Folding Perfect Thirds Using math

This week’s activity comes from James Tanton of the Global Math Project and G’Day Math. He’s also demonstrated this activity and similar problems for the Math Teachers’ Circle Network.

how to fold a tie into perfect thirds

Imagine you’re packing for a trip, and you’re planning on bringing your favorite tie. It’s too long to fit in your suitcase, even after folding it in half.  You would fold it into fourths, but you don’t want all of those creases ruining your tie. You’ve decided folding it into thirds will be the perfect length to fit in your suitcase without noticeable creases on your tie. However, you don’t have a ruler or any means of making sure your tie is folded into perfect thirds. Is there anything you can do about this?

If this particular setup to the problem seems a bit peculiar to you, you can imagine many different setups around the core question: Can you fold any object into equal thirds without a ruler or any special tools?

A Possible solution through math

Let’s start by “eye-balling” an attempt at folding off one-third of the tie. Don’t worry if your first attempt isn’t great. In fact, it’s much more interesting if it isn’t close to start!

Now let’s make our next fold by taking the right side of the tie and folding it in half.  If we had perfectly folded off one-third of the tie on our first attempt, then the right side would be two-thirds long in total, so folding it in half should give one third on each side of the new fold.

But since our first attempt was (intentionally) bad, we can clearly see our tie is definitely not in three equal pieces. However, our second fold might look a lot better than our first. Let’s replace our first fold by folding the left side in half. Using the same logic as before, the left side should be two-thirds long.

Our new folds might look even better, so let’s even try a fourth fold for good measure:

We can keep this process going. You may notice in the image above that our marks seem to be getting closer together, and if you kept going, you would see that our marks are continuing to get even closer and appear to be approaching something. But is this process leading to perfect thirds?

Illustration of the folding process in action.

What's happening?

As we said before, if we were perfect with our first attempt at one-third, then the second fold would have one-third on either side of it. However, we presumably weren’t perfect. We can think of our first attempt as having some amount of distance, or error, from the one-third mark. We can then see that our second fold would have half as much error. The image below shows the amount of error between the folds and the true one-third and two-third marks:

This occurs because we folded a length equal to two-thirds plus our error from the first fold, so our new fold on the far right will result in lengths of one-third plus half our original error. If we keep this process going infinitely, we will see that our folds will have smaller and smaller error each time. This means our folds are approaching the true one-third and two-third marks. 

So, indeed, it is possible to fold a tie (or any other object) into equal thirds without a ruler or any special tools!

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