I had the idea for this thread when talking about the Monty Hall problem.

Suppose you are teaching a class of mathematics students, and you ask them:

How do you know that an equilateral triangle has three sides?

Student 1:“I don’t know how many sides it has.”

Student 2:“Because it looks very pointy, and the pointier a shape the less sides it has.”

Student 3:“Because it resembles a knife-blade, and I know that those have three sides.”

Student 4:“Because this is what all mathematicians say, and they can’t all be wrong.”

Student 5:“Because you told us so last lesson, and as our teacher you wouldn’t lie to us.”

Student 6:“Because it’s named with the prefix “tri-”, meaning three.”

Student 7:“Because it contains a 180° angle.”

Student 8:“Because it’s a connected shape with three vertices.”

Student 9:“It doesn’t: since four equilateral triangles can be tesselated inside a square, and a square has four sides, each triangle only has one side.”

Student 10:“Isn’t it obvious? You can plainly see that it has three sides.”

Actually I don’t know about potential.
I picked what would be my answer. Since I didn’t got that far with my sturdy answers, probably number 10 isn’t the one with most potential.

I voted 8 because to me it seems their argument makes the most sense, and also they explained it in a short and clear statement. Being able to filter out unnecessary information is an important quality in Logic.

I also like the answers by Students 9 and Student 2, showing some creativity as well. Student 6 is applying linguistic, which is an approach I would never consider, but I respect the effort.

My answer (10) may be more about their potential in physics/engineering than their potential in mathematics. Maybe I’m not enough of a mathematician(theoretical) and too much of a physicist(practical) to judge potential in mathematics.

The correct answer would be that it’s by definition of what a triangle is.

9 doesn’t make any sense to me, how does one tesselate a square with four equilateral triangles in the first place?

I would go further and suggest that any answer that does not refer to the definition does not show mathematical potential. In general mathematics boils down to defining the right things, and using the definitions to prove theorems.

I say 1 and 11, students with most potential are usually those whom have no idea what the others are talking about and are first to declare it, and those who just take a last minute guess and still ace it.

That’s a bit harsh on the students though, don’t you think?

If we go by the definition of side of a polytope, then a triangle has 7 sides (3 vertices corresponding to 0-dimensional affine subspaces, 3 lines corresponding to 1-dimensional affine subspaces and 1 plane region corresponding to a 2-dimensional affine subspace).

I don’t think it’s harsh. The most important thing to learn is not to try and answer a question immediately, but to try and understand the concepts in the question. If you’re not using some definition of what a triangle is, then how could you explain why it has three sides? It needs to start somewhere.