Advanced Sudoku Solving Techniques

How to solve complicated sudoku puzzles

Sudoku puzzles can in a large number of cases be solved using just a couple of rules.

These rules require us to think "what numbers could possibly go in this cell" and "where could this number go in this region", whereby a cell is an individual square on the grid and a region is a row, column, or grid.

These are rule 1 and rule 2, and are usually all that's required to solve easy or moderate puzzles, the difference being how many possible paths to solving there are available at each time as you progress through solving the puzzle.

Rule 1, then says 'based on the values already placed in this region, what numbers are left as options for this cell', whilst rule 2 says 'which cells in this region can a certain number possibly go on, based on the other values in the respective regions'. This whilst rule 1 might allow for instance 1234567 to be placed in a cell, if there is only one cell in the region with the candidate '6' and it is this cell, we can place it straight off.

So these are the two main and intuitive rules used to solve sudoku, often called the donkey functions, because they do most of the work.

Some people leave it there, but there are other rules too. This article is going to look at two of these rules. They have various different names, so this is just going to explain the logic. They involve looking at sets of numbers, but don't let that put you off. The idea is simple, so we'll start with a simple example.

Imagine we have two cells in a row that have these candidates '4,5' and '4,5'. Now what this means is that these cells MUST be 4 and 5, whether it is 5,4 or 4,5 because if one is 4 the other must be 5 and if one is 5 the other must be 4.

This means that we can eliminate '4' and '5' as candidates from other cells in that region - and sometimes we do need to do that to progress to a solution of a puzzle without guessing. That case was obvious, as is say 123, 123, 123, in three cells in a region. But how about this:
24, 234, 34... well, it might be less obvious here, so work through in your head what happens in case? If the first cell is 2, then the other cells are 34. If it's 4, the third cell is 3 and the second cell is 2. So either way, we see the 234 in this region MUST go in these cells, and so we eliminate them from other cells in the region.

The rule abstractly says that if there are n cells that contain a set of different numbers, and 'n' is equal to the number of members of that set, then the members of that set must be placed in those cells and can therefore be removed from other cells in that region.

That rule as stated above may sound complex, but it's the same thing: we still just count. 23, 23 means there are two numbers that are candidates in a total of two cells, so we apply the rule. 123, 23, 13 has three numbers as candidates in three cells, so the rule applies. But 123, 23, 134 has four numbers as possible candidates in three cells, so we can't apply the rule.

The fourth rule we're going to look at pretty much reverses the process above. Whereas with the above rule, we remove candidates from OTHER cells in the region, this time around we remove values from the cells THEMSELVES.

Let's see this in action. Imagine we have this in a region:
1, 2, 346, 34678, 456789, 678, 456789, 34678, 678

Now, you can actually apply the rule above here (see if you can find which cells it applies too) but you have to work very hard to spot it!

However, there is something else we spot. Look at the numbers '5' and '9'. You will notice that they are only candidates in two cells in the region - and when we have two numbers only being candidates in two cells, we know that those two cells MUST contain those numbers, and therefore we can turn the region into this:
1, 2, 345, 34678, 59, 678, 59, 34678, 678

That looks better! And that's exactly the sort of logic you may have to apply sometimes when solving puzzles.

These patterns can be hard to spot, and harder again with jigsaw and toroidal regions which obscure a puzzle even more, but with practice you'll start to learn these patterns.

Combine the four rules above with region intersection and you should be able to solve the vast majority of 'fair' hard puzzles that you see in print. Any puzzle that is created using human solve algorithms should be solvable using these techniques, apart from a small number that use arguably fair techniques such as X-wings and effectively low level 'if this then contradiction, so this' type logic.

Why not try using some Advanced Sudoku Logic with the toroidal sudoku x puzzle that this links to!

Author: Dan

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Last Updated: Nov 1st 2007

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