~ Loops ~
Last week Dan Meyer announced a contest built around a loopy task from Anna Weltman in her new nonmath math book, This is not a Math Book.
I turned the task into an interactive walkthrough and loop laboratory, which you’re cordially invited to use as a demonstration for your students in conjunction with Anna’s book (Dan excerpted a few pages here and here). Or, of course, if you just want to get nerdsniped into messing around with loops for a little while.
Below I included a couple of problem sets I think would make for interesting extensions.
§ Faces and Vertices and Edges (Oh, my!)

(1) We saw that sometimes a loop only intersects itself at individual points (like the 234), while other times a loop will retrace a portion of itself (like the 352). Let’s call loops that retrace themselves clingy. Under what circumstances will a loop be clingy?

(2) A loop divides the plane into a bunch of regions that are walled off from the rest of the plane. Let’s call those regions cells. How many cells does the 214 loop have? (Hint: Don’t forget the big, huge, giant cell that’s outside the whole loop.)

(3) Each time a loop intersects itself at a point, it forms a little cross. How many crosses does the 214 loop have?

(4) If you start at a cross and trace your way along the loop, eventually you’ll either end up at another cross, or back at the same cross from a new direction. That portion of the loop between two crosses (or between a cross and itself) is called a blip. How many blips does the 214 loop have?

(5) Make a whole bunch of nonclingy loops. Can you find a relationship among the number of cells, crosses, and blips that a loop has?

(6) Find yourself a cube. How many faces does it have? How many vertices? How many edges?

(7) Find yourself a tetrahedron (admittedly a little bit harder). How many faces, vertices, and edges does it have?

(8) Get your hands on a bunch of convex polyhedra. (Unless you have some really nerdy math friends, you’re probably going to have to hit up the internet for this one). Can you find a relationship among the number of faces, vertices, and edges each one has?

(9) Convince yourself that this connection between loopdeloops and convex polyhedra isn’t an accident.
§ Loops Gone Wild

(1) We looked at some loops with three lengths (3loops), a loop with four lengths (a 4loop), and a loop with five lengths (a 5loop). The 3loops all returned to their starting point, so let’s call them tame. (The 5loop was also tame.) But the 4loop spun off to infinity. Let’s call loops like that wild. Are there any wild 3loops? Prove there aren’t or find a counterexample.

(2) Are there any tame 4loops? If so, are there any that a snooty mathematician wouldn’t call trivial? Why or why not?

(3) For what n are nloops guaranteed to be tame? Guaranteed to be wild? Neither necessarily tame nor necessarily wild?

(4) Let’s say that, instead of always changing direction in a counterclockwise way, at each step you pick a random direction (up, down, left, or right) in which to draw your line. Let’s call this new kind of loop a drunken loop. Are drunken 3loops more likely to be wild then their sober counterparts? Are drunken 4loops more likely to be tame?

(5) Remarkably (to me, at least), any drunken 1loop (like 111… or 9…) is tame. Prove that there are no wild drunken 1loops.

(6) Instead of restricting our loops to a 2D grid, let’s have them live in a 3D lattice. Now a drunken loop can head off, at each step, in one of 6 random directions in 3space. Are all drunken 1loops still tame? If not, what’s the probability a drunken 1loop will be tame?

(7) As n increases, what happens to the probabilty that a drunken 1loop is tame in R^{n}? Come up with a formula for that probability in terms of n.