A New Beginning

Hello! Right now, you are probably like, “Wait, didn’t this blog die like a year ago?” This is an astute and true observation, and I thus feel that I should reveal the reason for my prolonged absence.

Without further ado, this reason is… well, actually, there isn’t, I was just lazy. But I promise to stop being that lazy from now on, so you can expect a blog post coming out every week or two – yay posts!

Anyway, this last year was a pretty big year for me. I finished high school, moved from Russia to the USA, and began studying Computer Science in the University of Michigan. I also started and almost finished a huge side project (more on that in a future post). However, the single most important thing that happened to me in the past year was the fact that someone gave me this blue wibbly-wobbly thing:

It is a loop consisting of eighteen 90° arcs that are connected end to end with rotary joints. After playing with it way longer than I am comfortable admitting, I for no apparent reason decided to put it flat on a table. Unfortunately, my plan didn’t work out: every time some part of the blue thing refused to lay flat.

It looks like there should be some mathematical reason why this thing doesn’t want to be flat. And hey, this is a math blog after all, so let’s find out what this reason is!

We will approach this problem like real mathematicians, that is, in an inconsistent, confusing, and inefficient way. First of all, we declare that solving a single practical problem is too boring, so we generalize it:

For which positive integers N can a blue thing with N segments lie flat on a table?

Then we remember that the segments of the blue thing were actually 90° arcs. And you know what else has a ton of 90° angles? Right, a square grid! So we go ahead and put our blue thing on a square grid like this:

Now, instead of worrying about a blue thing lying flat, we only need to worry about the arcs on the grid forming a closed loop. Abstractions like this one usually make problems easier, so we can consider it to be a step forward, even if it means that we don’t get to play with the blue thing any more.

What should we do next? It would probably be best to try to draw a lot of loops on a square grid and see if it helps us. However, since we promised to be like real mathematicians, we will instead try to apply a random math technique that looks kind of relevant: coloring. Let’s color the square grid like a chessboard:

Wow, it worked! As we go along the loop, we alternate between green and yellow tiles, which means that the number of tiles must be even for the loop to close. Hooray!

But wait. At this point, you might have noticed a small problem, namely the fact that the original blue thing had an even number of segments (18) and still didn’t form a flat loop. What are we missing?

To answer this question, we should probably stop being like real mathematicians and start being like good mathematicians. Let’s draw a bunch of loops:

(Yes, a single square can totally hold two arcs) How long are these loops? There is one with 4 segments, one with 8, then 12, 16… Those look suspiciously like multiples of four. And what if we try to draw loops with lengths that are even but not multiples of four?

Right, they don’t work.

Now, if we can prove that loops can only have lengths that are multiples of four, it will be great! But I won’t post the proof now – it’s your turn to do math! Share your solutions in the comments, and I will share mine in the next post.