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Multi Colouring

Warning! The Multi Colouring technique is an advanced technique, meaning that it is only used when you ask Puzzle Tiger for a "Very Hard" puzzle. As such you will almost certainly never come across a published puzzle in a newspaper or magazine which requires this or any of the more advanced techniques. If, however, you're interested in really stretching yourself, read on...

The multi-colouring technique is a candidate elimination technique, meaning that it eliminates possible values from cells. All X-Wings are special cases of this technique. It is a very powerful technique, and it is also Puzzle Tiger's most advanced technique. To use the multi-colouring technique, as with the Single Colouring technique, you need to construct a pairwise colouring of the digit you're interested in. This time, however, when you run out of locked pairs for the first set of labels (A and A*), if there are any locked pairs left, you continue labelling them with the next set of labels (B and B*) and so on. Here is a pairwise colouring example:

There are four chains of locked pairs of "6"s in this example, which I've labelled A and A*, B and B*, C and C* and D and D*. Sometimes elements of different chains share a group. The reason why we can't just continue the same chain is because in every case, there are more than 2 cells in the group, so there isn't a locked pair to continue the chain.

Type 1 Multi Colouring

The type 1 multi colouring rule is: If any two labels from different chains share a group, then any cell sharing a group with both their conjugates cannot contain the digit. To understand how to use this rule, here is the above pairwise colouring, with the eliminations marked:

The digit "6" can be eliminated from the cells highlighted in red. First, let's look at the red cell in the left middle box. It shares a box with a cell labelled A and also shares a row with a cell labelled C*. Now look at the top left box, which is shared by a cell labelled A* and a cell labelled C. Because A* and C share a group, at least one of them isn't a "6". This means that at least one of A and C* MUST be a "6". So anything that can see them both can't be a "6" - hence we can eliminate the "6" from our cell.

Looking at the red cell in the right middle box, we see that it shares a row with A and a column with D*. Looking at column 7, we see that A* shares with D. So we can eliminate "6" from this cell as well.

Type 2 Multi Colouring

The type 2 multi colouring rule, like the type 2 single colouring rule, allows for the elimination of a complete set of candidates. The rule is that: if one type of label from a conjugate pair can see both of another conjugate pair, then that label can be eliminated. A conjugate pair is a pair like A and A* or B and B*. An example is shown below:

In this example, the digit "7" can be eliminated from all cells highlighted in red or orange. There are two conjugate chains in this example, labelled A, A* and B, B*. Look at the cells highlighted in orange. These are both labelled A. The top orange cell shares a row with a cell labelled B, and the bottom orange cell shares a row with a cell labelled B*. Because one of B and B* must be a "7", and because cells labelled A can see both of them, none of the As can be "7"s.

There are many more techniques that can be used to solve sudoku puzzles - swordfish, jellyfish, turbot-fish, y-wing chains, empty rectanges, remote pairs, nice loops, multi-digit multi-colouring, dual implication chains, triple implication chains, nishio and so on. Puzzle Tiger does not, however, implement any of these (because they are far beyond what the vast majority of players would ever want to use), so if you have mastered all the techniques in this reference, you will be able to solve any of the puzzles that Sudoku Tiger generates. Have Fun!

Copyright © Adam A. Brown, 2006, All Rights Reserved. www.sudokutiger.com