How to solve sudoku: tips and techniques for beginners

How do you solve sudoku? You never guess. You look for a square where only one digit can go, or a digit that has only one square left in a row, column or box, and you fill it in — then look again, because every digit you place makes the next one easier to find. That is the whole method for beginners. The rest of this guide shows you how to find those squares quickly, and what to do on the day they run out.

The rules, in one sentence

Fill the nine-by-nine grid so that every row, every column and every three-by-three box contains each digit from 1 to 9 exactly once. The digits you start with (the “givens”) never change.

Notice that there is no arithmetic anywhere in that sentence. The digits could just as well be nine colours or nine letters; sudoku is a puzzle about placement, not sums. If maths makes you nervous, you have already cleared the only hurdle that was never really there.

A well-made sudoku has exactly one solution, and that solution can be reached by reasoning alone. That matters, because it means that when you are stuck, there is always something on the board you have not seen yet. Every grid on BrainBreak is checked for both before it is dealt.

Start by scanning: cross-hatching

The fastest way into a fresh grid is to pick one digit and chase it through the boxes. Say you pick 5. Find every 5 already on the board. Each one rules out its whole row and its whole column. Now look at a box that does not contain a 5 yet, and imagine lines drawn through it from every 5 in the neighbouring rows and columns. That is cross-hatching.

Often the lines, together with the squares already filled, leave only one empty square in the box. The 5 goes there. Then move on to the next box, or the next digit. Beginners usually find that working through 1 to 9 like this, before doing anything cleverer, fills in a surprising amount of an easy grid.

Start with the digits that appear most often. A digit with seven givens already placed has only two left to find, and the lines from those seven do most of the work for you.

Hidden singles: only one square is left

What cross-hatching finds has a name: a hidden single. A digit has only one possible home in a row, column or box, so it must go there — even if that square could otherwise have held three or four other digits. It is “hidden” because if you only looked at the square itself, you would never spot it. You find it by asking about the digit, not the square.

Hidden singles are the workhorse of sudoku. On easier grids they solve most of the board, and even on hard ones they are what you fall back on between the cleverer moves.

Naked singles: only one digit fits

The opposite question is to pick a square and ask which digits could go in it. Rule out everything already in its row, its column and its box. If eight digits are ruled out, the ninth is the answer. That is a naked single — “naked” because the square is plainly down to one candidate for anyone who looks at it.

Naked singles are easiest to see in crowded areas: a row that is already seven-ninths full, or a square where the row, the column and the box between them hold almost every digit. Once a region fills up, switch from scanning digits to checking squares.

Pencil marks: when you cannot hold it all in your head

On a harder grid, singles run out. That is the moment to start writing pencil marks (also called notes or candidates): the small digits in a square that list everything still possible there.

A few habits make them useful rather than cluttered:

• Mark late, not early. Pencil-marking a fresh grid wastes effort on squares that scanning would have solved in seconds. Take the singles first.
• Be complete within a unit. Half-finished marks are worse than none, because they tempt you into thinking a square has two candidates when it has three.
• Tidy as you go. Every time you place a digit, remove it from the marks in its row, column and box. Most of the later techniques are only visible on a tidy grid.

On BrainBreak, the Notes button switches to pencil marks, and pressing it twice fills every empty square with every digit still possible there. Placing a digit clears the marks it rules out automatically, so the tidying habit is done for you.

Naked pairs

Here is the first technique that places no digit at all and is still enormously useful. Suppose two squares in the same row both have exactly the same two candidates — say, 3 and 7 — and nothing else.

You do not know which square gets the 3 and which gets the 7. But you do know those two squares will use up the 3 and the 7 between them. So no other square in that row can be a 3 or a 7, and you can erase those candidates from the rest of the row. That is a naked pair. It works the same in a column or a box, and if the two squares share a box as well as a row, you can clear the box too.

Often, erasing those candidates leaves another square with one digit left — a fresh naked single that was not there a moment ago. That is the rhythm of harder grids: an elimination, then a placement.

Pointing pairs

Look inside one box for a single digit, say 4. Suppose every square in that box that could still hold a 4 lies in the same row. The 4 for this box has to be somewhere in that row, then — and since the row can only hold one 4, no square in that row outside the box can be a 4. Cross it off everywhere else along the row.

That is a pointing pair (or pointing triple, if three squares are involved): the candidates in the box “point” along the line and clear it. The same works for a column.

There is a mirror image worth knowing, usually called box/line reduction. If, in one row, every possible square for a digit lies in the same box, then the digit for that box must be in that row, so you can clear it from the rest of the box.

A routine for when you are stuck

1. Run through the digits 1 to 9 once more, cross-hatching every box. Most “stuck” moments are a hidden single you skimmed past.
2. Check the fullest rows, columns and boxes for naked singles.
3. Fill in pencil marks, if you have not already, and look for two squares in one unit with the same two candidates.
4. For each box, look at each digit’s candidates. Are they all in one row or column? That is a pointing pair.
5. After every elimination, go back to step 1. New singles appear constantly.

If that still gets you nowhere, the Hint button in our sudoku fills in the next square that logic can reach, not a random one — so it is the square you were about to find, and it is worth looking at why it works before carrying on. A wrong digit is marked and counted, and that is all: there is no three-strikes rule, and you can always take it back out.

How our grids are graded

Many sudoku sites grade by the number of givens. That turns out to be a poor guide: a grid with few givens can fall apart with singles alone, and one with plenty can need a pair early on. BrainBreak grades each grid by solving it with the same techniques described above and adding up how much attention they cost. A naked single is a glance; a hidden single is a sweep of a whole unit; pairs and pointing ask you to hold several squares in mind together.

In practice that means Easy and Medium grids ask you to look at one square or one digit at a time, while Hard and Expert grids expect you to hold two or three squares in mind together — the pair and pointing techniques above, and on the toughest Expert grids a few beyond them. The daily grid, which is the same for everyone, climbs through the week: easy early on, the real test on Sunday. When you finish, the game tells you the hardest technique that grid needed, which is a nice way to notice you have just used one.

Where to go next

If sudoku clicks for you, the same “what must be true?” thinking drives Nonogram, where the deductions paint a picture instead of filling in digits — our guide to solving nonograms walks through it. For a wider selection, see logic games that make you think.