Cross Sums

Cross Sums are the mathematical equivalent of crossword puzzles. In principle, they are integer programming problems, and can be solved using matrix techniques, although they are typically solved by hand.

Standard Play

The canonical puzzle consists of a grid of rows and columns, with provided row sums and column sums. Solution entries are filled with digits from 1 to 9. No row sum entry or column sum entry can duplicate a digit, and the entries must ultimately sum to the respective row sums and column sums.

Possible Sums

All sum combinations can be represented using a bitmapped representation. This representation is useful for determining missing and required values using bitwise logic operations.

Solving Techniques

• Brute Force Supposition
• Constant/Offset Constraints
• Minimum/Maximum Sum Constraints
• Missing/Required Sum Values

Symmetry

There are two kinds of symmetry in readily identified in cross sums. Minimum and maximum constraints are duals, as are missing and required values.

Variants

There are a number of variants. Some allow for limited duplication of digits or for larger sums.