors, allowing only rotations" (Sloan A006528).]
  Posted by Dej Mar on 2008-03-05 10:53:52
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