Advanced Techniques for Cross Sum Puzzles
Here are some techniques for use in solving more difficult puzzles.
Pocket Arithmetic
Most puzzles have several dead-end "pockets" of white cells that are approached by a single entry point. This technique allows you to calculate the value of the cell at the entry point, giving you an important clue for solving in towards the pocket, and out towards the rest of the puzzle.
Consider the classic "P-shaped pocket" shown below. The total value of all five cells in the pocket can be had by adding the column totals. (In this case, 8+19=27.) The value of the four cells in the pocket can be had by adding up the row totals. (16+4=20) The difference between these two totals (27-20=7) must be the value of the entry cell, because that is the only cell that is counted in one total and not the other.
(Hover over each step in turn, to see the technique illustrated.)
You can apply this technique however large or oddly-shaped the "pocket", although you might have to do more arithmetic. Keep an eye out for open passages that turn into pockets as you solve the cells at the other entrance.
Min-Max Technique
This technique often works when other techniques have failed to eliminate enough possibilities, especially in relatively open areas.Start with a cell in the area. Now work out both the smallest and largest value that can go in that cell, according to the row that it is contained in. The minimum value for a particular cell is found when all the other cells in the row or column are all at their maximum, when uniqueness is considered. (For example, consider a row of 4 cells adding up to 28. The maximum value for the three other cells in a row might be 9, but they can't simultaneously be 9. The highest they can achieve is some combination of 9,8, and 7, for a combined total of 24. Which makes the minimum for the first cell 28-24 or 4.
Now turn 90 degrees and consider the smallest and largest values for the column that cell is in. Often, this yields a different range which only partly overlaps the range determined by the row. This narrows the range to several values, some of which can be eliminated because they would conflict elsewhere.
Every time you narrow a range within a row (or column) you end up narrowing the range of its partners elsewhere in the row or column. By working back and forth, you can eventually hit upon the only solution.
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