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).