Room square
Encyclopedia
A Room square, named after Thomas Gerald Room
, is an n × n array filled with n + 1 different symbols in such a way that:
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.
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.
Thomas Gerald Room
Thomas Gerald Room was an Australian mathematician who is best known for Room squares.-Biography:Thomas Room was born on 10 November 1902, near London, England. He studied mathematics in St John's College, Cambridge, and was a wrangler in 1923...
, 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:
| 0,7 | 1,5 | 4,6 | 2,3 | |||
| 3,4 | 1,7 | 2,6 | 0,5 | |||
| 1,6 | 4,5 | 2,7 | 0,3 | |||
| 0,2 | 5,6 | 3,7 | 1,4 | |||
| 2,5 | 1,3 | 0,6 | 4,7 | |||
| 3,6 | 2,4 | 0,1 | 5,7 | |||
| 0,4 | 3,5 | 1,2 | 6,7 |
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 Anstice to solve Kirkman's schoolgirl problemKirkman's schoolgirl problem
Kirkman's schoolgirl problem is a problem in combinatorics proposed by Thomas Kirkman in 1850 as Query VI in The Lady's and Gentleman's Diary...
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
Thomas Gerald Room
Thomas Gerald Room was an Australian mathematician who is best known for Room squares.-Biography:Thomas Room was born on 10 November 1902, near London, England. He studied mathematics in St John's College, Cambridge, and was a wrangler in 1923...
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.
See also
- Combinatorial designCombinatorial designCombinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties....
- Latin squareLatin squareIn 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...
- Magic squareMagic squareIn recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
- Square matrices