Sudoku Generator

A Unique Solution is the defining property of a valid Sudoku puzzle. When a puzzle has only one solution, every digit placement can be determined through pure logical deduction - no guessing is ever required. If a puzzle had two or more possible solutions, then at some point a solver would reach a fork where neither choice could be ruled out by logic alone, making the puzzle fundamentally ambiguous and unfair.

Professionally published puzzles - and every puzzle generated on this tool - are verified to have exactly one solution. The verification works by running the solver algorithm twice: once to find the first solution, and then again from a different starting path to confirm no second solution exists. Only if both checks agree is the puzzle accepted.

Puzzles with multiple solutions are sometimes called "broken" or "invalid" puzzles. They appear most often when too many numbers are removed from the grid, leaving the remaining clues (called Grid Givens) insufficient to uniquely constrain all 81 cells.

The Backtracking Algorithm is a systematic trial-and-error technique used in computer science to solve constraint-satisfaction problems. For Sudoku generation, it works in two stages.

Stage 1 - Full Grid Generation: The algorithm fills the board cell by cell, left to right and top to bottom. For each empty cell, it tries digits 1 through 9 in a randomly shuffled order. If a digit is valid (does not conflict with any row, column, or box constraint), it is placed and the algorithm moves to the next cell. If no digit works, the algorithm "backtracks" - it erases the previous cell's value and tries the next candidate instead. This recursion continues until a completely filled, valid 9x9 grid is produced.

Stage 2 - Clue Removal: Numbers are removed from the full grid one at a time in a random order. After each removal, the solver is run to confirm the puzzle still has exactly one unique solution. If removing a number creates ambiguity (two or more solutions), that number is put back and the algorithm tries a different cell. This process continues until the target number of Grid Givens (pre-filled clues) for the selected difficulty is reached.

The result is a minimal puzzle where every given is necessary - removing any single additional clue would break the unique-solution guarantee.

Pencil Marks (also called candidates or notes) are small numbers written inside a cell to track which digits are still possible for that position. When solving Easy puzzles, the answer to each cell is often immediately obvious. But on Hard and Expert puzzles, you frequently reach cells where 3, 4, or even 5 different digits are all technically possible - without pencil marks, it becomes impossible to track the logic.

Professional solvers write in pencil marks from the very beginning of a hard puzzle, then eliminate candidates systematically as new digits are placed. When a cell is finally filled, all pencil marks in its row, column, and box that match that digit can be erased, often triggering a cascade of new deductions.

This tool's Pencil Mode toggle activates a mini 3x3 grid inside each cell where you can record your own candidate notes. Toggle it on, then click a number to add or remove it from the selected cell's notes rather than placing a final answer.

Candidate Elimination is the core process of narrowing down which digits can legally go in each cell. The two most fundamental techniques are Naked Singles and Hidden Singles.

A Naked Single occurs when only one candidate digit remains in a cell after all conflicts have been removed. This is the most direct solve: place that digit immediately. A Hidden Single occurs when a digit can go in only one cell within a row, column, or box - even if that cell still has other candidates listed. Scan each group (row, column, 3x3 box) and ask: "Where can the digit 5 go in this row?" If there is only one legal position, place it there.

To improve speed, develop a scanning rhythm: sweep each row left to right, then each column top to bottom, then each box left to right. During each sweep, look for Naked and Hidden Singles before moving to more advanced techniques like Pointing Pairs or X-Wings. Consistent scanning patterns prevent missed deductions and dramatically reduce the time spent on each puzzle.

Constraint Propagation is the formal name for this ripple effect - when placing one digit eliminates candidates elsewhere, which in turn reveals more placements, and so on. Strong solvers chain these deductions together rapidly across the entire board without backtracking.

The minimum number of Grid Givens required to guarantee a unique solution is 17. This was proven mathematically in 2012 by Gary McGuire, Bastian Tugemann, and Gilles Civario after an exhaustive computer search. They checked billions of 16-clue configurations and found that none had a unique solution, while valid 17-clue puzzles do exist.

In practice, most published Expert-level puzzles use between 17 and 22 givens, as puzzles at the 17-clue boundary require extremely advanced solving techniques well beyond ordinary logic. The Expert difficulty on this generator targets 17-21 givens - genuinely challenging but solvable without computer assistance using the strategies described above.