(Gonna reply my very own query, as is inspired.)

To set the stage: an arc (or a Jordan arc) is a non-self-intersecting curve with two distinct endpoints. (For those that are conversant in topology, it is a subset of the airplane homeomorphic to $[0,1]$.) So any of those, mainly; you are allowed fractals if you would like, so long as they do not intersect themselves and have two particular endpoints.

(I’m calling them “arcs” reasonably than one thing easy like “curves” is as a result of (a) in arithmetic they generally go beneath the identify Jordan arcs and (b) I wish to emphasize that I do not depend loops.)

The multiset sum of two units is like their union, however counting multiplicity. As an instance, the multiset sum of a / form and a form could be a X form the place the middle has multiplicity two and all different factors have multiplicity one. The multiset sum of > and < might make the identical form.

Show that it’s doable for the multiset sum of two arcs to equal the multiset sum of three arcs.

(Note that breaking a line into two items will not work as a result of the multiplicity will not be right the place they be part of.)