How to Find Solutions to Cross Sums

A cross sum has interlocking rows and columns filled with numbers instead of letters.

A cross sum is a mathematical puzzle game based on the popular crossword puzzle, retaining the grid of interlocking rows and columns while replacing words with addition facts. Strategy for solving cross sums consists of eliminating potential values in each square, row and column using logical deduction, as well as calculating the values of certain special squares using basic mathematics.

Instructions

1. Eliminate potential number values in a row, column or cell using the rule that a number may not appear more than once in a row or column and the rule that only the numbers 1 through 9 may be used. For example, if a three-square row has a sum of 8, the only possible combinations are "1-3-4" and "1-5-2." All other three-number combinations use a number twice. If a three-square row has a sum of 21, then every square has a value of at least 4, because if one square is a 3, the remaining two squares add to make 18, which can only be done with 9 + 9.

2. Eliminate potential number values in a square using the more advanced strategy of contingencies. This strategy involves guessing the values of a row, column or cell and following the logical consequences as far as possible. If you reach a contradiction, then those values are incorrect. If not, then the values are possibly, but not definitely, correct. For example, if you know that a square is either a 1 or a 2, pretend that it's a 2. This excludes all other values in that square's row and column from being a 2. Suppose this forces another square in the row to be a 3 instead of a 2. If this is impossible because there is another 3 in that square's row or column, then the original guess of the first square being a 2 is incorrect.

3. Calculate the value of special squares that are part of a row or column in a region but not both. For example, if a region with three rows with sum 15, 20 and 8 and three columns with sum 21, 11 and 6 has a square that is in one of the rows but not one of the columns, you can calculate its value by subtracting the sum of the columns from the sum of the rows: (15 + 20 + 8) - (21 + 11 + 6) = 41 - 38 = 3. The square has a 3 because it is the only square in the region that is part of the row sum 41 but not part of the column sum 38.

References

• Jim Loy's Home Page: Solving Cross Sums; Jim Loy
• Kevinpluck.net: Kakuro Cross Sum Combinations