[UP: Sudoku Solving Techniques]
[PREV: Y - Wings]
[NEXT: Multi Colouring]

Single Colouring

Warning! The Single 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 single colouring technique is a candidate elimination technique, meaning that it eliminates possible values from cells. It is actually a VERY powerful technique. All intersection reductions are special cases of this technique, but it goes far beyond this. To use the single colouring technique, you need to construct a pairwise colouring of the candidates of a particular digit.

Pairwise colouring

To construct a pairwise colouring, you need to identify locked pairs of the digit you're interested in. A locked pair is a pair of cells which are the only places in a group where a digit can go. The example below shows a pairwise colouring of the digit "7":

In this case, the cells marked A and A* have been pairwise coloured. To see how this is done, first label the cell in the top left A. Because there is only one other place on that row where a "7" can go, these cells form a locked pair, and we can label the other cell on that row A*. Now, we see that our A* cell shares a box with only one other possible "7", so we label that A. It also shares a column with only one other possible "7", so we label that one A as well, and so on.

Type 1 Single Colouring

So what use is all this? As it turns out, it allows us to eliminate some potential "7"s, using the Type 1 Single Colouring rule: If any cell shares a group with a label and its conjugate for a digit, the cell cannot take that digit. So in our example, if a cell can "see" cells labelled both A and A*, it cannot be a "7". Here are the two eliminations highlighted:

The reason this works is because if any cell labelled A contains a "7", all of the As MUST contain a "7", and NONE of the cells labelled A* can contain a "7". The reverse is also true. So if a cell can see a two cells with opposite labels, then no matter whether the "7"s occur in the A cells or the A* cells, we can eliminate a "7" from that position.

Type 2 Single Colouring

Type 2 single colouring is an incredibly powerful tool, allowing you to eliminate an entire set of candidates from the puzzle. The rule for type 2 single colouring is: if two of the same label for a digit share a group with each other, then both must be false, and all cells with that label can have the digit eliminated. Below is an example:

The single colours (type 2) pattern shown above allows the digit "8" to be eliminated from all the cells highlighted in red. I have established a pairwise colouring on the digit "8", using the labels A and A* as before. Remember that if we can show that one of these labels isn't a valid position for an "8", then there cannot be an "8" in any cell with the same label. Now look at the bottom middle box. Notice that there are two cells labelled A in that box. Because A's are either all "8"s or all not "8"s, and because two As share a box, we know that none of the As can possibly be an "8", and we can eliminate all of them.

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