Combinatoric techniques[ edit ] Although brute-force guessing is possible, a more efficient approach is the understanding of the various combinatorial forms that entries can take for various pairings of clues and entry lengths. The solution space can be reduced by resolving allowable intersections of horizontal and vertical sums, or by considering necessary or missing values. 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.

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