Curves on potatoes

On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler:

The puzzle is attributed to the book "The mathemagician and the pied piper", and appears on p. 56 in a chapter on "Puzzles from around the world" by Richard I. Hess (so presumably this problem did not originate in this book).

The point of this question is how does one make this puzzle precisely formulated mathematically? Specifically, does it hold if one interprets "potato" to mean a compact surface embedded in $$\mathbb{E}^3$$ (3-dimensional Euclidean space), and "curve" to mean an embedded loop, and “identical” to mean differing by an orientation preserving isometric of $$\mathbb{E}^3$$?

I don't want to give any hints as to the "solution" to the puzzle, so I will leave further discussion to the comments (so don't look there if you don't want a hint).

• Robin Houston's comment that "the solution is incredibly simple" is a great hint! – Nik Weaver Jun 23 at 19:39
• The most obvious mathematical interpretation of the puzzle is that the surface of the potatoes should be smoothly embedded spheres. Then one can show by transversality that there will be a translate of one potato that intersects the other in a smooth closed curve. However, this argument doesn't work in the topological case. One could imagine that the surface is an Alexander horned sphere, or even worse an Alford sphere, and that the intersections between translates might be indecomposable continua, or maybe with non-trivial $\pi_1$ but no closed embedded curve, or a topologist's sine curve. – Ian Agol Jun 23 at 19:40
• @ycor: what other interpretation can you think of? That would help me clarify the question if it is ambiguous. – Ian Agol Jun 23 at 21:11
• Well, "smooth compact surfaces" and "embedded loop" should be enough. Just move one surface toward the other and let them intersect transversally. Probably Sard theorem is necessary at a certain point in order to formalize things, but the idea seems to work – Francesco Polizzi Jun 23 at 22:03
• I always skip potatoes with infinite surface area at the store - they take too long to peel. – Per Alexandersson Jun 25 at 10:43