New Gozinta Boxes Trick

22nd December 2025, 10:16 am

Imagine you’re watching a magician. She pulls out two perfectly ordinary boxes — or so it seems. One box is inside the other, like a set of nesting dolls. So far, nothing suspicious.

Then she removes the smaller box, closes the larger one, and slides the larger box inside the smaller one. Ta-da!

The name comes from the way one box goes into the other. Would you like to know the secret? The two boxes are actually identical. Moreover, they are not cubes but cuboids. The inner box is fully closed, while the outerbox is slightly expanded, and the inner box is rotated relative to the outer one.

I first heard about Gozinta Boxes at the Gathering for Gardner conference in 2024. Ivo David gave a talk and presented his new trick: Triple Gozinta Boxes, which you can now buy at TCC Magic. He can place three boxes inside one another — and then repeat the trick in the reverse order.

During his presentation, David mentioned that he knew how to prove that you cannot have more than ten Gozinta Boxes. My immediate reaction was that ten must be overkill. So I decided to give the problem to my STEP students as a project.

We proved that in three or higher dimensions, the maximum number of boxes is three. We also showed that in two dimensions, the maximum is four. You can find all the details in our paper Mathematics of Gozinta Boxes, posted on the arXiv. But we didn’t stop there. We invented a new trick. We constructed three boxes such that not only can they be nested in one order — say, ABC — and in the reverse order, CBA, but they can also be nested in three additional orders, for example ACB, BAC, and BCA. We also proved that achieving all six possible orders is impossible. You can see the trick by following the link for A New Gozinta Boxes Trick.

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