The Cross Sums is a logic puzzle that is often referred to as a mathematical transliteration of the crossword. In principle, Cross Sums puzzles are integer programming problems, and can be solved using matrix techniques, although they are typically solved by hand. Cross Sums are regular features in most, if not all, math-and-logic puzzle publications in the United States; Dell Magazines uses the Cross Sums name, which was formerly unique to them but is now in common use among various publishers (although some other names, such as Cross Addition, are still used). In Japan, the puzzle is called Kakuro, an abbreviation of kasan kurosu (addition cross); its popularity there is immense, second only to Sudoku among Nikoli's famed logic-puzzle offerings. In an international tapdance, Kappa reprints Nikoli Kakuro in the United States, in GAMES Magazine under the name Cross Sums. The Guardian in Britain began printing the puzzle under the name Kakuro in September 2005; since then many other British papers have followed suit and now also print daily puzzles.
Variants of Kakuro
A relatively common variant of Cross Sums is its logical successor, Cross Products (or Cross Multiplication), where the clues are the product of the digits in the entries rather than the sum. Another variant is Arrow Numbers, where the combinations for each clue value cannot be repeated within the grid.
The final puzzle of the 2004 United States qualifier for the World Puzzle Championship is titled Cross Number Sums Place: it is a Cross Sums where every row and column of the grid (except the top row and leftmost column as usual) contains exactly nine white cells, none of which - even across multiple entries - are allowed to use the same digit twice, like a Number Place (Sudoku); in addition, small circles are printed on the borders between some white cells; numerically adjacent digits must be placed astride those circles, and may not appear orthogonally adjacent when not astride a circle.