Pair of Squares

Pair of Squares is a browser-based HTML5 mathematical combination puzzle. Every tile placement must satisfy n₁ + n₂ = k² — the sum of two adjacent integers must equal a perfect square (4, 9, 16, 25, 36…). The classic 1-to-18 linear sequence has exactly one unique Hamiltonian path solution, making random placement statistically unviable. The engine instantly validates each placement against the perfect square array and rejects invalid moves.

The correct solving discipline starts with the most-constrained numbers: 18 has only one valid partner within 1–18 (7, since 18+7=25), so it must be placed first. Follow each forced single-partner continuation from there — the chain always converges on the unique solution. The game also offers variant modes including a memory product matrix (matching x to x²), a two-player geometric territory grid, and a prime pair alternative where sums must equal prime numbers instead of perfect squares.