Sierpinski pyramid by Solkoll
“Arithmetic! Algebra! Geometry! Grandiose trinity! Luminous triangle! Whoever has not known you is without sense!” -Comte de Lautréamont
When you think about it, it’s amazing that our physical Universe makes sense at all. The fact that we can observe what’s happening, determine the laws that govern it, and predict what will happen under the same or similar circumstances is the most remarkable power that science has. If that’s what you’re doing in any aspect of your life, congratulations, you are a scientist. But that doesn’t tell us, fundamentally, what the Universe is like at its most basic level. Are we made up of point-like particles? Or are they geometric constructions? Are we ripples in the Universe itself? In a way, They Might Be Giants might be pondering exactly this in their song that I present to you this weekend,
At the root of all of this is mathematics, which is in its own way beautiful, elegant, and happens to be our foundation for making sense of the Universe. And in what appeared to be a simple puzzle, I saw an image similar to this one floating around the internet and making the rounds on facebook.equilateral triangle with three extra lines coming out of two of the vertices, along with a question of “how many triangles?” can be found in this image.
Try solving it yourself, if you like, before reading on, where I’ll explain for you the correct answer, and show you a fun and beautiful math pattern that’s in there, too.
As can be expected, I saw a large number of attempts at answering this, including some fairly sophisticated erroneous ones.
This one was particularly bothersome, because — spoiler alert — 64 is the right answer, but this diagram is totally wrong, missing some triangles that are actually there, and counting a number of triangles twice. (For example, look at the fifth row, at the red triangle in the first column, and how that’s the same as the green triangle in the sixth row, second column.)
When someone gets the right answer for the wrong reason, it’s particularly aggravating, because it takes multiple mistakes to make that happen. So I’d like to show you a foolproof method for showing you all the unique triangles in this diagram, and when we’re finished, we’ll see a pattern and get a formula to learn something fun and beautiful.
Each time we do, we’ll count all the new unique triangles by using the new, intersecting point and one (or both) of the two base vertices at the bottom of the triangle. In order to avoid double-counting, we’ll only create triangles using points below our current point, ensuring that we’ll never count the same triangle twice. You’ll also notice that some points — labeled 2 and 3, 4 and 5, 6 and 7, 9 and 10, 11 and 12, and 14 and 15 — are mirror-reflections of one another, so those sets better give us the same numbers of triangles.
Let’s go through these points, from 1 to 16, and see what we get.
Easy enough, so it’s on to the next one(s) up.
So let’s move up to points 4 and 5.
Let’s move on up, and hit points 9 and 10.
Now, 64 is an interesting number: it’s a perfect square (82 = 64), it’s a perfect cube (43 = 64), and you might wonder if it’s related to the number of extra lines coming out of those two base vertices. Well, it is, but the pattern is really fantastic. Let’s show you what we get if we count the number of new triangles we were able to create — using each new point as a necessary vertex — as we moved up the triangle.
If we only had one, we’d only have the lowest line from each vertex, meaning we’d only get 1 triangle.
If we only had two, we’d have the two lowest lines from each vertex, getting a total of 8 triangles: 1 x 1 + 2 x 2 + 3 x 1 = 8.
If we only had three, we’d get the three lowest lines from each vertex, for a total of 27 triangles: 1 x 1 + 2 x 2 + 3 x 3 + 4 x 2 + 5 x 1 = 27.
And as you can see, for four, we get 64: 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 3 + 6 x 2 + 7 x 1 = 64.
And, as you may have noticed, 13 = 1, 23 = 8, 33 = 27, and 43 = 64, so that’s how the pattern goes! So go ahead and draw a triangle with an arbitrary number of lines coming from each vertex; you’ll not only now know the pattern, including how many triangles you can generate as each vertex as you move upwards, but you now know an awesome way to generate the perfect cubes of numbers! What a fun and beautiful little bit of math, and I hope it helps bring you not only a great weekend, but peace of mind, and closure to this epic triangle riddle!