Tutorial 4

Pairs, Triplets and Groups

We can make some interesting deductions about the possible contents of a square by looking at pairs of possible numbers contained in other positions in the same row, column, or 3×3 block. Let’s have a look at the row below:

We’ve already counted out all of the possible numbers for each unsolved square, so we can see that both the fourth and the seventh column (marked in green) can contain either 3 or 7, and nothing else. There are only two possible cases: the column four could hold 3 and column seven could hold 7, or column 4 could hold 7 and column seven could hold 3. There are no other possibilities for those two squares. In neither of these cases could any of the other squares hold a 3 or a 7. So, we can eliminate and 3s or 7s we find in any other squares in this row – in this case the first and eighth squares (marked in pink).

Methods like this may not necessarily give you the full solution to a square, but just being able to eliminate another possibility is very valuable when you get on to the harder puzzles.

Triplets and Beyond

The same principle applies to triplets: If you can find unsolved squares in a row (or column, or box) which contain the same three possible numbers, then you can be certain that whatever the actual solution may be, those three numbers occur only within those squares – and thus can be eliminated from consideration anywhere else within this row (or column, or box). To make this a little more concrete, look at the following row:

There are three squares (marked in green) containing only the numbers 2,3 and 7. By the same logic as before, this means that 2,3, and 7 are restricted to these three boxes, and can be deleted wherever else they are found in this row (or column, or box). In this case, the first and sixth squares (marked in pink) contain entries which can be eliminated.

This principle can be extended from pairs and triplets up to groups of four, five – theoretically up to eight. As the number increases, however, the situation becomes not only progressively more unlikely, but harder and harder for the human eye to spot. Pairs will be constantly helpful to you, triplets sometimes helpful, and as for the rest – well, it might happen…

One Last Complication

The three green squares still contain only the possible numbers 2,3 and 7 between them, but this time no single square contains all three numbers.

The most general case of this rule is: if n squares in a row, column, or block contain only a given group of n numbers between them, when n can be between 2 and 8, then any other occurences of those n numbers elsewhere in the same row, column, or block can be eliminated. That’s a bit of a mouthful, but I hope we’ve lead up to it in sufficiently reasonable steps.

What do you do if you’ve tried every technique there is and you can’t go on? Giving up and having a nice cup of tea is probably the best thing, but if you don’t want to do that, you could read Tutorial 5