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The problem of determining if a partially filled square can be completed to form a Latin square is NP-complete. [22] The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that nine particular 3×3 adjacent subsquares must also contain the ...
For example, if 45 eggs are to be put into 12-egg cartons, then after the first 3 cartons have been filled there are 9 eggs remaining, which only partially fill the 4th carton. The answer to the question "How many cartons are needed to fit 45 eggs?" is 4 cartons, since = + rounds up to 4.
For classical Sudoku, the number of filled grids is 6,670,903,752,021,072,936,960 (6.671 × 10 21), which reduces to 5,472,730,538 essentially different solutions under the validity-preserving transformations. There are 26 possible types of symmetry, but they can only be found in about 0.005% of all filled grids. An ordinary puzzle with a ...
Describe how all Latin subsquares in multiplication tables of Moufang loops arise. Proposed: by Aleš Drápal at Loops '03, Prague 2003; Comments: It is well known that every Latin subsquare in a multiplication table of a group G is of the form aH x Hb, where H is a subgroup of G and a, b are elements of G.
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The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...