Sudoku Strategy
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Probably the biggest factor in a puzzle's difficulty is the type of logic needed to find a solution. An easy puzzle requires fairly basic logic, while a tough puzzle depends on advanced logic such as Disguised Pair or X-wing.
If a Sudoku puzzle had 80 given (or seed) numbers, the solution would be trivial. If a puzzle had fewer than 10 starting numbers, even the most sophisticated computer program would be hard-pressed to find a solution. So the quantity of seed numbers comes into play the most at either extreme.
A Sudoku puzzle with many given numbers could be difficult if simultaneous fill-ins are not readily found. That is, if the next fill-in depends on the one before it, then arriving at a solution requires more exhaustive searching. Some people call it a "bottleneck."
Disguised Pair
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Sometimes referred to as “Hidden Pair,” this is a form of what graph theorists call Pile Exclusion. A Disguised Pair happens when the only instances of two numbers are in two cells, but those cells have other occupants as well. Because one of the two numbers must reside in one of the two cells, and the other number must reside in the other cell, any other occupant is a straggler and can be erased. As an example, consider the row below. Notice that the numbers 2 and 4 only occupy the third and sixth cells, along with a few other inhabitants.
Because the 2s and 4s are only found in two cells, stragglers (numbers besides 2 and 4) can be erased from the third and sixth cells. See the simpler, resulting row below.
Exposed Pair
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Sometimes called “Naked Pair,” this is a form of what graph theorists refer to as Chain Exclusion. If, for any given row, column or box, you have a pair of cells with only two matching numbers, you can eliminate any other occurrences of those numbers in the same row, column or box.
Say, for example, that numbers 2 and 4 are the only candidates in the third and sixth cells of a given row. Because there aren’t any other options in either of the two cells, the 2 and 4 couldn’t be candidates elsewhere in that row. See the partial puzzle below.
You can see in the graphic above that because 2 and 4 are the only residents of the third and sixth cells, they don’t belong anywhere else in that row, and “extras” can be safely erased. See the resulting row below.
X-wing Logic
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X-wing logic in Sudoku gets its name from the fact that occasionally two rows will contain only two occurrences of a candidate number, and those occurrences happen to share the same two columns; in other words, their arrangement forms a rectangle or square, and interconnecting lines drawn along the diagonals would make an “X” shape. The wing part of the term comes from the fact that only one of the two diagonals is possible, but we don’t know which one yet. See the graphic below.
If you look at the second and eighth rows, you can see how 4 is an option only in the second and fifth cells. These four 4s have two rows in common and two columns in common, which means that either the upper-left and lower-right cells are the correct locations, or the upper-right and lower-left cells are the correct locations. To satisfy the row requirement for a 4, one and only one of these can be true. In light of this, we can safely erase any other occurrence of a 4 in the second and fifth columns. See the resulting partial puzzle below.
In the picture above, you can see how we’ve kept the four original 4s, indicated by the arrows, and removed any other 4 from the second and fifth columns.