Complete Resolution of the {0,1} Case of Guy's Problem F24

We prove that if the decimal expansion of n² uses only the digits 0 and 1, then n = 10ᵃ for some integer a ≥ 0. This settles the {0,1} case of Guy's Problem F24, a problem that has remained open for decades in the literature. The proof uses a p‑adic valuation analysis to partition the problem into four families of Diophantine equations with parameters in exponents. Two families are excluded by a self‑contained "island" argument, which first forces u' < 10ᵗ via a size bound and then excludes that case via an island lemma. The remaining two families are shown to be equivalent, through the construction of a Pell equation and a complete case analysis of the 2‑adic and 5‑adic valuations of the factors, to the quadratic families already excluded.