Nine Beginner Mistakes That Make Sudoku Harder Than It Is

Most beginner trouble comes from one habit, and swapping it takes a minute. You look at an empty cell and ask what could go in it. Ask the other question instead. Pick a digit, pick a box, and ask where in that box the digit is allowed to sit.

That swap turns an easy grid from a search into a scan. The eight mistakes after it are about pencil marks, about what you write down before you have proved it, and about what you do when the grid stops moving.

None of this is about speed. A beginner who makes none of these will finish an easy puzzle sooner than a practised solver who makes three.

On a phone? Sudoku Master can highlight the row, column and box of whichever cell you tap, which is the quickest cure for mistake 2 below. Free, and no ads while you solve.

The nine, on one screen

Mistakes 1 to 3: how you read the grid

1. Hunting for the cell instead of the digit

An empty cell on an easy grid usually has three or four possible digits. Working out which means checking its 20 peers: the eight other cells in its row, the eight in its column, and the four still open in its box. Do that for 45 empty cells and you have read an enormous amount of grid for a handful of digits.

Now the other direction. Take one digit, take one box, and ask where the digit can go. A digit already on the grid blocks its whole row and its whole column, which cuts across three boxes at a time. The digits do the work instead of you.

Here is a fragment. Only the digits that matter are shown, so it is not a complete puzzle. Where does the 7 go in box 1, the top-left nine cells?

53..7...9
...195.7.
....4..6.
8...6...3
47.8.3..1
....2...6
7....2...
..3.....5
...9.....

Four digits settle it.

• The 7 at r1c5 rules out the whole of row 1, so r1c1, r1c2 and r1c3 are gone.
• The 7 at r2c8 rules out row 2, so r2c1, r2c2 and r2c3 are gone.
• That leaves the bottom row of the box: r3c1, r3c2, r3c3.
• The 7 at r7c1 rules out column 1. The 7 at r5c2 rules out column 2.

One cell survives. Write a 7 in r3c3. You read four digits, not twenty-four cells, and you did not need a single pencil mark to do it.

That is cross-hatching, and on an easy grid it will place most of the puzzle by itself. Work one digit at a time, all the way through, before you move to the next one. There is a fuller treatment in cross-hatching and scanning.

2. Checking the row and the column, but forgetting the box

Sudoku has three constraints and beginners reliably use two. The row and the column are visible: they run across the page and your eye follows them. The box is the one you have to remember, because nothing about the shape of it pulls your attention.

Watch it decide a cell on its own. Only the digits bearing on one cell are shown here, so it is not a complete puzzle.

...4.....
...5.....
.........
123...89.
.........
.....6...
.........
.........
.........

What goes in r4c4?

Row 4 already holds 1, 2, 3, 8 and 9, which leaves 4, 5, 6 and 7 to place. So the candidates start as {4,5,6,7}. Column 4 holds a 4 at r1c4 and a 5 at r2c4, and both are out. You are left with {6,7}.

Stop there and the cell looks undecided. It is not. Box 5, the middle nine cells, already contains a 6 at r6c6. That kills the 6 and r4c4 is a 7.

The row and the column together got you two thirds of the way and then said nothing. If you had stopped, you would have written the cell off as unsolvable and gone looking somewhere else. Three constraints, always.

3. Reading the clue count as the difficulty

Counting the givens tells you almost nothing, and beginners lean on the count because it is the only number printed on the page. The thing you want to know is which technique the grid will demand before it moves. A clue count cannot carry that, and reading it as though it could talks you out of puzzles you could finish and into puzzles you cannot.

I built Sudoku Master, so I can put honest numbers on this. The hard puzzles in it ship with 26 to 28 givens and the extreme ones with 21 to 27. Those ranges overlap. An extreme grid with 27 clues is harder than a hard grid with 26, because the extreme one has a point where nothing but an X-Wing will move it, and the hard one does not.

The floor is worth carrying as well. Seventeen givens is the fewest a puzzle can have and still pin down one answer, and 16 does not work at all. Those 17-clue grids are brutal and you will never meet one in a newspaper.

The words on the cover are no more reliable than the count, because every publisher grades against its own ladder and there is no shared one to appeal to. See what actually makes a Sudoku easy or hard.

Mistakes 4 to 6: what you do with a pencil

4. Filling in every candidate before you have taken the free digits

An easy grid arrives with about 45 empty cells. Writing the full candidate list into all of them takes several minutes, produces a grid you can barely read, and most of it is wasted: the first six or seven placements from a plain scan will delete a third of what you wrote.

Do it in the cheap order.

1. Scan each digit, 1 through 9, through every box. Place what falls out.
2. Repeat the scan. Placements from step 1 open cells that were closed before.
3. Keep going until a full pass through all nine digits places nothing.
4. Only now pick up the pencil, and only for the cells that are still open.

Step 3 is the one people skip. A single new digit changes three units at once, so a scan that found nothing five placements ago can be productive now. Two clean passes with nothing found is the signal to start marking.

5. Leaving a candidate on the grid after it has died

You write a 4 into r5c9. Somewhere in that row, that column and that box sit cells with a 4 pencilled in, and every one of those is now a lie. Cross them out straight away, while you still know which cells you are looking at. Coming back in five minutes means re-deriving the same thing twice.

Here is what stale marks actually cost you. Say r6c2 reads {3,8} and r6c7 reads {3,8}. Two cells in row 6 with the same two candidates is a naked pair, which is a real and useful pattern, so you settle in to work out what it eliminates. Except that you put an 8 into r4c8 ten moves ago, and r4c8 is in box 6, and so is r6c7. The 8 in r6c7 died back then. That cell is a plain 3, and once you write it, r6c2 has to be the 8.

You spent a minute proving something smaller than the two digits that were sitting in front of you.

Worth knowing exactly how this hurts, because it is not the way people assume. A candidate list that is too big will not make you write a wrong digit. Anything you deduce from it is still true, since the real answer is somewhere inside the list you kept. Stale marks hide answers and waste time. They do not lie to you.

The dangerous error is the opposite one: a candidate you never wrote down, or rubbed out by accident. That one can produce a naked single that is not there, and you will place a wrong digit with complete confidence. When you tidy your marks, remove only what a digit on the grid has actually killed.

This is why the app has an auto-remove-notes setting. Place a value and every matching note in that row, that column and that box goes with it. On paper the pencil does the same job, and the discipline is to do it in the same second rather than later.

6. Refusing to write anything down

The opposite failure, and it turns up in people who did easy puzzles for a month and then hit a wall. A hard grid will not go all the way on scanning. Around ten or twelve placements in, the scan stops paying, and from there the puzzle is made of patterns you can only see when the candidates are written somewhere.

You do not have to mark everything. Mark where the information is:

• Rows, columns and boxes with two or three empty cells. A unit missing three digits is nearly solved and the marks are short.
• Cells that already look tight. If you can see the cell has only two or three options, write them. They are your pairs.
• One digit at a time, if the grid is very open. Mark only where the 4 can go, everywhere. It finds hidden singles and pointing pairs that a full marking hides in the noise.

The system that keeps marks readable is in how to use pencil marks, and the first patterns they earn you are in naked pairs, triples and quads.

Mistakes 7 to 9: when the grid fights back

7. Guessing when it stalls

A properly formed Sudoku has exactly one solution. That is not a nicety, it is the definition, and it carries a promise: every digit in the grid can be reached by a rule. If you are guessing, one of two things is true. Either the puzzle is broken, which is rare in print and less rare from a random generator, or there is a step in front of you that you have not spotted.

The second is almost always the case. Before you guess anything, run this:

1. Re-scan all nine digits through all nine boxes. Slowly. Most stalls end here.
2. Look for a cell with one candidate left. That is a naked single.
3. Look for a digit with one home in a row, a column or a box, even where that cell has other candidates. That is a hidden single, and it is the step beginners miss most.
4. Check for a pair of cells in one unit sharing the same two candidates.
5. Check whether a digit's places inside a box all sit in one row or one column. If they do, that digit comes out of the rest of that row or column.

A guess is expensive in a way that is easy to underrate. Guess at cell 40 of 81 and every digit after it is provisional. When the contradiction appears twenty moves later, you cannot tell whether the guess was wrong or your logic was, so you have to unwind the whole lot. Solving without ever guessing goes through the full ladder, and the stuck checklist is the version to keep beside the puzzle.

8. Writing a digit that only looks right

Mistake 7 is a guess you know you are making. This is a guess you do not.

The two sentences below feel the same at the time and are not remotely the same.

• "A 4 works in this cell."
• "Only a 4 can work in this cell."

The first says you checked one digit. The second says you checked all nine, or that a rule eliminated the other eight for you. Beginners write the first and believe they wrote the second, usually while tired, usually late in the grid when the cell has been staring at them for a while.

The cost is that a digit is never wrong quietly. It is a peer of 20 other cells, and it will be used as evidence in every deduction you make from now on. Twenty moves later the grid is inconsistent and the cause is three-quarters of an hour behind you.

Before each placement, name the reason out loud. "Only cell in the box that can take a 7." "Only digit this cell has left." If no sentence like that comes, the digit is not ready. There is more on both patterns in naked singles and hidden singles.

9. Starting over when you hit a contradiction

Two 5s in a column, or a cell with no candidates at all, and the instinct is to rub the whole thing out. Do not. The puzzle is still mostly correct, and the error is findable.

Work it like this.

• If it is a duplicate, you have two candidates for the wrong digit and one of them is right. Check each against the givens. A given never moves, so any digit that conflicts with one is yours.
• If a cell has run out of candidates, list the digits that killed each option and find which cell each one lives in. Your bad digit is in that list.
• If neither works, find the last placement you can still justify out loud and rub out everything after it. Painful, and it is a fraction of restarting.

Sometimes an early error on paper really is unrecoverable, and you will lose the grid. That is the argument for mistake 8, not an argument for the eraser. Solving on a screen removes this problem entirely, since undo exists and a duplicate is flagged the moment you make it.

Which band teaches which fix

Give yourself grids that ask for the thing you are practising. An extreme puzzle is not practice if you cannot yet see a hidden single. It is just staring.

The givens are from the set I ship; the right-hand column is the standard ladder, and it varies from puzzle to puzzle. Move up a band when you can finish the one below without stalling, not when you get bored of it.

On your phone: Sudoku Master is free on iPhone and Android. The link sends you to whichever store your phone uses.

Questions people ask

What is the fastest way to get better at Sudoku?

Do easy grids until the digit-first scan is automatic, then move up one band. Speed comes from recognising a shape you have seen before, not from thinking harder. Ten easy puzzles finished teach more than one extreme puzzle abandoned.

Is using pencil marks cheating?

No. Everyone uses them on hard grids. The only rules are the puzzle's rules: one digit per cell, no repeat in any row, column or box. What you write in the corners is your business. Marks become a problem only when there are so many that you stop reading them.

Do the digits in a Sudoku have to add up to anything?

No, and nothing in classic Sudoku is ever added or compared. A digit is a marker for "not one of the other eight", which is all the puzzle asks of it. Sums belong to Killer Sudoku, a different puzzle with cages drawn over the grid.

Can a Sudoku puzzle have two answers?

Only a badly made one. Uniqueness is the standard every publisher works to, and it is what lets you prove a step instead of picking. Two endings that both check out means a clue went missing in generation or in typesetting, and no technique would have rescued the grid.

Why does an easy Sudoku take me so long?

Nearly always mistake 1. Searching cell by cell makes you read the whole grid to place one digit; scanning digit by digit makes you read one digit to clear a whole box. As a pace rather than a target, the estimates I attach to the daily challenges in the app are 6 minutes for easy, 12 for medium, 22 for hard and 35 for extreme. More on this in how long a Sudoku should take.

Should I do Sudoku in pen or in pencil?

Pencil, until mistake 8 has stopped happening. Pen sounds like a discipline and mostly produces unfinished puzzles. If you are using one anyway, write only digits you can give a reason for, and put candidates in the margin rather than in the cell.

Where should you start on a Sudoku grid?

On the digit that appears most often already, and on the fullest box. Count how many 1s are on the grid, how many 2s, and so on. Each copy of a digit blocks a whole row and a whole column, so the digit with six copies on the board is the one closest to placing a seventh.

Why do I keep finishing easy puzzles but stalling on medium ones?

Because scanning alone finishes an easy grid and does not finish a medium one. The wall is the point where you need candidates written down and a hidden single or a naked pair found. That is a new skill, not a failure of the old one.

Keep reading

• How to play Sudoku: the complete rules, the ruleset underneath every habit on this page.
• Your first Sudoku, solved one move at a time, which is this advice applied to a whole grid from empty to full.
• Cross-hatching: how to read a grid quickly, the full version of mistake 1.
• How to use pencil marks without drowning in them, for mistakes 4, 5 and 6.
• Stuck on a Sudoku? Work this checklist in order, the page to open instead of guessing.
• The full Sudoku guide, rules through to the hard patterns, in reading order