It’s been a while since I gave cute problems, so here are a few with a theme: solving mathematical problems with physics (I’m not talking about the other way around, which is why I said “physical mathematics” instead of “mathematical physics”).
- Take a triangle ABC. Prove that the point X inside the triangle which minimizes AX + BX + CX is a point such that the angles AXB, BXC, and CXA are all degrees.
- Consider an ellipse with ratio of the major to minor axes equal to . From either foci, start enlarging a circle until the circle is tangent to the ellipse from the inside. It is obvious that the number of tangent points will be or and that this only depends on . Given an , figure out this count.
- Take a convex polyhedron and any point inside of it. To every face, drop a normal line from the point. Note that it is both possible to land inside the face or outside. Construct a polyhedron where every such normal line drops outside of the corresponding face, or prove such a polyhedron cannot exist.
These are really delightful problems and among my all-time favorite – if you haven’t seen them before, I highly recommend thinking about them before the jump. =)
Here are the solutions:
- Take a table and drill holes so they form vertices of ABC. Now, take three strings and attach one end from all three at a single point X. At the other end of each of the three strings put a weight of mass . Put the weights through the holes. This system will seek to minimize potential energy, which happens when AX + BX + CX is minimized (if you don’t see this immediately, consider the potential energy is just proportional to the negative of the sum of the three taut parts of the string under the table). However, this means the three forces from the three weights must balance at X. Since the forces have the same tension (all ), the forces must be at degrees from each other.
- It doesn’t matter what is – the answer is always . This nonintuitive answer comes easily with some optical thinking: draw a tangent line to one of the tangent points and fire a ray of light from the chosen focus to . Note that because we have an ellipse, the reflection must go through the other focus. However, because the line is also tangent to the circle, the reflection must also go through the center of the circle, which is the focus we started with! Since the reflection must go through both foci, the tangent must have been orthogonal to the major axis, meaning we only had contact point.
- This polyhedron cannot exist. Suppose we put it on a flat table with bottom face . Since by construction the normal line hits outside of , the polyhedron must roll onto another face (yes, or edge, but we can spent basically zero energy to push it onto a face using that edge instead, so the argument still works), but then the same thing happens and we get perpetual motion. As we recall, perpetual motion cannot exist in our world =)
The first problem is fairly canonical – it has been around for a while and I might have seen it first in Titu Andresscu and Razvan Gelca’s “Mathematical Olympiad Challenges” back in the Olympiad days. The second problem met me in middle school – it became one of my favorite problems ever to appear in a contest after I found it one day practicing old ARML/NYCIML problems (sadly I cannot remember which contest this was from). I learned of the third problem from Tadashi Tokieda’s (seems like a really fun guy, by the way) article “Mechanical Ideas in Geometry” in an old Monthly.
The prompt for this post is that I recently rediscovered the first problem in Mark Levi’s gorgeous “The Mathematical Mechanic.” Though I was actually a bit sad since I’ve been working on a collection much like his to share them in blog/book form at some point, I think it is really a great thing for the world that he put these problems out there in organized form, so everyone should go read that book at some point and I’ll be happy vicariously through him. Besides, the second and third problems I gave are not in that book, so I can still share them here with you.
Have a nice summer, everyone,