Foundations

Scanning, Cross-hatching & Last Digit

Scanning is solving by eye, without pencil marks: pick one digit and a box that still needs it, then extend the row and the column of every copy of that digit across the box. If exactly one empty cell of the box is left uncrossed, the digit goes there — that is cross-hatching, the visual form of the Hidden Single. Its companion is the last digit: a row, column or box that already holds eight digits takes the one it is missing.

What you need to see on the grid

Start every puzzle by scanning, and come back to it after each placement: a new digit changes one row, one column and one box at once. Sweep one digit at a time, starting with the digit that appears most often, through every box that still needs it. Four terms carry the method:

Cross-hatching: Pick a digit and a box that lacks it. Every copy of the digit elsewhere rules out its whole row and its whole column, so extend those lines across the box. The empty cells no line crosses are the only places left for the digit in that box; when exactly one is left, the digit goes there.
Hidden Single: The name the app's hint and the later guides give to what cross-hatching finds: a digit with only one place left in a row, column or box. The cell itself may still have several candidates — the digit is hidden among them — which is why you find it by following the digit, not by staring at the cell.
Last digit: A row, column or box that holds eight digits has one empty cell and one missing digit, and that digit goes in that cell. Some players call it a full house. The app's hint reports it as a Naked Single, since the cell has a single candidate left.
Digit view and cell view: Two ways to read a grid. Scanning by digit asks where this 2 can still go in this box, and leads to Hidden Singles. Scanning by cell asks what R5C1 can still take, and leads to Naked Singles. You need both; this guide works mostly by digit.

How to apply it, step by step

One easy puzzle runs through all five steps, exactly as printed: nothing has been written in yet. Every diagram shows the whole grid; the cells a step needs are drawn, the others are left blank and hatched, so you know you can ignore them. Steps 1 and 2 show no pencil marks, because scanning does not need any; from step 3 on, the box or row being decided shows its full candidate lists, so you can check every claim.

Step 1Pick a digit, and a box that still needs it

Scanning works one digit at a time. Start with the digit that already appears most often: every copy rules out a whole row, a whole column and a whole box, so a frequent digit leaves the fewest places to check. Here that is 2, which appears six times — more than any other digit.

Six 2s fill six boxes, so three boxes still need one: boxes 4, 5 and 9. Take box 5, in the centre. It has five empty cells — R4C4, R4C6, R5C5, R6C5 and R6C6 — and the only question for the next step is which of them can still hold a 2.

Box 5 and the six 2s already on the grid. No pencil marks: scanning reads only the digits.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1						2
R2		2
R3									2
R4					4
R5				3		5
R6				6			2
R7
R8			2
R9				2

Digit already on the grid
Box 5: no 2 yet
Not needed for this step

Filled in: box 5; cells R1C6, R2C2, R3C9, R6C7, R8C3 and R9C4. The other 66 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 2Extend the line of every 2 that reaches the box

Look along the rows and columns that run through box 5 — rows 4, 5 and 6, columns 4, 5 and 6 — and find those that already hold a 2. The 2 in R6C7 sits in row 6, so no other cell of row 6 can take a 2: strike R6C5 and R6C6. The 2 in R9C4 strikes column 4, which removes R4C4, and the 2 in R1C6 strikes column 6, which removes R4C6 (and R6C6 a second time). Rows 4 and 5 and column 5 hold no 2, so they strike nothing.

That is cross-hatching: lines laid across the box from two directions. Only the empty cells count — R4C5, R5C4, R5C6 and R6C4 are already filled. One empty cell is crossed by no line at all: R5C5.

Box 5 with the three lines that carry a 2 — row 6, columns 4 and 6 — drawn in full. Every empty cell of the box except R5C5 lies on one of them.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1				7		2
R2				5
R3						8
R4					4
R5				3		5
R6		9		6			2
R7				1
R8
R9				2

Digit already on the grid
Box 5
Line through a 2: no other 2 along it
The only empty cell no line crosses
Not needed for this step

Filled in: row 6; columns 4 and 6; box 5. The other 54 cells are left blank: they are not needed for this step. Nothing is removed yet: this step only reads the grid.

Step 3Write the 2 in the cell that survives — a Hidden Single

Box 5 must contain a 2, and R5C5 is the only place left for it, so R5C5 = 2. You did not write a single pencil mark.

This diagram adds the other cells' candidate lists, to show what you just found: none of them lists a 2. R5C5 itself could still hold a 7 — nothing in its row, column or box rules the 7 out — so looked at on its own the cell is not decided, and staring at it would never give you the 2. Only the question asked of the whole box — where can the 2 go? — does. That is a Hidden Single, a digit with one place left in a row, column or box, and cross-hatching is the way to find one by eye. The Hidden Single guide asks the same question of rows and columns too, with pencil marks.

Technique: Hidden Single

Box 5 with every candidate, and the three 2s whose lines cross it. No other cell of the box lists 2, so R5C5 = 2.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1						2
R2
R3
R4					4
R5				3	2	5
R6				6			2
R7
R8
R9				2

Digit already on the grid
Box 5: one place left for 2
Line through a 2
The cell that must be 2
Digit placed
Not needed for this step

Filled in: box 5; cells R1C6, R6C7 and R9C4. The other 69 cells are left blank: they are not needed for this step. Placed: R5C5 = 2.

Step 4Check the row the new digit fills: the last digit

Every digit you place adds to one row, one column and one box, so look at those three first. Row 5 now holds 4, 1, 3, 2, 5, 6, 9 and 8 — eight digits and one empty cell. Count through 1 to 9: the missing digit is 7, so R5C1 = 7. Before the 2 went in, R5C1 could still have been 2 or 7; the 2 settled it.

This is the last digit, which some players call a full house: a row, column or box with one empty cell takes the one digit it lacks, and nothing outside that unit needs checking. The app's hint reports it as a Naked Single, because R5C1 has one candidate left — but when a unit is this full, you see it without listing candidates at all.

Technique: Naked Single

Row 5 after step 3: eight digits and one gap. 7 is the only digit missing, so R5C1 = 7.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1
R2
R3
R4
R5	7	4	1	3	2	5	6	9	8
R6
R7
R8
R9

Digit already on the grid
Row 5: eight digits, one empty cell
The last empty cell
Digit placed
Not needed for this step

Filled in: row 5. The other 72 cells are left blank: they are not needed for this step. Placed: R5C1 = 7.

Step 5Follow the new digit, then keep sweeping

The 7 you just wrote is a new line for the 7s: column 1 runs up through box 1, which still needs a 7. A moment ago 7 could go in three cells of that box, R2C1, R3C1 and R3C3; the first two now lie on column 1, and the 7 in R1C4 already strikes row 1. One cell survives, so R3C3 = 7 — another Hidden Single, found at once because you knew where to look.

That is the scanning habit. After each placement, check its row, column and box, and the boxes its lines cross. When nothing new appears, go back to sweeping digit by digit, most frequent first, through every box that still needs it — then through rows and columns, which hide singles too. The 2s are not finished either: box 9 gives up its 2 at R7C8 in exactly the same way.

Technique: Hidden Single

Box 1 with every candidate after step 4, the 7 in row 1 and the new 7 in column 1. 7 is left in R3C3 only.
	C1	C2	C3	C4	C5	C6	C7	C8	C9
R1			5	7
R2		2	8
R3		6	7
R4
R5	7
R6
R7
R8
R9

Digit already on the grid
Box 1: one place left for 7
Line through a 7
The cell that must be 7
Digit placed
Not needed for this step

Filled in: box 1; cells R1C4 and R5C1. The other 70 cells are left blank: they are not needed for this step. Placed: R3C3 = 7.

Common mistakes

Striking a line from memory

Cross-hatching is only as good as its lines. Strike a row that holds no copy of the digit and you erase its real place; miss a copy and you write the digit where it does not belong. Before you write, point at the digit behind every strike — in step 2, the 2s in R6C7, R9C4 and R1C6 — not just the ones you remember.

Counting empty cells instead of surviving cells

Two counts get mixed up: the empty cells of a box, and the empty cells that survive the lines for the digit you are scanning. Box 5 had five empty cells and still forced its 2, because only one survived. A box that already contains the digit tells you nothing about it, however few gaps it has — only a unit down to one empty cell is a last digit.

Sweeping on without looking back

Each placement changes a row, a column and a box at once, and often hands you the next move: the 2 of step 3 completed row 5, and that 7 opened box 1. Players who finish a fixed round of all nine digits before looking again walk straight past moves that opened two placements earlier.

Try it on a real puzzle

Keep going

Hidden SingleA digit with only one place left in a row, column or box — even if that cell lists other candidates.
Naked SingleOne cell, one surviving candidate — the safest placement on the grid.

Learn the move, not just the digit

In a real game, the Guide hint in the app picks the next cell itself and explains it. On the puzzle above it starts with this very move: it places the 2 in R5C5, and its explanation names the row and the two columns that rule out every other empty cell of box 5 — your cross-hatching lines, in words.

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