Puzzles/Logic puzzles/Sudoku/9 by 9/Approach to solutions

For some people, developing a personal method of solving Sudoku is much of the pleasure, so consider whether or not to read this.

There are many techniques for solving 9x9 Sudoku. Some use pencils and erase a lot, some use colored pencils, some write tiny numbers in the cells while working the puzzle.

The approach given here will assume that the solver only has an ink pen and cannot erase. For this reason, with very difficult Sudoku, where it becomes necessary to test a possibility that has not been logically proven, it is possible that the puzzle will become an unreadable mess. However, most Sudoku, except for a few very difficult puzzles in some books -- and some books don't have these very difficult puzzles -- can be solved with this technique, with no guessing at all. It still takes keen observation and painstaking care, because one mistake can create many later incorrect entries, and it might not be discovered that one has created a contradiction until much later. However, it is possible to recover, using ink and the technique described here, from a single error, see /Mistakes. If a mistake is made in the recovery process, perhaps it's time to give up on that puzzle; the time for this is when it really cannot be clearly read any more.

A relatively fine-point pen is preferred, and gel pens work well even on relatively flimsy paper. Instead of writing numbers in the cells, the solver will use dots. Imagine each square is divided into nine sections, like a little image of the 9 x 9 grid of the Sudoku itself. A dot in the upper left corner of the square will represent a possible "1" there, in the upper right corner a "3", in the lower left corner a "7", and in the lower right corner a "9", with the other numbers in the obvious intermediate positions.

One might look at the numbers in any sequence, the numbers in a Sudoku have no particular significance as to their importance. Some will look at the numbers first that occur most frequently, because these will most commonly lead to obvious answers for some cells. We will assume that the solver, however, goes through the numbers from 1 to 9, in sequence, at first.

For terminology, a cell will hold one number in the end, and there are nine boxes of cells, each containing nine cells.

So for each number, look at each box where the number isn't found and see what cells within the box could contain that number. If it is only one, write that number in the cell, write it large so that it can be very clearly seen. If it is two, mark two dots, one in each cell, for a possible "1", this would be the upper left corner of the cell. If it is more than two, ignore it for the moment. At this point you are only looking for boxes with two possible positions for the number, so two dots to be placed.

At some place outside the box, possibly lined up with the 9 cells along one side, but at some distance, so that, later, numbers can be written next to the boxes, and if one has found all the occurrences of a particular number, write an X to show that this number, aligned with the first row or column for one, the second row or column for two, etc., has all been marked. If one has marked a dot for this number for all the cells, write the number, showing that no more dotting is needed for this number, though it remains to be completely solved. (This list of numbers will end up as the numbers 1 through 9, only with some positions having an X instead of the number. (Or the number X'd out).

Proceed through all the numbers this way. Watch for situations where two cells in a box contain the same pair of dotted positions, and no other cells contain those represented numbers. This is easy to see if you have only been marking two cells for a number in each box. Mark a diagonal line across a corner of these cells, which will signify that, though the exact number has not yet been found for each of the two cells, no other number can occupy this pair of cells. Then, in considering possible positions for other numbers in a box, these cells can be excluded. (Very often, one will find that there is another number written in that pair of cells. Since at this point, the other number is only written in two cells, its position has been solved. Cross off the "intruding" dot and mark the number in the remaining cell of its pair.

If there are many dots, it can be much harder to see. The more you can solve in a simple way, the easier it will be to see the more difficult steps before the solution comes unravelled and it all gets extremely easy.

Notice that if only two cells or three cells in a column or row contain two or three optional numbers, and all cells have been dotted for that set of numbers, and they are all in one column or row, the combination functions to exclude that number from other cells aligned with the column or row, even if they might otherwise have been dotted as possible. So those extra dots can be crossed off.

Crossing a dot off, a small X can be put over it. It turns out this is easy to distinguish from a dot.

Pass through all the numbers until one does a pass where no more particular cell solutions have been found, and no more pairs of dots can be placed.

In continuing, for each pass, one need only consider the numbers which are blank in the list of numbers one has made on outside the puzzle.

Then pass again through the numbers and do two things: for each number, if there are only three positions which could hold a number, mark the number outside the cell with an underscore. If more than three positions can contain the number, just write that number next to the cell. Leave room for all the possible numbers you might need to write! Perhaps write small! For the center box, use another position on the page to mark those numbers. The point is to quickly see what numbers remain to be placed in the cell. When you have dotted a number in a cell, blot out the number. If possible cell positions are reduced to three in a box, underline the number on the outside. When a box has been fully dotted or solved, write a small check next to the box or, for the center, in it or in the little box you drew outside.

What this does is to show what work remains to be done in the first stage. Some simple Sudoku may already be solved by the time this is done.

When you finish dotting a number where you wrote an underscore, write the number in the list, and if you finish solving all cells for that number, cross it off.

If you have completed finding all positions for a number, draw an X over that number in the list outside.

Suppose you only have 1 number remaining to be placed in a box, and only one cell with no dots. The number must go in that position, but check it! Maybe you have made a mistake! Suppose there are two numbers remaining, and two cells. You can place the dots for those numbers in those two cells, and cross off the number you wrote on the side, because it's now dotted.

The same is true for rows and columns, but you must remember to be sure that all the cells in the row or column have been dotted for the numbers being considered. You can quickly see this if you have completed writing the numbers outside the cells as described.

Keep looking for any combination of two numbers or three numbers (or, rarely, four) that only occur in 2, 3, or 4 cells, respectively, in a box, row, or column.

Notice that if, with any two columns in a group of three (a "box column"), or the same with box rows, a number is, in two of the three boxes, only in those two columns, the number must be in the third column in the remaining cell in the box column, so you can so dot it or place it as appropriate.

This stage is complete when all remaining cells have been fully dotted, which would mean that all of the small numbers outside the boxes have been blotted out. Continue looking for the patterns that reduce possibilities to a single one.

If you come up against a contradiction, almost certainly indicating an error, see /Mistakes.

This technique can easily solve Sudoku up through intermediate difficulty, it simply requires being painstaking and not making mistakes. See /Mistakes for how to handle errors with an ink process.

For more difficult Sudoku, approaches will be found at /Advanced solution techniques/. The technique here will prepare the puzzle for those techniques to be applied.

/Personal reflections/ will report why this author does Sudoku and what they tell him about his mental state.