Influence of entries in critical sets of Room squares

We consider structures which have rules for completion such as balanced incomplete block designs, Latin squares, Rooms squares, F-squares, Youden squares, regular graphs, colourings, finite geometries and difference sets. In particular we are concerned with the problem of unique completion of structures given partial information. If the partial structure can be uniquely completed then this partial structure together with the rules contains the same information as the final structure. In this paper, we study the information inherent in partial Room squares, where it is not possible to uniquely complete the square. We study the influence and power of parts of the partial square on the unique completion of larger partial squares containing those parts. That part of Room square, called the strong box, which is inaccessible to all the q-subsets of a critical set may be thought to contain the secret information. We study the size of the secret which will be used to model secret sharing schemes.