Fill every cell with a digit from 0 through 9.
Each row must contain every digit from 0 through 9 exactly once. A digit may repeat in a column.
Identical final digits may not touch horizontally, vertically, or diagonally. Cells that share a side or a corner are adjacent; cells beyond the board edge are not.
The digits in each column must add to the target shown beneath that column.
The puzzle is solved only when the grid is full and satisfies every row, adjacency, given, and column-sum rule. Every published puzzle has exactly one valid completed grid.
Every row holds all ten digits exactly once, so the digits already showing tell you precisely which ones are left. Write that shrinking set down before you reason about anything else — most deductions on this board start from it.
Example: A row showing 4, 7, 1, 9, 0 and 3 has 2, 5, 6 and 8 to place, in four cells and nowhere else.
A column’s target minus the digits already in it is exactly what the empty cells in that column must add up to. That remainder is often small enough to name the digits outright.
Example: A column of five cells targeted at 21 already shows 9, 6 and 4. The last two cells must add to 2 — so they are 0 and 2, or 1 and 1 if they do not touch.
Each empty cell can hold at most 9 and at least 0, so a column with several blanks still has a reachable range. If the remainder is bigger than 9 times the blanks, you have made a mistake; if it is small, the big digits are all ruled out.
Example: Three blanks needing 4 between them cannot include a 5 or more, so every candidate above 4 comes out of all three cells.
A 0 is a real answer that happens to add nothing to its column. It still takes its place in the row, still blocks its eight neighbours, and still has to be written in before the puzzle is complete.
Example: A column whose remainder is 0 with one blank left is not finished — that blank holds the digit 0.
The moment a digit is certain, cross it out of all eight cells around it — including the diagonals in the rows above and below. It is the rule players forget, and the one that most often unblocks a stuck column.
Example: A 7 placed in row 3 bars a 7 from the three cells above it, the three below it, and the cell on either side.
Rather than asking what a cell can hold, ask where a digit can go. A row owes each of its missing digits exactly one cell, so if a digit is barred from all but one of the blanks, it belongs there.
Example: A row still needs a 5, and every blank but one has a 5 touching it diagonally. The remaining blank is the 5.
The strong deductions come from a cell that two constraints meet in: a row that owes a short list of digits, and a column whose remainder only a few of them fit.
Example: A row owing 2, 3 and 8 crosses a column that still needs 15 from two cells. The 2 cannot go there — even a 9 alongside it falls short.
Each target shows the running total against the goal. It turns green when the column is full and exactly right, and red when the column can no longer reach its target — so a red gutter tells you the mistake is in that column, before you have filled the rest of the board.
Example: A gutter reading 34 / 31 has gone red: something in that column is too big, and it is quicker to check the four cells above it than to hunt the grid.
A cell with two or three possibilities is worth noting rather than guessing. Notes never count as answers, so a cell full of candidates leaves the puzzle exactly as unfinished as it was — but the pattern of them is what shows you the next certainty.
Example: Two cells in one row noted as exactly 3 or 6 mean no other cell in that row can be a 3 or a 6.