Learn Sudoku - Pencil Marks

This article assumes you know the basics of Sudoku. If you need a quick introduction to the game, check out the Introduction to Sudoku.

All of the red words in this article are dynamic. Click on them to get helpful clues!

The more difficult a puzzle, the more numbers you need to keep track of in your head. You often need to juggle dozens of candidates before making a breakthrough. Pencil marks are a great way to help you manage those numbers when you first begin treading deep water.

When you use the Single Position or Single Candidate techniques, you are methodically figuring out all of the candidates for the empty cells in a puzzle. But it can be very difficult to keep track of these candidates. Some people find it easier than others to remember a large number of candidates. We all have a limit to how many we can remember. The more puzzles you solve, the more you will raise this limit. But when you first begin playing, your limit may be very small. That's where pencil marks come in. As you figure out the candidates for an empty cell, write those candidates into the empty cell using a pencil. Write the numbers very small so that you can fit a lot of candidates into one cell. As the puzzle progresses you will be able to logically rule out certain candidates. When you have eliminated a candidate just erase it or mark it out.

One great consequence of this is that if a cell only has one candidate written in, then by the rules of the Single Candidate technique we know that the written candidate must be the answer for that cell.

Take a look at the puzzle to the left. This puzzle is quite a bit more difficult than the ones we have used in previous articles. When first working with a more difficult puzzle, if you find that you are having a hard time juggling the candidates then start writing them down in the empty cells as you go. Click here to see what this puzzle would look like if you were to fill in all of the candidates for every empty cell. That's a lot of information! Now you can see why juggling the candidates in your mind is difficult. Pencil marks can truly be your best friend!

Pencil marks can also help you find problems with your puzzle before you get too far. If you fill in the wrong answer for a cell, then at some point down the road you will run into a contradiction. If you are using pencil marks and you encounter an empty cell that has no candidates, then you know something is wrong. Think about this for a moment. If a cell has no candidates then there is no correct answer to it! When you encounter this situation then you know you have messed up somewhere and it's time to backtrack. You can try this for yourself right now by clicking on the candidates in the puzzle above. Just click on random candidates and eventually (probably sooner than later) you will run into a cell with no candidates. That's because you weren't choosing the correct answers and a contradiction appeared.

So when should you use pencil marks? Is it cheating? These are good questions that you will have to answer for yourself. For me, I judge when I need to use pencil marks and try not to use them until I have to. This way I can get better over time at juggling numbers, but I don't have to spend hours and hours on a particularly difficult puzzle just because I'm stubborn. There are some purists out there who feel that using pencil marks is cheating, but I think this is extreme. Cheating is when you utilize some external resource so that you don't have to figure something out for yourself. When you use pencil marks you are figuring everything out on your own... you are just recording those calculations. Imagine a mathematician who never jotted down notes while making progress on a difficult problem. Kinda silly, huh?

In short, if you want to improve your mental capacity and skills, then only use pencil markings when you absolutely have to. Otherwise, if you just like to solve Sudoku puzzles and you don't want to spend a lot of time doing the same calculation over and over, then use pencil markings!

Most of the articles on more advanced techniques will utilize pencil marks to help you visualize the technique. Also, as an aid to puzzle solvers like you, all of the puzzles on this website can have the pencil marks automatically filled in for you. Just click on the mark button at the top right of the puzzle to turn it on. Click it again to turn the pencil markings off. Is this cheating? Yes, it is! But sometimes it is very helpful to have this feature when you are practicing difficult techniques and you don't want to be bogged down in the basics.