“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.
How many triangles are in this image? 92.6% of Americans get this question wrong!
It’s pretty straightforward: an
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.
Image credit: source unknown, retrieved from Irena Haj.
It makes sense to try and construct triangles from each of the points
where lines intersect, but you have to be careful not to
double-or-triple count triangles. The number here is too high, as the
answer isn’t seventy.
Image credit: Patryk Solarczyk.
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.
All of the points of intersecting lines within our triangle.
We’re going to start at the bottom of the triangle, with the two base
vertices. As we move up the diagram, we’ll progressively run into
points where two lines intersect, labeled above in the order we’ll run
into them.
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.
Point #1 as a necessary vertex in each triangle.
For the first point we come to, there’s only one possible triangle
using the points below it: there are three points in a triangle and this
triangle uses all of them.
Easy enough, so it’s on to the next one(s) up.
Points #2 and #3 as a necessary vertex in each triangle.
As you can see, each of those new points can make two new triangles,
one using both base vertices and one using our intersecting point #1,
which is now an option in making a triangle. This pattern will continue
as we continue to move upwards, as all lower points now become fair
game.
So let’s move up to points 4 and 5.
Points #4 and #5 as a necessary vertex in each triangle.
There are three new triangles we can construct for each of those, as
you can see. This is pretty straightforward, as are points 6 and 7,
below.
Points #6 and #7 as a necessary vertex in each triangle.
Four new triangles apiece, using all of the allowable, lower points
as possible vertices. So far, so good: no double-counting, and no missed
triangles. And moving up one more, to intersecting point #8, finally
gets a little interesting.
Point #8 as a necessary vertex in each triangle.
Why is this — point #8 — interesting as compared to the others?
Because, for the first time, we can build successful, new, unique
triangles that connect to
either one of the base vertices, something that we’ll have to keep in mind for all of our subsequent points.
Let’s move on up, and hit points 9 and 10.
Points #9 and #10 as a necessary vertex in each triangle.
Points 9 and 10 give us four new, unique triangles each, connecting
to either (or both) base vertex (or vertices), as appropriate.
Points #11 and #12 as a necessary vertex in each triangle.
And for points 11 and 12, we get five each. Feel free to check: all
of these triangles, so far, are unique, and encapsulate all of them.
We’ve only got four intersecting points left, so let’s take them all
down!
Point #13 as a necessary vertex in each triangle.
Five more for intersecting point #13…
Points #14 and #15 as a necessary vertex in each triangle.
Six each for points #14 and 15, and for the final, uppermost point…
Point #16 as a necessary vertex in each triangle.
Seven! All told, we can add these up, and get 1 + 2 + 2 + 3 + 3 + 4 + 4 + 3 + 4 + 4 + 5 + 5 + 5 + 6 + 6 + 7 =
64, and so there are, in fact, 64 unique triangles here.
Now, 64 is an interesting number: it’s a perfect square (8
^{2} = 64), it’s a perfect cube (4
^{3} = 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.
Number of triangles created at each new vertex, going upwards.
Now, that’s a beautiful pattern, and it happens to be
very closely related to the number of lines — in this case, 4 — coming out of each base vertex of 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, 1^{3} = 1, 2^{3} = 8, 3^{3} = 27, and 4^{3} =
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!