Kakuro - Solving Techniques

Solving Techniques

Although brute-force guessing is of course possible, a better weapon is the understanding of the various combinatorial forms that entries can take for various pairings of clues and entry lengths. Those entries with sufficiently large or small clues for their length will have fewer possible combinations to consider, and by comparing them with entries that cross them, the proper permutation—or part of it—can be derived. The simplest example is where a 3-in-two crosses a 4-in-two: the 3-in-two must consist of "1" and "2" in some order; the 4-in-two (since "2" cannot be duplicated) must consist of "1" and "3" in some order. Therefore, their intersection must be "1", the only digit they have in common.

When solving longer sums there are additional ways to find clues to locating the correct digits. One such method would be to note where a few squares together share possible values thereby eliminating the possibility that other squares in that sum could have those values. For instance, if two 4-in-two clues cross with a longer sum, then the 1 and 3 in the solution must be in those two squares and those digits cannot be used elsewhere in that sum.

When solving sums which have a limited number of solution sets then that can lead to useful clues. For instance, a 30-in-seven sum only has two solution sets: {1,2,3,4,5,6,9} and {1,2,3,4,5,7,8}. If one of the squares in that sum can only take on the values of {8,9} (if the crossing clue is a 17-in-two sum, for example) then that not only becomes an indicator of which solution set fits this sum, it eliminates the possibility of any other digit in the sum being either of those two values, even before determining which of the two values fits in that square.

Another useful approach in more complex puzzles is to identify which square a digit goes in by eliminating other locations within the sum. If all of the crossing clues of a sum have many possible values, but it can be determined that there is only one square which could have a particular value which the sum in question must have, then whatever other possible values the crossing sum would allow, that intersection must be the isolated value. For example, a 36-in-eight sum must contain all digits except 9. If only one of the squares could take on the value of 2 then that must be the answer for that square.

A "box technique" can also be applied on occasion, when the geometry of the unfilled white cells at any given stage of solving lends itself to it: by summing the clues for a series of horizontal entries (subtracting out the values of any digits already added to those entries) and subtracting the clues for a mostly-overlapping series of vertical entries, the difference can reveal the value of a partial entry, often a single cell. This is possible because addition is both associative and commutative.

It is common practice to mark potential values for cells in the cell corners until all but one have been proven impossible; for particularly challenging puzzles, sometimes entire ranges of values for cells are noted by solvers in the hope of eventually finding sufficient constraints to those ranges from crossing entries to be able to narrow the ranges to single values. Because of space constraints, instead of digits some solvers use a positional notation, where a potential numerical value is represented by a mark in a particular part of the cell, which makes it easy to place several potential values into a single cell. This also makes it easier to distinguish potential values from solution values.

Some solvers also use graph paper to try various digit combinations before writing them into the puzzle grids.