Room squares and subsquares

Abstract

If a Room square of side s contains a Room subsquare of side t, then s≥3t+2. For t=3 or 5, there is no Room square of side t, yet one can construct (incomplete) Room squares of side s "missing" subsquares of side 3 or 5 (the same bound s≥3t+2 holds).It has been conjectured that if s and t are odd, s≥3t+2 and (s,t) ⊄ (5,1), then there exists a Room square of side s containing (or missing, if t=3 or 5) a subsquare of side t. Substantial progress has bee made toward proving this conjecture. In this paper we show that there exists a Room square of odd side s containing or missing a subsquare of odd side t provided s≥6t+41. For odd t≥127 and odd t≥4t+29, there exists a Room square of side s containing a subsquare of side t.