Purdue POW

Backdrop:

My friend Abhishek is of the opiniun that good professors are able to assign problems/tasks to their students based on their students’ ability. Thus a good professor will not give a really hard problem to a student who will not be able to handle it. It’s true the other way round also. Accordingly, he gives me simple problems in mathematics that I will enjoy solving¹. Quite a few of them are sourced from Purdue POW. He rarely solves them² but forwards them to me. The following is one such problem and the discussion we had on it.

The problem(link):

A tetrahedron has a base which is an equilateral triangle of
edge length one, and is placed so that the base is sitting flat on
a table. The other three faces are congruent isosceles triangles
with one edge an edge of the base (of course) and the other
edges of length three. Find the length of the shortest path,
lying in the union of the three isosceles faces, which starts and
ends at the same vertex of the base, and which meets every
line segment drawn from the top vertex to the perimeter of
the base triangle.

The discussion (it can be a spoiler to the problem)³:

A (Abhishek): I have a problem which is custom made for you.

(That’s usually a good sign. Shortly after that I opened the link to the problem.)

S (Shashank): Yeah, seems too easy to be solved using a program. Of course you are expecting a simple and elegant solution.

A: I think it can be solved with 1 variable calculus.

S: I think I see the solution. You open the 3 larger faces of the tetrahedron…

A (interrupting abruptly, jokingly of course): No that’s not your solution. It’s mine. So you open up the faces of the tetrahedron and lay them flat. But wait, what about the angle?

S: Aha! So you haven’t seen it completely! You simply…

(And here starts the ‘telepathy’ as Raju noted once, which is a characteristic of the hundreds of conversations I had with Abhishek.)

A: Obviously! Sorry, my bad!

S: Not just bad. Criminal bad!

(Don’t worry if you find the above conversation boring – that’s because it is boring. Here’s the link to my solution.)

1. Those who can see through the apparent condescension, will find a really great friend in Abhishek.
2. Purdue POW are UG level problems form their mathematics department. Being a Ph.D. candidate Abhishek has far too challenging problems than to care about the POWs. Would you drive a Ford Mustang if you can drive a Nissan GTR? Would you listen to Kumar Sanu if you can listen to Rafi? Would you settle for something impure if you can easily have something that is pure? Unfortunately, for some, the answer is yes. Suum Cuique!
3. It’s just a part of discussion. Not verbatim, but mostly similar.