How We Wasted Time Before the Internet – Mathematics Edition

Or more precisely, how I wasted too much time long before the Internet. Here’s the picture:

SquareWhen I was a little kid, somebody, probably my father, drew a picture like that and challenged me to draw the same thing without lifting my pencil from the paper. That seemed like a pretty easy thing to do. It wasn’t.

Many years passed before my first and last attempt. Not that I spent days and nights continuously working on this, but there were a lot of classes and then some meetings to sit through. And I guess I was competitive, stubborn and/or obsessive.

But one thing that kept me going off and on through the years was the belief that I had successfully met the challenge once, couldn’t remember how I did it and should be able to do it again (too bad I didn’t keep notes).

Of course, I eventually concluded that this was a false memory. The thing cannot be done!

What brought all this back to me was an article at Three Quarks Daily called “A Square Peg for Every Round Hole”. It’s about mathematical puzzles, the most famous being Fermat’s Last Theorem (“I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”). In particular, it’s about 

another enticing mathematical morsel which is still unsolved: the Square Peg Problem (SPP). The history is a bit murky, but it is generally credited to Otto Toeplitz in 1911. The SPP is the conjecture that if you draw a curve on a sheet of paper without picking up your pencil and which begins and ends at the same place, then you can find four points on the curve which form the corners of a square. 

For example, I drew this wavy curve in black and was then able to overlay a square with its four corners intersecting the curve. 

Square 2

(Ok, I cheated and put in the square first and then drew the curve. The other way may be mathematically possible in every case (or not) but it’s not that easy and my little obsession lies elsewhere.)

Maybe mathematicians proved long ago that it’s impossible to draw a picture like the one at the top of this post. Maybe there’s even a name for this particular “mathematical morsel”: the Square With Lines Around It and a Cross in the Middle Problem (SWLAIAACITMP).  

On the other hand, if you know a way to draw the damn thing without lifting your pencil or pen from the paper or your index finger from the mouse – or know why it can’t be done – please let me know.

Update: That didn’t take long. A person going by the name of “X” gave the answer in the comments at Three Quarks Daily after I described the SWLAIAACITMP problem:

This is a cool problem called “Euler Paths”. You can prove it’s impossible for this graph because there are too many vertices with an odd number of edges coming out of them. So there will always come a time when you go into a vertex and can’t get out. This page has the rules: Euler’s Graph Theorems.

Thank you, X, whoever you are.