Last Remaining Cell & Full House
Last Remaining Cell is the simplest and most intuitive Sudoku solving technique, making it the perfect starting point for beginners. The idea is straightforward: when eight of the nine cells in a row, column, or 3x3 box are already filled, the remaining empty cell must contain the one missing digit. Since every row, column, and box must contain the digits 1 through 9 exactly once, there is only one possible value for that last cell.
This same concept is sometimes called a Full House, which is simply another perspective on the same idea. Rather than scanning the entire grid for easy placements, Full House emphasizes looking at one unit at a time — a single row, column, or 3x3 box — and asking whether it has exactly one empty cell. If it does, you identify the missing digit and fill it in. The logic is identical: every unit must contain each digit exactly once, so when eight are present, the ninth is forced. Whether you think of it as finding the last remaining cell or completing a full house, the underlying principle is the same.
To apply this technique, scan each row, column, and 3x3 box in the grid and count how many cells are already filled. When you find a unit with exactly one empty cell, identify which digit from 1 to 9 is missing and place it there. This process requires no complex logic or candidate tracking — just simple counting and observation.
Last Remaining Cell is often the first technique you will use when starting a new puzzle, and it frequently triggers a cascade of further placements. After filling one cell, another row, column, or box may now have only one empty cell remaining, allowing you to continue solving. This chain reaction effect is especially common in the late stages of a puzzle when most cells are already filled. Building the habit of quickly spotting these opportunities keeps your solving momentum going and prevents you from overlooking easy placements while hunting for more complex patterns. Mastering this technique builds the foundation for all other Sudoku strategies and helps you develop the habit of systematically scanning the grid.
Try It Yourself
Walk through each step of the last remaining cell & full house technique on a real puzzle. Follow the instructions and try entering the correct value when prompted.
Look at row 1 (the top row). Eight cells are already filled with the digits 1 through 8. Count them to confirm only one cell is empty.
Step-by-Step Guide
Choose a row, column, or 3x3 box to examine.
Count the number of filled cells in that unit.
If exactly eight cells are filled, identify the one empty cell.
List the digits 1 through 9 and cross off each digit already present in the unit.
The one remaining digit is the answer — place it in the empty cell.
Check whether filling that cell creates a new Last Remaining Cell in an intersecting row, column, or box.
Continue scanning until no more units have exactly one empty cell.
Think of it like a jigsaw puzzle with one piece left — you do not need to figure out what it looks like, because it is the only piece that fits in the last remaining hole. Or imagine a shelf with nine labeled slots where eight items are already in place — there is only one empty slot left, so the last item's position is obvious.
Sudoku requires each unit (row, column, or box) to contain the digits 1 through 9 exactly once — a direct application of the pigeonhole principle. When eight of the nine slots are occupied by eight distinct digits, the ninth digit is uniquely determined because it is the only value from the set {1–9} not yet assigned. No further reasoning or cross-referencing is needed; the constraint alone forces the answer. This holds whether you view it from the cell's perspective (Last Remaining Cell) or the unit's perspective (Full House) — the mathematical guarantee is identical.
When to use: Scan for any row, column, or box that has exactly eight cells filled. This is the very first thing to look for whenever you start a puzzle or place a new digit. It is especially common near the end of a puzzle when most cells are already filled.
Common Mistakes to Avoid
Miscounting the filled cells and thinking a unit has one empty cell when it actually has two.
Always count all nine positions deliberately. Point at each cell or use your finger to track across the row, column, or box.
Forgetting to check all three unit types — only scanning rows but neglecting columns and boxes.
Make a habit of sweeping rows first, then columns, then boxes systematically so you never miss an opportunity.
Confusing this technique with Naked Singles — a Naked Single requires checking three units simultaneously, but Last Remaining Cell and Full House only examine one unit.
If you are looking at one row, column, or box in isolation and it has one gap, that is a Last Remaining Cell / Full House. Save cross-referencing for Naked Singles.
Skipping the verification step and placing the wrong digit because of a hasty count.
Tick off digits 1 through 9 against the filled cells to confirm exactly which digit is missing before writing it in.
More Examples
See last remaining cell & full house applied in different puzzle configurations to strengthen your pattern recognition.
Highlighted cells show the last remaining cell & full house pattern
Practice Puzzles
Apply the last remaining cell & full house technique on these mini challenges. Tap a highlighted cell and enter the correct digit.
Quick Reference
- Pattern:
- A row, column, or box has only one empty cell
- Action:
- Place the missing digit in the empty cell
- Look for:
- Units with 8 of 9 cells filled