Nishio

Nishio is the trial-and-contradiction technique that sits at the contested edge of pure deduction. The procedure: pick a candidate, assume it's the digit at that cell, then propagate the consequences for that digit alone through the grid — placing it in cells where it's forced, eliminating it from cells it can't be in. If the propagation leads to a contradiction (a row, column, or box where the digit can no longer be placed at all), the original assumption was wrong, and the candidate can be eliminated.

Why it sits at the edge of deduction

The technique is named after Tetsuya Nishio, the Japanese puzzlist who formalised it in the 1990s. Some communities reject it on principle: assuming a value and following its consequences feels closer to trial-and-error than to logical deduction. Others accept it as a legitimate single-digit technique because the propagation rules are exactly the same as the rest of Sudoku's logic; you're just running them under a hypothesis.

The middle position is pragmatic. Nishio is mathematically equivalent to certain forcing chains and to many AIC patterns when the AIC is constrained to a single digit. The same eliminations a Nishio produces can usually be re-derived through formal chain logic — but doing so by hand often takes longer than running the trial directly. Solvers who want speed accept Nishio; solvers who want a clean deduction trail re-derive via chains.

How it differs from a full forcing chain

A forcing chain explores multiple digits and looks for convergence or contradiction across the puzzle. A Nishio is restricted to one digit at a time: take a candidate of digit d, assume d there, propagate d's placements and eliminations only, ignore every other digit. If d fails to fit somewhere, eliminate the original candidate.

The single-digit restriction is what gives Nishio its character. It's both more limited (some forcing-chain eliminations don't show up under a single-digit trial) and more disciplined (the propagation graph stays small enough to verify by eye).

A cousin to uniqueness reasoning

Nishio shares its uneasy reception with unique rectangle and BUG reasoning. All three lean on something one rung removed from the no-repeats rule: Nishio leans on the puzzle's consistency under a hypothesis, UR and BUG lean on the puzzle's uniqueness. Strict solvers reject all three; pragmatic solvers accept all three. The line between "real deduction" and "supported guessing" turns out to be philosophical, not mathematical, and reasonable solvers draw it in different places.

For Sudoku Mountain's puzzles, Nishio is safe to use — every published puzzle is checked for uniqueness, and the trial propagation only fails on candidates that genuinely cannot be the answer. Whether to call it a technique or a fancy guess is up to you.