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A Latin square is said to be reduced (also, normalized or in standard form) if both its first row and its first column are in their natural order. [4] For example, the Latin square above is not reduced because its first column is A, C, B rather than A, B, C. Any Latin square can be reduced by permuting (that is, reordering) the rows and columns ...
To reduce the problem somewhat, a Latin square can always be put into a standard form known as a reduced square. A reduced square has its top row elements written in some natural order for the symbol set (for example, integers in increasing order or letters in alphabetical order). The left column entries are put in the same order.
A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s, t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells ...
A Sudoku is a type of Latin square with the additional property that each element occurs exactly once in sub-sections of size √ n × √ n (called boxes). Combinatorial explosion occurs as n increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved, as illustrated in the following table.
In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the Loops (Prague) conferences and the Milehigh (Denver) conferences.
2. Equivalence class: given an equivalence relation, [] often denotes the equivalence class of the element x. 3. Integral part: if x is a real number, [] often denotes the integral part or truncation of x, that is, the integer obtained by removing all digits after the decimal mark.
Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Then, the upper-right square of any polyomino of size n can be attached to the bottom-left square of any polyomino of size m to produce a unique (n+m)-omino. This proves A n A m ≤ A n+m.
In mathematics, specifically abstract algebra, a square class of a field is an element of the square class group, the quotient group / of the multiplicative group of nonzero elements in the field modulo the square elements of the field.