Uniqueness

This wiki page discusses the use of uniqueness in sudoku, including the debate surrounding whether uniqueness can and should be assumed when solving or when setting sudoku puzzles.

Uniqueness in Sudoku

Uniqueness in sudoku refers to the concept that a well-formed sudoku puzzle should have only one valid solution. This idea is central to various solving techniques known as Uniqueness Techniques which rely on the assumption that the puzzle has a unique solution.

Uniqueness Techniques

Uniqueness Techniques (UT below) are strategies that exploit the unique solution assumption to make deductions while solving sudoku puzzles. These methods identify configurations that would lead to multiple solutions, thereby contradicting the unique solution assumption. By eliminating these configurations (see also Guardians), solvers can make progress towards the correct solution.

Effectiveness of Uniqueness Techniques

Assuming a puzzle has a unique solution, this solution can always be obtained through Trial and Error steps or equivalent reformulations, such as complex nested chains or other specialized solving techniques. Consequently, the use of Uniqueness Techniques (UT) is never strictly required to solve any sudoku puzzle.

The effectiveness of UT spans a wide spectrum: from cases where UT can be applied but may not offer a significant advantage over traditional solving techniques, to situations where using UT considerably simplifies the puzzle or even transforms an otherwise seemingly unsolvable puzzle into one that can be solved by humans. Template:Citation needed

Using Uniqueness for Solving

The use of UT for solving sudoku is not universally accepted, as it hinges on the assumption that a puzzle has a unique solution. While rare, this assumption may not hold true for all sudoku puzzles, particularly those not designed with a unique solution in mind (for example puzzles with relaxed rules such as Schrödinger Cells or similar concepts; see also Fractional Dimension).

Solving a sudoku puzzle without using UT offers a stronger validation of the puzzle's well-formedness. When a solver completes a puzzle without relying on UT, they inherently prove that the puzzle has a unique solution. Conversely, using UT may lead to a solution that is only one of many, without providing information on whether the solution is, in fact, unique.

This distinction is particularly important in test solving and providing feedback to the puzzle setter. Solving a puzzle without using UT allows the solver to confirm that the puzzle is indeed well-formed, an essential quality check for the setter.

Puzzle setters intending the use of UT should include clear language in the puzzle rules, explicitly stating that the puzzle has a unique solution, thereby legitimizing and hinting at the use of UT for the solvers. Puzzle rules can include phrasings such as: "This puzzle has a unique solution." or more explicitly, "This puzzle has a unique, computer-verified solution. Feel free to use this fact during your solving process."

In the absence of such wording, it is not always safe to assume that the setter intended the use of UT. Therefore, by avoiding UT and relying solely on logical deductions within the puzzle's constraints, solvers not only ensure that the puzzle has a unique solution but also maintain the integrity of the solving process by avoiding shortcuts not intended by the puzzle setter.

However, in some communities or forums where the assumption of uniqueness is commonly agreed upon, solvers might feel more comfortable using UT even without explicit instructions.

Speed Solving and Competitions

In speed solving and sudoku competitions, the use of UT is common. These events often feature puzzles designed with unique solutions, making the uniqueness assumption more justified. Additionally, the competitive nature of these events encourages solvers to use all available techniques, including UT, to achieve the fastest solving times.

Summary

Pros

• UT can simplify and speed up the solving process, allowing solvers to eliminate possibilities more efficiently.
• In many cases, sudoku puzzles are designed with the intention of having a unique solution, making the use of UT valid and appropriate.
• Exploiting uniqueness can provide interesting and satisfying logical deductions for the solver.
• In some communities, uniqueness might be a shared implicit assumption and not using UT can result in unintentionally difficult solves.

Cons

• Solving a puzzle without using UT proves a stronger result (the existence of a single solution rather than the existence of a solution), which solvers may find more satisfying.
• Relying on UT may lead to incorrect deductions for puzzles without unique solutions.
• Using UT may shortcut the intended solving path, possibly resulting in a quicker but less satisfying solve.
• Some solvers argue that using UT detracts from the pure logic aspect of sudoku, as these methods depend on a meta-assumption about the puzzle rather than working strictly within the puzzle's constraints.