1. Sudoku 7

Su Dokuis usually played on a 9 by 9 board, divided into 3 by 3cells. You can generalize this into playing on an M×M squareboard broken into non-overlapping rectangular cells, each containingM squares (an obvious question is to how define a puzzle where M is aprime number). The solution of the puzzle is to place M symbols on theboard such that each row, column or cell contains each symbol exactlyonce, without moving the initial clues.

Puzzles with M9 are called.Technorati tag:Related topic:Thanks to the following people who pointed out errors and generallyhelped me keep this page of tips in tip-top shape: Larry (LA, CA,USA), Justin Pearson, Tom Schoenemann (Michigan, USA), Mike Fleming(New Ipswich, New Hampshire, USA), Mike Godfrey (Manchester, UK), G B(whose name I wish I knew). In this page:.How to solve Su Doku: tips The mathematics of Su Doku. werethe first set I ever found. They take you from nothing into formalreasoning methods with colourful names like 'three in a bed'at your own pace. What I liked best about this site are the animatedtutorials. from the Daily Telegraph is by now somewhat of a classic. Itis a very well-written systematic explanation of the basic techniques forsolving Su Doku.

It introduced the nomenclature 'Ariadne's thread'. web site has a veryfine page of Su Doku tips, starting with the most basic element: find thesingletons, and progressing to complicated and bizarrely named rulesof Su Doku logic like the 'Swordfish'. also does a good job ofexplaining the rules of inference, one per blog.

Since it is a blog, ittakes each rule by itself and spends time explaining it. It contains someof the best explanations I have seen of the more complex rules. has a nice tool:it can give hints on a puzzle, solve one step while telling you howit was done, or solve the full puzzle and explain each step. You can playgames that it generates, or type your own puzzle into it.

Why another tutorial? Because you don't really need to know many tricks. Ishow this using a relatively hard puzzle by Wayne Gould, who createsof London. These are rated in difficulty from mild (thesimplest) to fiendish (the one on the left). Gould that none of his puzzles ever need trial anderror solutions. If you follow this example through you will find that you neverreally need very complicated tricks either.

Another way of solving this very puzzleis given by Roger Walker in one of his tutorials. Our methods differ: I try toillustrate some often-used tricks in this example. 'When you have eliminated the impossible whatever remains, howeverimprobable, is the truth', said Sherlock Holmes. This is the principleby which we put the 3 in the top row. 1, 2 and 7 are eliminated by theclues in the row; 4, 5, 6 and 9 by those in the column, and 8 by the cell.This leaves the truth.

I don't see it as very improbable; but one mustgive the master some poetic license. This rule may or may not be useful tobegin things off, but it is indispensible in the end game (especially whenit is coupled with the hidden loner rule of Step 8). Let's see how to place a 4 in the bottom right cell.

The blue lines show that it mustgo right into the bottom-most row, because the other two rows already have a 4 in them.These are the slices. Merging. Now one of the three squares in the bottom row of the cell alreadyhas a clue in it. The other square is eliminated by dicing. The green line shows thatthe middle column is ruled out, because it already contains a 4 in another cell. So wehave finished the second move in a fiendish puzzle and found out what slicing and dicingis.Step 3: Applied 'slice and dice'. Angus Johnson has this to say about hidden pairs: 'If two squares in a group containan identical pair of candidates and no other squares in that group contain those twocandidates, then other candidates in those two squares can be excluded safely.' Inthe example on the right, a 2 and a 3 cannot appear in the last column.

Sudoku 7

So, in themiddle rightmost cell these two numbers can only appear in the two positions wherethey are 'pencilled in' in small blue font. Since these two numbers have to be in these twosquares, no other numbers can appear there.Angus Johnson again: 'Sometimes a candidate within a cell is restricted to onerow or column. Since one of these squares must contain that specific candidate, thecandidate can safely be excluded from the remaining squares in that row or columnoutside of the cell.'

Since the hidden pair 2 and 3 prevent anything else fromapearing in the first two columns of the middle rightmost cell, an 8 can only appearin the last column. Now we apply the locked candidates rule. Next we extend the logic of the locked candidates. The 5th and 6th rowsmust each have an 8: one of them has it in the middle left cell, and the other in thecentral cell. Therefore the 8 in the middle right cell cannot be in either of theserows. From what we knew before, the 8 must be in the top right corner square of thecell, as shown in the picture on the right. This is almost magical.

Putting togetherimprecise information in three different cells, we have reached precise informationin one of the cells. And now the final step of the bootstrap is shown in the picture onthe left. The placement of the 8 dictates that the 6 must be justbelow it, and therefore the 7 in the remainingsquare. The diabolical magic is complete: reason enough for thisto be classified as a fiendish puzzle. One of Roger Walker's tutorialsis a solution of precisely this puzzle, by a different route. But beforegoing there, I invite you to try your hand at completing the solutionwhich we have started upon here.Step 7: The beginning of the end. The last rule, I promise.

And it is hardly one, although you could call it the'hidden loner' rule. The only reason one should give it a name isthat it fixes this very useful method in one's mind.

So here is the example:In the 6th row there's more than one choice in each square. However there is onlyone place where the 5 can go (it is excluded from the squares with X's in them).So there is a loner hidden in this row: hence the name. I stop here, but youcan go on to solve a fiendish puzzle by the simplest tricks exclusively.Not so fiendish? Mike Godfrey wrote to me to point out a much simpler way of solving thisparticular puzzle. After step 3, as before, one can fill in the 6 shown inblue in the figure here, by noting that all other numbers can be eliminatedby requiring that they do not appear in the same row, column or block. Afterthis the remaining puzzle can be solved by spotting singles.Mike writes that this puzzle 'is not too fiendish perhaps'.

Perhaps.But that opens up the question of how to rate puzzles. I haven't foundmuch discussion of this aspect of the mathematics of Su Doku: partly becausecommercial Su Doku generators (by that I mean the humans behind the programs)are not exactly forthcoming about their methods, but also because the problemis not.

This is a wide open field of investigation.Constraint programming. The minimum Su Doku shown alongside (only 17 clues) requires only two tricks tosolve: identifying hidden loners and simple instances of locked candidates. Thekey is to apply them over and over again: to each cell, row and column. Theapplication of constraints repeatedly in order to reduce the space of possibilitiesis called in computer science.' Pencilling in' all possible values allowed in a square, and thenkeeping the pencil marks updated is part of constraint programming.This point has been made by many people, and explored systematically by.Non-polynomial state spaceThis is where much of the counting appears.

Before clues are entered into aM×M Su Doku puzzle, and the constraints are applied, there areM M 2 states of the grid. This is larger than anyfixed power of M (this is said to be faster than any polynomial in M).If depth-first enumeration were the only way of counting thenumber of possible Su Dokus, then this would imply that counting Su Doku is ahard problem. Applicationof constraints without clues is the counting problem of Su Doku. As cluesare put in, and the constraints applied, the number of possible statesreduces. The minimum problem is to find the minimum number of clues whichreduces the allowed states to one.

The maximum problem is analogous.Many known hard problems are of a type called.In this class, called NP, generating a solution of a problem of size M takes longer thanany fixed power of M, but given a solution, it takes only time of order somefixed power of M to check it (ie, a polynomial in M). If enumeration were the only way of countingthe number of Su Doku solutions, then this would be harder than NP. If someone tellsme that the number of Su Doku solutions is 60, I have noway to check this other than by counting, which I know to takes time largerthan polynomial in M.