Last Remaining Cell

A number with nowhere else to go.

Last Remaining Cell is the mirror image of Last Free Cell. Instead of looking for a region with only one empty cell left, you pick a specific digit and look for a region — a row, column, or 3×3 box — where that digit still needs to be placed.

Among the region's remaining empty cells, some are already blocked: the digit can't go there because it already appears elsewhere in that cell's row, column, or box. If every empty cell except one is ruled out this way, the digit is forced into the one cell left standing, even though other cells in the region are still technically empty. This technique works cell by cell for one digit at a time rather than scanning an entire region for its last gap, which makes it more broadly applicable once a board fills in enough digits to start blocking rows and columns from multiple directions at once.

How It Works

Choose a digit. Scan a region. If all other empty cells in that region are already forbidden from holding that digit — because the digit already appears in their row, column, or box — then only one cell can hold it.

  1. Pick a digit (1–9).
  2. Find a region where that digit must still appear.
  3. Eliminate cells in that region where the digit is already blocked by its row, column, or box.
  4. If exactly one cell remains, place the digit there.

Example

Below board presents an example of the last remaining cell for digit 5.

digit 5 elsewhere — blocks its row or column
only cell left for 5 in this box

Look at the center box. A 5 already sits in row 4 and row 6 elsewhere on the board, ruling out both rows inside the box. A 5 also sits in column 5, ruling out that column too. The box's own 9 occupies another cell. The only cell in the box in which 5 can be placed is the one colored in green.

On Evil Boards

Finding the last remaining cell for a digit usually takes two or three copies of that digit already placed elsewhere on the board, blocking rows and columns until only one cell survives. That density of clues rarely exists until the mid or late game on an Evil board. Still, it costs nothing to check for it right after the puzzle is generated — on the rare occasion the technique already applies.