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Saturday, March 5, 2011

KenKen Strategies

Most of the strategies used to solve Sudoku rely on the interplay of boxes and lines. Therefore, it is not surprising that they are not transferable to KenKen® (registered trademark of Nextoy LLC).

To see what Sudoku strategies do apply to KenKen, I will first define the concept of segment. A segment is a collection of cells belonging to a cage that are also on the same line (row or column). In the following example with four cages, the horizontal segments (along a row) are: A, BC, DE, F, G, H, IJ, KLM, and N, while the vertical segments (along a column) are: AF, B, CH, DI, EJN, GL, K, and M.

What follows is a list of Sudoku strategies adapted to KenKen.

‘Naked single’: If a cell only has a single candidate, that candidate must solve the cell. This is obvious: if there are no other possible candidates in a cell, the only one present must be it.

‘Unique’: If a candidate for a particular number is found in a single cell of a line (row or column), it must solve the cell.

‘Cleanup’: When you solve a cell, the digit that solves it cannot be a candidate anywhere else in any of the lines to which the cell belongs.

‘Naked pair’: If two cells in the same line but belonging to different cages only contain the same two candidates, it means that one of those candidates will solve one of the two cells, and the other candidate will solve the other cell. Therefore, the same candidates can be removed from all other cells of the line.

And now a couple of KenKen-specific strategies.

‘Pointing segment’: If you know one or more digits of a segment, even if you don’t know which cells they individually solve, you can remove candidates for the same digits from all other cells of the line to which the segment belongs.

‘Solving order’: Solve first all the cages that only contain one cell. Then write down the candidates of the cages with two cells that can only be solved by one, two, three, or four pairs. As next, look at cages with high totals obtained by multiplication, because they are likely to have a low number of possible combinations. Straight cages are always easier to solve than cages with bends.

Look for example at the following puzzle:

You can immediately write 8 in cell (0,4) and 4 in cell (8,5). Then, notice that the cages [(1,8) (2,8)], [(2,5) (3,5)], [(3,8) (4,8)], [(4,7) (5,7)], [(6,2) (7,2)], and [(7,3) (8,3)] all admit only one pair of combinations. You can therefore fill them with the pairs of alternatives [9 1], [9 8], [8 5], [9 8], [8 1], and [7 3].

At this point, also the cage [(5,8) (6,8)] can only be filled with a single pair of alternatives. Normally, to obtain a total sum of 13 you can add any of the pairs [9 4], [8 5], and [7 6], but in this case, as the digits 9 and 8 have already been allocated to other segments of the same column, only the pair [7 6] remains. As an example of a pair with more alternatives, I filled in the cage [(4,5) (5,5)} with the possible alternatives [6 1], [3 2], [2 3], and [1 6].

The square cage at the top [(0,5) (0,6) (1,5) (1,6)] requires a product of 200. The prime factors of 200 are 2 2 2 5 5. This means that the four cells must hold either [8 5 5 1] or [5 5 4 2].

The cage [(7,4) (7,5) (7,6)] can only contain [9 7 5], because the factors 0f 315 are 3 3 5 7. This means that the 7 that we had considered possible for the cell (7,3) has become impossible.

Similarly, the cage [(4,1) (4,2) (5,1)] can only contain [1 7 7], because 49 only admits the two factors 7 7. As a result, we can remove the pair [1 6] from the cage [(4,5) (5,5)].

At this point, the puzzle looks as follows:

Incidentally, I have found out how to make my CleverClever puzzles solvable analytically. At least, I have managed to solve the first ten I generated, although in a couple of cases only on the second attempt.

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