When Does a Square Board Split?

We study integer-board graphs whose edges have Euclidean length\(1/\lambda\) or \(\sqrt2/\lambda\); their moves form a union of leaperorbits and the components of the graph on \(\mathbb{Z}^2\) are the cosets of aGaussian ideal. Write \(\lambda^2=1/k\) or \(2/k\) in lowest terms and\(r(k)=\min_{x^2+y^2=k}\max(|x|,|y|)\). Our main result classifiescompletely the square boards whose components are their nonemptyintersections with these cosets: if \(k\) has a prime divisor\(1\pmod4\), they are, apart from trivially small sides, exactly thesquares of side at least \(2r(k)\). Otherwise, every square qualifies.The same holds for the graph whose moves are the integer vectors ofsquared length \(k\). In particular, if every prime divisor of \(k>1\)is \(1\pmod4\), a square of side \(N\ge2\) is connected if and only if\(N\ge2r(k)\). The hardest case,\(\lambda=1/(c\sqrt2)\) with \(c\) divisible by a prime \(1\pmod4\), inwhich the steps of least maximum absolute coordinate are diagonal, issettled by contracting diagonal pairs and arranging the columns of thecontracted grid in a single cycle. Further results are a self-containedproof of the two-component bound stated by Jelliss for half-freeleapers, the limiting distribution of the ratio of primitive and generalradii along powers, a classification of two-row boards by Pell equations,and a finite connectivity test for each width.