[You might find it safer to use toothpicks instead, but I prefer matchsticks, and fire.]
This investigation is split into 2 parts. I will post the 2nd part tomorrow
In this investigation, we want to figure out the number of matchsticks that we need, to form several (unit) equilateral triangles. For simplicity, we will be restricting ourselves to a 2-D plane. (If you are brave, go ahead and try the 3-D version. It is quite challenging)
1) What is the minimum number of matchsticks that we need to form 1 equilateral triangle?
Clearly, we need at least 3, and 3 are sufficient.
2) What is the minimum number of matchsticks that we need to form 2 equilateral triangles?
Well, we could do it with \( 2 \times 3 = 6 \). But if we allow them to share a common side, then we only need 5 matchsticks!
3) What is the minimum number of matchsticks that we need to form 3 equilateral triangles?
Continuing the above (almost like a snake), we see that we need \( 5 + 2 \) matchsticks. Can we do better than that? I don't think so.
4) How can we (easily) form \(N\) equilateral triangles using \( 2N+1 \) matchsticks?
Hint: Do you spot a pattern above? Can you explain in detail how it works? Mathematicians use "Induction" as a way to formally express the pattern that they see.
5) What is the minimum number of matchsticks that we need to form 6 equilateral triangles?
From the previous question, we see that \(13\) is enough. Can you do better than that?
Hint: Yes we can!
This investigation continues in Part 2.