# 1 eg. Given the digits in black, we know that digit 1 must be at Row 2 Column 9. X stands for any digit other than 1.

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1 Sudoku Solving: Dot method and more. David A Paterson (aka mollwollfumble) 2 Apr 204 This is a quick summary giving examples of how I solve Sudokus of all difficulties. It contains useful mental and written methods, some of which haven t appeared on the web before. Low hanging fruit Fill in any digits that jump out at you. Don t spend too long on this, if you spend more than two seconds looking for the next digit then you re wasting time eg. Given the digits in black, the digit must be at Row 4 Column 4. Scanning Scanning is where you go through each digit in turn, eg. to 9, and look along rows and columns for new places to put each digit. We all do it. X eg. Given the digits in black, we know that digit must be at Row 2 Column 9. X stands for any digit other than.

2 While scanning, you will come across distractions in the form of digits that can be filled in out of sequence. Don t allow yourself to be distracted, come back to these at a later time. Keep cycling through each digit in turn until you re sure that you can do no more. While scanning, I use three tricks that find more digits than simple scanning alone without appreciably slowing the scanning process but. I call these Arrows, Left-Middle-Right, and Casual Dots. Arrows and Left-Middle-Right are mental techniques whereas Casual Dots is a written aid these are described below. After you ve passed through the scanning phase of solution into more advanced techniques, come back to scanning every now and then to search for more digits. Arrows Mentally draw lines on the Sudoku grid along rows and columns that are eliminated by each digit in turn. I find it handy to add a mental arrowhead to these lines so I know in which direction they re going. X X X X X X X eg. Given the digits in black, following rows and columns allows you to place a new digit in Row 2 Column 9. Left-Middle-Right While scanning, I keep up a mental commentary of Left-Middle-Right and Top-Middle- Bottom. You can say this out loud if you like. For each digit scanned and for each group of three rows or columns keep watch on which of the three is a possible place. See example.

3 Top Mid X X X X X Bot eg. Along the bottom three rows, reading from left to right, say the possible positions for digit as top-bottom, top-middle-bottom, top-bottom. The centre-bottom box is the only one with a middle so digit must be in the middle row of the centre-bottom box. A challenge in solving Sudokus is knowing when NOT to search, when all digits findable using that search method have already been found. The method of Left-Middle-Right gives an immediate knowledge of when scanning for any particular digit is complete. X eg. A commonly encountered case where the method of Left-Middle-Right says that no more digit s can be found by scanning. Reading down the three left columns gives middle-right, middle-right, left so there is no unique middle or right. Reading across the top three rows gives middle-bottom, top, middle-bottom so there is no unique middle or bottom. The presence or absence of a digit marked X in the top left box has no influence on the result. Dot Method You will already know about using pencil marks in a cell. The dot method is a more compact way of showing the same information. Imagine the digits,2,3,4,5,6,7,8,9 placed in a cell with,2,3 across the top, 4,5,6 across the middle, and 7,8,9 across the bottom. Instead of writing the digit in pencil, place a dot where the digit would be. Also, rather than pencilling

4 in digits that are possible, place a dot wherever that digit is eliminated. In this way, there is never any need to rub out dots and I recommend putting the dots in with a pen, not a pencil, unless you re doing an extremely difficult Sudoku using trial and error. Just touch the pen to the paper once for a dot, don t waste time drawing a filled circle. When you finally discover the unique digit for a cell, write that digit over the top of the dots, it will be easily visible eg. The nine positions within a cell and the nine dots at those positions eg. Pencilled digits in a cell and the equivalent information using the dot method. Saying that, 3, 4, 8 & 9 are not eliminated is the same as saying that 2, 5, 6 & 7 are eliminated. 8 eg. A digit written over dots is easily visible. I ve been told that some people never make mistakes. I m not one of them. If you add a dot in the wrong place by accident, strike it out by putting a single diagonal line through it. Later if you find that that digit is indeed eliminated, put a second diagonal line through it to form a cross. eg. Correcting accidental errors. a) Digit is eliminated. b) Oops, I made a mistake and digit is still possible. c) Digit is indeed eliminated. Advantages of the dot method over pencilled digits are: you never need to rub out or scrub out a digit, the more compact form allows you to do Sudokus with smaller printed cells, the final digits are perfectly visible when written over the top of dots, and certain more advanced techniques such as groups of four are more easily visible using the dot method than when using pencilled digits. I ll come back to the last of these later.

5 The Dot Method for Puzzles other than Sudoku. The dot method is also useful for a whole range of puzzles other than Sudokus. The dots can replace letters or symbols equally as well as replacing digits. For puzzles that have other than nine symbols I use the following dot placements eg. Dot placements when there are three, four, five, six or sixteen possibilities. It is rare to have a puzzle format where it becomes easier to mark where a digit may be rather than where it can t be. Such puzzles are those where we are given information on the jump from one digit to another, see example below. I mark a possible location using a small circle rather than a dot. And then, when the digit is finally eliminated, place a dot in the centre of the circle. Don t use this technique for Sudoku. 2 eg. On this non-sudoku puzzle, the puzzle rules say that number is a knight s jump away from the number 2. Possible positions for the number are marked by circles. When the position is eliminated place a dot in the circle. Casual Dots. Pencilled digits can be casual, placed wherever you feel like it, or complete, placed in every cell. Casual dots can be placed while scanning without slowing down the process much. For Sudokus you know in advance will be difficult, place dots every time a digit is eliminated when there is no matching digit in the same row, column or box.

6 X X X eg. Two examples of the placement of casual dots. Casual dots go in places where digits are eliminated despite there being no matching digit in the same row, column or box. More often than not, placing casual dots doesn t help you find any extra digits, but occasionally it does, the following is an example. In addition, placing casual dots saves a lot of time if you need to progress to complete dots eg. An example where casual dots may help. Three dots appear in the cell in Row Column 3. When combined with the other digits present in the same row, column and box, there is only one candidate left, the digit 7. Quick Checks. Some quick checks help to find digits that have been accidentally missed. While still in the scanning phase of solving a Sudoku, count the number of times the digit appears in the puzzle. If it s eight times, find the ninth. Once all nine occurrences of a digit are found, write that digit outside the puzzle, as a quick way to avoid later searching for digits that have all already been found.

7 Another type of quick check is illustrated in the following example eg. I know that I know the digit at Row 6 Column 7, because the three other digits in Column 7 are fixed by the digits in the bottom right box, but what is it? A quick check comparing the digits in the box to the digits in the column finds the digit 6. Line End Digits. My sister-in-law likes solving black-belt Sudokus. She introduced me to the method of Line End Digits. At the end of each row or column that you are interested in, outside the puzzle, write all possible unplaced digits in that row or column, or part of row or column. This is most useful when the printed Sudoku is too small to pencil in digits within the cells eg. In Column 5, the top 3 digits form a group of three digits listed at the top of the column. The remaining possibilities are listed at the bottom of the column. The only overlap between the digits and 3456 is the digit 6, so the digit at Row 6 Column 5 is 6.

8 n-blank. Go through the digits to 9 in turn for each row or column or box with n blank cells, mentally marking each possible digit by bending a finger of the left hand starting with the little finger. Then for each blank cell within that row, column or box mentally figure out which of these is eliminated, looking for cells where there is only one candidate as well as for groups of two and three related cells. See example below. When n is, 2 and 3 this is a trivial mental exercise. When n is 4 it is still fairly easy using the fingers of the left hand. When n is 5 and 6 it is still possible. After n = 3 or 4, I tend to turn to the written method of complete dots given below eg. 3-blank and 4-blank. For Row mark off the only possibilities, 2, 5 using fingers of the left hand. Row Column cannot be 2 or 5 so must be. For Column mark off the only possibilities 3, 7, 9 using fingers of the left hand. Row 6 Column cannot be 7 or 9 so must be 3. For the left middle box mark the only possibilities 2, 6, 7, 9 using fingers of the left hand. Row 6 Column 3 cannot be 2, 7 or 9 so must be 6. Complete Dots. This is the use of the Dot Method where every blank cell is filled with dots. It helps a lot if casual dots have been previously used. Visibility is reduced if pencilled digits have previously been used. Line end digits are an unwanted distraction. Be systematic in adding dots. I fill in the dots box by box, by first eliminating (by placing a dot) every digit that casual dots would have picked up, then every digit in the box, then every digit in the row and in the column. The method of complete dots carries exactly the same amount of information as the method of pencilling digits in each cell.

9 eg. The top Sudoku is the original puzzle. The second is after scanning with casual dots and after 3-blank. The third is with complete dots. On Row 6, Column 3 must be digit 6, so Row 6 Column 2 is digit 2 and Row 6 Columns 7 & 8 can t be digits 6 or 2. On Row 7, Columns 3 and 7 form a pair using up digits & 3, so Row 7 Column 9 must be 2 and Row 7 Column 6 is 9. On Row 8, Columns 6 and 8 form a pair using up digits 4 & 5, so Row 8 Columns 4 & 7 can t have digit 5. Working from there, the Sudoku is easily solved without trial and error.

10 With the method of complete dots, groups of 2, 3 and even 4 connected cells can be seen immediately. More advanced techniques such as X-wing can also be seen. eg. An X-wing. Row 4 Column 2 cannot be digit 3, because then that would give the Sudoku multiple solutions. Line Trial and Error. About midway through my Sudoku career I came upon a Sudoku that defeated me utterly. Six months later I learnt that someone had solved it with minimal difficulty using the method of Line Trial and Error. I haven t been completely defeated by a Sudoku since. Select a row or column and try in turn all the different ways to complete it. I usually choose a row or column that can be completed in six to twelve different ways. I use pen for all dots and digits before trying trial and error and use pencil only for trial and error

11 eg. At top is the original Sudoku puzzle. Below gives complete dots. There are four possibilities for Column 3 reading from top to bottom these are 75293, 75392, 75932, Try each in turn. It will be found that is the correct solution. Sudoku difficulty vs method It is frequently said that you can t tell how difficult a Sudoku puzzle will be in advance. This isn t quite true. If there are more than 28 digits in the puzzle then it s likely to be very easy. Further, Sudokus with digits arranged along diagonal lines tend to be more difficult than those with digits arranged along vertical or horizontal lines. If you have a fair idea how difficult a Sudoku will be, you can choose an appropriate solving method in advance. For example, with a Sudoku you know in advance will be extremely difficult, don t waste time with pencilled digits, line end digits or n-blank for n > 3. Go straight from scanning with casual dots to the method of complete dots and then use row trial and error. Solution methods for Sudoku puzzles from trivially easy to extremely difficult in turn, in order of increasing difficulty: Low hanging fruit Scanning Scanning with arrows Scanning with left-middle-right Casual dots Casual pencilled digits Line end digits n-blank Complete pencilled digits Complete dots (skip pencilled digits) Line trial and error with complete dots Never balk at the prospect of copying the Sudoku onto a new grid. It s seldom needed but it often helps.

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