Samin_Mustakim's blog

Forming incongruent triangles
By Samin_Mustakim, history, 2 hours ago, In English

Problem: There are $$$n$$$ sticks of length $$$1,2,3,...,n$$$. How many incongruent triangles can be formed by using three of the given sticks?

My solution: After solving the problem, I found the number of triangles, $$$x$$$, to be:

$$$\begin{align}x = \sum_{i=1}^{n} i \cdot \left\lceil \frac{i+1}{2} \right\rceil + 2 \cdot \sum_{i=1}^{n} \left\lceil \frac{i+1}{2} \right\rceil - \sum_{i=1}^{n} \left\lceil \frac{i+1}{2} \right\rceil^2 - n(n+1)\newline \end{align}$$$

However, I am not satisfied with this expression. Can it be simplified? (Can we express this equation without using the summation sign?)

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I am following up to know the answer

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