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These are some of the best contributions made to the forum. Your tips could appear here if you have joined the free Sudoku community at Sudoku Discussions

A new rule from Dave Richards?
I downloaded your file Solving Sudoku. I had already arrived at all of the approaches it contains independently (and, very interestingly, I also hit on the term "bifurcation" independently).

You rightly identify the following situation as a subset of simple triplets which excludes all other instances of the triplet's components in Row 1 (and of course the same rule applies in respect of columns or boxes) [see also John MacLeod on this page,

Simple Triplet 1

You correctly go on to extend the principle to the more general case of what I call the "simple subset", eg,
Simple Triplet 2

- these numbers being a subset of the "quadruplet" 1357, they similarly preclude any instance of those component numbers in that "domain" (my generic term for rows, columns and boxes).

The Complex Subset

But here's another rule - not a common one but which seems worth noting, that I call the complex subset.

This sort of combination allows you to eliminate the value 4 from the whole of the rest of box 1 (since 4 must occur in either square 1 or 3)... The essential condition is that two of the triplets occur in two intersecting domains with a duplet subset in only one of those domains. In principle the rule is extensible to quadruplets etc - but such elaborate combinations seem likely to be uncommon. - Dave Richards

Paul Bowden has some tips for all those solvers who have set up spread-sheet programs as work sheets:

Further to all of the Excel-based tips, and how best to unwind to an earlier position after a wrong guess, the following make might things easier.

Don't forget that ctrl-Z will unwind the last action. Simply tap ctrl-Z until you get back to where you were.

You can make it a bit easier to get back to where you were by setting a spare cell, with =sum(A1:I9), assuming that your grid is in A1 to I9. Before you start your guess, note down the current total. Simply unwind with ctrl-Z until you get back to where you were. If you overshoot, ctrl-Y will wind you forward.

This tip comes from Telegraph solver David Brook:

I think my tip might help solvers a lot. It is a little laborious the first time, but the gains are enormous. The Impossible Diabolical in the Sudoku Experiment took me about 2.5hrs utilising my method. My tip is to set out nine grids using blank worksheets. Allocate one grid to each of the values 1 to 9 and work from there. Eg, say the given starting grid contains a 5 at grid point 2,3, then working on the new "5" grid one can blank out all of row 2 and all of column 3 and all of square 1. Then you can blank out the 2,3 grid point on the other 8 grids. Proceed in this way for all the given data and the subsequent steps in the solution become much easier to see - and the 'what if' thoughts easier to remember.

Steve Edmonds writes:

Further to John MacLeods useful tips, I too use an Excel grid, (see first spreadsheet below) but colour fill the cells. The given numbers have bright yellow fill and then all numbers derived from the given numbers have a paler yellow fill. When it comes to a decision point in the "Tough" and "Diabolical" puzzles , then copy the grid within the spreadsheet and use another colour, that way you can then get back to your decision point if that particular option/decision was incorrect.

Now after a few weeks since the Sudoku puzzles first appeared, I only have to resort to the spreadsheet for Tough and Diabolical puzzles to determine the solution.

Further to Malcolm Sevren's comments about "showing the logic track that was used to complete the grid" the second spreadsheet below shows where the cells have 2 numbers in them; the 1st larger font size number being the solution and the 2nd smaller font size number being the sequence of derivation... I guess the down side is that it would take up too much space in the paper to be legible.

Here's a tip in a similar vein from a Daily Telegraph sudoku solver, Brian Thubron for all pc owners:

Recovering from wrong guesses can be laborious with pencil and paper. Here's a tip to get back to the point before the "wrong" guess.

Create a blank grid with a spread sheet program such as Microsoft Excel or Lotus 123.
Fill in the given numbers then save as a new file, say "puzzle1.123", this will preserve your blank.
I use a large point size font for the given and solved numbers and a smaller point size for the rest. This allows you to insert more numbers in the cell.
When you get to a point where you have to guess, save the file as another name, eg, "puzzle1x.123" and continue with the puzzle.
If you come to a dead end, simple load the file puzzle1x.123 and take the other "guess".
Before changing anything, save the file again as, say, "puzzle1y.123". The reason for saving as another name is that "autosave" keeps the current copy up to date and the file "puzzle1x.123" will be unchanged.
Each time you need to make a guess, save the file as a checkpoint, to give you a rapid wind-in of the silk!

I was very pleased to receive the following solving tips from John MacLeod. Having the website to look after and a bad cold put me behind on my own tips. John has looked at a subject I was about to cover, but in much greater depth than I had planned. Here it is:

The Strategy of "Pairs" and "Trios"

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!

Sudoku - the holiday companion

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