A Mathematician Ties a Tie

I learned to tie a tie before I became a mathematician.

I was still just a kid when I learned. My grandparents taught me to make a basic Windsor knot. I could get that part right, but the length would invariably be a bit off.

– “How do I fix it if it’s too long or too short?”

– “Just untie it and try again.”

And so I did. For the next two decades, tying a tie was essentially a procedure of guess-and-check.

And then one day, it dawned on me that I’ve learned some things about problem solving in the last 20 years that might make this task easier. There is a saying in mathematics that “One must always invert.” In order to solve a problem, we stop and think about what a solution would look like, and then we figure out how to get something like it.

So, rather than making my knot and hoping that the length works out, I can invert the problem: get the length right, and then make the knot.

 

First up, we decide the length we want. If the length is going to be right in the end, then the knot has to end up where my left hand is pinching.

Getting the knot there is a bit tricky. We have to plan ahead for the fact that we’ll be wrapping the top half around the back side to make the knot, otherwise the length will be too short in the end. So, we need to leave about three-and-a-half times the width of the back side to give room to go under, over, up to and through the hole.

Once we do that and pull it tight, we get the length just right.
This technique is just a straightforward application of the mathematical strategy of inverting a problem. However, if you’ve spent considerable time with mathematicians, you understand that it is still not widely-known. Please help spread the word.