The Prisoner's Dilemma

D17. Transrotation

Contributed by Akiva Weinberger, Yale University
Figure 1: Dissecting a square into a triangle.

Often you can slice up one shape and reassemble the pieces to make a different shape, like the famous square-to-triangle dissection shown in Figure 1. Notice that in this rearrangement you are forced to rotate some of the pieces when you put them back together. Sometimes, though, you can do the assembly by only translating the pieces, i.e. sliding them without rotation, as in the square-to-cross dissection shown in Figure 2.

Figure 2: Sliding pieces of a square to make a cross.

But is it possible to start with a square and “rotate by translation?” In other words, can you dissect a square into finitely many pieces, and translate each of them without any rotation, to reassemble them into the shape of a square that has been rotated (by some angle other than a multiple of 90 ∘ )? This goal is schematically shown in Figure 3. Note that although this problem was suggested to the Prisoner’s Dilemma by Akiva, it was apparently first posed by Hugo Hadwiger way back in 1950.

Figure 3: Can you rotate by slicing and sliding?

This problem originally appeared in the Prisoner’s Dilemma in the 2023 Fall issue of the PMP Newsletter. Solutions are no longer being accepted for this Dilemma.

Show solution?  
Solution.

The proposer Akiva Weinberger provided a twist on the classic “proof on the back of a napkin,” namely, a “solution using a napkin”:

In case it’s not quite clear how to rearrange the pieces, here are the “before” and “after” diagrams with the corresponding pieces numbered.

If you can find a way to do it with four or fewer pieces, that will be a new discovery indeed! (It may not be possible, but on the other hand, the Problem Warden is not aware of any proof of impossibility.)