8 November 2006

Sudoku Tips

Sudoku Tips from sudoku.org.uk: 
There is a further strategy that can be carried out between those set out on the web in Week 1 and Week 2, before embarking on those set out as strategies to solve Diabolical puzzles. The strategy is an extension of that of twins and triplets, but cannot easily be performed by visual examination alone.

It is first necessary therefore to bite the bullet, and once the initial tasks of filling those cells into which one and only one number can be put are complete, the solver should transcribe the work so far onto a separate piece of paper containing a larger 9x9 grid.

Then they should write in each cell ALL possible numbers that could legitimately go into that cell. Then they should look for what I call ‘pairs’ and ‘trios’.

In its simplest form, a ‘pair’ comprises two cells on the same row (or column), each containing two and only two possibilities, those two possibilities being the same.

For instance, 3 7 and 3 7. Obviously if the 3 in the first cell is correct, then the 7 in the other cell is correct, and vice versa. So the 3 and the 7 can thus be crossed off wherever they occur as possibilities in any other cell in the row (or column) concerned.

The two cells can be anywhere in the row or column. But if they appear in the same box, then all instances of the two numbers, 3 and 7 in this example, can also be crossed of as possibles in any other cell in that box.

There may be instances of a ‘pair’ of cells occurring as two cells in a box, but not on the same row or column - separated diagonally in other words. By the same reasoning, all other instances of the two numbers can be removed wherever they occur as possibilities in any other cell in that box.

Trios

The same reasoning can be applied where there are three cells in the same row or column or box, sharing the same three numbers, and only those numbers. The simplest example is 3 7 - 3 9 - 7 9; but 3 7 - 3 9 - 3 7 9 , 3 7 - 3 7 9 - 3 7 9 and even 3 7 9 - 3 7 9 - 3 7 9 are further examples. Here, all three numbers would be eliminated from other cells in the row or column and/or box affected, just as described above in the case of ‘pairs’.

Hidden Pairs

Sometimes a ‘pair’ can be hidden.

Consider by way of example two cells 3 - 7 and 3 - 7 - 9 occurring on the same row, column or cell. Then, provided neither the 3 nor the 7 occur in any other cell on that same row, column or cell, it must be that the two cells concerned are 3 and 7, though which way round is uncertain. The 9 can thus be eliminated. This means that the ‘hidden pair’ is turned into a pure ‘pair’ and can then be treated as above.

By the same reasoning, 3 - 7 - 8 and 3 - 7 - 9 could also conceal a hidden ‘pair’; in which case both the 8 and the 9 would be eliminated

Note that a hidden pair can be found within a box, but even though its two cells share the same row (or column), they do not form a ‘hidden pair’ as far as that row (or column) is concerned, because other 3s or 7s appear in that row (or column).

Once however the pair has been turned into a pure pair by virtue of its belonging to a box, it will then become a pure pair within the row (or column) in which it might also appear, and can then be treated as such. One should not therefore ignore the possibility of two cells being a hidden pair just because they do not qualify as such when considered in the context of the row (or column) on which they appear. They should be further considered if they also appear in the same box.

By the same reasoning, ‘trios’ can be hidden; in which case they would be treated in the same way as hidden ‘pairs’

The knock-on effect of applying these strategies will help unblock many closed avenues. These strategies will therefore go a long way in helping solvers to arrive at complete solutions to puzzles, without their having to resort to the trial and error solutions described under the heading of Diabolical puzzles in the tips so far presented on the web.

The strategies are not however guaranteed to find a complete solution, as the Daily Telegraph puzzle of March 11th will confirm!