Sudoku tips and strategies

Every technique below, in the order it becomes useful. Learn them one at a time and stop when the grids you play stop resisting — there is no prize for knowing the ones you do not need yet.

Scanning

The first technique and the one you will use forever. Rather than picking an empty square and asking what fits, pick a digit and ask where it can go.

Take a digit that already appears four or five times. For each box that is still missing it, cross out every empty square that shares a row or column with an existing copy of that digit. Often exactly one square is left, and you have a placement — this is sometimes called cross-hatching.

Do all nine digits before you conclude the grid is stuck. Beginners typically scan two or three, hit nothing, and start pencilling marks they did not need.

Naked and hidden singles

Two shapes of the same idea, and knowing which one you are looking at saves time.

A naked single is a square with only one candidate left: everything else has been ruled out by what it can see. You find these by looking at squares.

A hidden single is a digit with only one square left in some unit — the square may still have three or four candidates of its own, but nowhere else in that row, column or box can take this digit. You find these by looking at units, and they are much easier to miss.

Pencil marks that stay honest

Everything past this point needs candidates written down. Three habits make them worth having:

• Fill a unit at a time, not the whole grid. Notes for one box are usually enough to unlock it, and are a fraction of the work.
• Keep them current. A stale candidate is worse than no candidate. Placing a digit here clears it from the notes of every square it can see, automatically — which is the commonest source of stale marks removed for you.
• Read them as a set. The value is not in any single square's list; it is in noticing that two squares in a unit have the same two candidates, or that a digit appears in only one row of a box.

Locked candidates

The first genuinely eliminating technique, and the one that unlocks most medium grids.

Pointing: if within one box a digit's only candidates all lie in the same row, then that digit must be somewhere in that row inside the box — so it can be removed from that row everywhere outside the box.

Claiming: the mirror image. If within one row a digit's only candidates all lie inside a single box, the digit can be removed from the rest of that box.

Neither places a digit. Both remove candidates, and removals are what make the next single visible.

Naked pairs and triples

If two squares in a unit have exactly the same two candidates — say 4 and 7 — then between them they will use up both digits, whichever way round it turns out. So 4 and 7 can be struck from every other square in that unit.

Triples are the same idea with three squares and three digits, and they do not need every square to hold all three: {2,5}, {5,9} and {2,9} in one unit is a valid naked triple on 2, 5 and 9.

Hidden pairs

The inverse. If two digits can each only go in the same two squares of a unit, those two squares must hold those two digits — so every other candidate can be struck from them.

Hidden pairs are easy to walk past because the squares involved often carry four or five candidates and look unremarkable. The way to find them is to count, per unit, how many squares each digit could occupy. Two digits with the same pair of squares is your pattern.

The X-wing

The first technique that reaches across the grid rather than within a unit, and the hardest thing Hard will ask of you.

Look for a digit — 3, say — that in two different rows has exactly two possible squares, and those squares fall in the same two columns. The four squares form a rectangle. Whichever way the 3s resolve, one goes in each of those two columns, so the 3 can be removed from those two columns everywhere else on the board.

It works identically with rows and columns swapped. Two eliminations, often in far corners, and usually enough to restart a stalled grid.

Discipline, not cleverness

Most failed grids are not failures of technique:

• Never place a digit you cannot justify. One guess contaminates everything after it, and the contradiction may not surface for twenty moves.
• Re-scan the three units you just changed. A placement affects a row, a column and a box at once. The next move is very often in one of them.
• Work eliminations, then look for placements. On sparse grids nothing is placeable until something has been eliminated. Alternating the two is the rhythm.
• Take the break. Pause the clock, look away, come back. The pattern you could not see is frequently the first thing you see on return.

Which grade needs what

If a grade is asking for a technique you have not learnt yet, that is the signal to drop back one and get quicker there first. Nothing above is worth memorising until the grid in front of you needs it.