Last Remaining Cell: Solve One Cell at a Time

The Last Remaining Cell technique shifts your perspective from the unit to the cell. Instead of scanning rows, columns, or boxes for missing numbers, you pick a single empty cell and ask: which numbers are already taken by its row, column, and box? Whatever number remains — the one not yet used in any of those three units — is your answer. It's called "Last Remaining" because it's the last candidate standing after you eliminate all the numbers that can't go there.

How It Works

Every empty cell in Sudoku is the intersection of three constraints: the 8 other cells in its row, the 8 other cells in its column, and the 8 other cells in its 3×3 box. Together, those three units can theoretically contain up to 24 different cells — but in practice, there is significant overlap (cells that are in both the row and the box, the column and the box, etc.). The total distinct cells that constrain any given cell is exactly 20: 8 in the row, 8 in the column, and 4 additional unique cells in the box.

Here's the process broken down:

• Step 1: Choose any empty cell on the board.
• Step 2: Scan its row from left to right. Which numbers (1-9) do you see? Mentally cross those off your list of possibilities.
• Step 3: Scan its column from top to bottom. Cross off any additional numbers not already eliminated by the row scan.
• Step 4: Scan its 3×3 box. Cross off any remaining numbers.
• Step 5: If exactly one number remains uncrossed, that's your value. Fill it in.

A Concrete Example

Let's work through a real scenario. Pick cell R3C5 (row 3, column 5). You scan row 3 and see the numbers: 1, 2, 3, 4, 5, 6. You scan column 5 and find 7 and 8. Scanning the 3×3 box that contains R3C5, you find the number 9. Together, you've spotted 1 through 9 — every number except one. If, say, 4 is the only number you didn't see in any of those three units, then R3C5 must be 4. You've solved the cell.

In practice, you rarely need to scan all three units thoroughly. After scanning the row and column, you'll often already have 7 or 8 numbers eliminated. A quick glance at the box confirms the last constraint. As you gain experience, this technique becomes almost automatic — your eyes will naturally sweep across the intersecting units without conscious effort.

How It Differs from Last Free Cell

Beginners sometimes confuse Last Remaining Cell with Last Free Cell. Here's the key difference: Last Free Cell starts with a unit (row, column, or box) and finds the missing number. Last Remaining Cell starts with a cell and deduces the number from all three constraints combined. Last Free Cell only works when a unit has 8 cells filled. Last Remaining Cell works even when no unit near the cell has more than 3 or 4 filled cells — as long as the combined constraints from all three units eliminate 8 of the 9 possible numbers.

When to Use This Technique

Last Remaining Cell is your go-to beginner technique when no row, column, or box has 8 cells filled. It's slightly more powerful than Last Free Cell because it pools constraints from three intersecting units. On Easy puzzles, it works from the very first move. On Medium puzzles, you'll use it alongside Last Possible Number to systematically work through the grid. Even on harder puzzles, it remains a valuable finishing tool — once the board fills up, many cells reduce to a single candidate, and Last Remaining Cell confirms the answer without pencil marks.

A practical tip: start with cells that sit at the intersection of heavily populated rows, columns, and boxes. Cells in the middle of the board often have richer constraints than corner cells, making them more likely to be solvable with Last Remaining Cell.