A Room square, named after
Thomas Gerald Room, is an ''n'' × ''n'' array filled with ''n'' + 1 different symbols in such a way that:
# Each cell of the array is either empty or contains an unordered pair from the set of symbols
# Each symbol occurs exactly once in each row and column of the array
# Every unordered pair of symbols occurs in exactly one cell of the array.
An example, a Room square of order seven, if the set of symbols is integers from 0 to 7:
It is known that a Room square (or squares) exist if and only if ''n'' is odd but not 3 or 5.
History
The order-7 Room square was used by
Robert Richard Anstice to provide additional solutions to
Kirkman's schoolgirl problem
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Rev. Thomas Penyngton Kirkman in 1850 as Query VI in '' The Lady's and Gentleman's Diary'' (pg.48). The problem states:
Fifteen young ladies in a school walk out three abre ...
in the mid-19th century, and Anstice also constructed an infinite family of Room squares, but his constructions did not attract attention.
Thomas Gerald Room reinvented Room squares in a note published in 1955, and they came to be named after him. In his original paper on the subject, Room observed that ''n'' must be odd and unequal to 3 or 5, but it was not shown that these conditions are both necessary and sufficient until the work of W. D. Wallis in 1973.
Applications
Pre-dating Room's paper, Room squares had been used by the directors of
duplicate bridge
Duplicate bridge is a variation of contract bridge where the same set of bridge deals (i.e. the distribution of the 52 cards among the four hands) are played by different competitors, and scoring is based on relative performance. In this way, eve ...
tournaments in the construction of the tournaments. In this application they are known as Howell rotations. The columns of the square represent tables, each of which holds a deal of the cards that is played by each pair of teams that meet at that table. The rows of the square represent rounds of the tournament, and the numbers within the cells of the square represent the teams that are scheduled to play each other at the table and round represented by that cell.
Archbold and Johnson used Room squares to construct experimental designs.
There are connections between Room squares and other mathematical objects including
quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...
s,
Latin square
In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin sq ...
s,
graph factorization
In graph theory, a factor of a graph ''G'' is a spanning subgraph, i.e., a subgraph that has the same vertex set as ''G''. A ''k''-factor of a graph is a spanning ''k''- regular subgraph, and a ''k''-factorization partitions the edges of the gra ...
s, and
Steiner triple systems.
See also
*
Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These co ...
*
Magic square
*
Square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
References
Further reading
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*{{mathworld, title=Room Square, urlname=RoomSquare, mode=cs2
Combinatorial design