Metal donuts

Let’s say you have a metal donut.

A donut (a circle with a concentric circle removed)

Donut

We put the metal donut in an oven, and it heats up. As we all know, metal expands when it gets hotter.

Does the donut hole get bigger or smaller? Read on for the answer, and how Martin Gardner, who first introduced me to this problem via his book, took my mental innocence.

Ready?

OK.


I first encountered the metal donut problem when I was about 10 or 11 or so, when my dad (or maybe my mom) gave me a book by Mr. Gardner. Gardner was a pretty cool guy; he’s probably best known for the books he wrote, which collected all sorts of logic, physics, and mathematical puzzles and games. (He died in May 2010). I tried this puzzle out later on my school mates at school, and this was the first moment I really realized: I’m smarter then some of these kids. And some of the kids are smarter then me.

Sure, I had long earlier realized some kids run faster then me. (Actually, at that time, probably almost everyone ran faster then me – I was kind of a fat kid!) And I had realized that I could read faster then some kids, too. But it didn’t hit home for me until that moment in the school cafeteria.

Some kids I was friends with just didn’t get it: they refused to believe my explanation, even when I resorted to repeating it, verbatim, out of the book. Others figured it out way faster then I did. I was kind of shocked, in those cases: my neat little lunch-time party trick was boring for them!

It was after this, I think, I started partitioning what I said to who: geeky nerdy things to one group, tv and chitchat to another. Innocence lost.

Of course, I was far too young to figure out most of the other problems in the book. I kept working at them, off and on, till I was much, much older; I don’t think I finished until I had graduated college. I still have the book around somewhere, completely torn up and cover missing; it’s with the rest of my worldly possessions in storage. (And that’s a topic for another post!)

As for the answer: ignore the outside circle in the donut. The inner circle is all that matters. Imagine that inner circle as a very thin piece of metal. When it heats up, what will happen is intuitive: the loop gets bigger, because the length of the metal making up the circle gets longer. In other words, the circumference has to increase, so the diameter must also increase.

So, if the inner circle gets bigger, the answer is: the donut hole gets bigger. The outer circle is just noise. It’s a red herring.

PS: it is, however, true that the ratio between the donut hole diameter and the diameter of the donut remains the same as the metal expands. This is what trips up people up. They imagine that if the outside expands, the inside must also expand at the same speed; both cancel out, leaving the donut hole the same size.

PPS. I might do more selected math puzzles; some of them are super interesting!

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