The Existence of Incomplete Room Squares

It has been conjectured by D. R. Stinson that an incomplete Room square \((n, s)\)-IRS exists if and only if \(n\) and \(s\) are both odd and \(n \geq 3s + 2\), except for the nonexistent case \((n, s) = (5, 1)\). In this paper we shall improve the known results and show that the conjecture is true except for \(45\) pairs \((n, s)\) for which the existence of an \((n, s)\)-IRS remains undecided.