Squares, three fleas, sporadic integer sets, and squares

Abstract

In the plane, three integer points ("fleas") are given. At every tick of time, two of them (say, P,Q) instantly jump to two vacant points R,S, so that PQRS is a square with that order of vertices. Description of all integers and half-integers which occur as areas of spanned triangles turns out to be unexpectedly intricate problem. For example, if one starts from a triple (0,0), (2,0), (4,1), these positive integers are missed: 2*{0, 1, 4, 15, 16, 20, 79, 84, 95, 119, 156}, with no other up to 3*10^6 (and seemingly, none at all); half-integers which are missed seem to form a 39-element set (the largest of them being 11365/2). However, there exist certain starting setups which have an "integrable" component as part of the answer. We demonstrate that for the initial triple (0,0), (2,1), (3,2), the integers missed as areas are perfect squares, and the the sporadic set {5, 29, 80, 99, 179} with no other elements up to 3*10^6 (most likely, no other at all; all half-integers serve as areas).