Trial and Error (Bifurcation)
advancedSystematic guessing and backtracking as a last resort technique
Trial and Error (also called Bifurcation or What-If Analysis) is the technique of last resort. When no logical pattern can be found, you pick a bi-value cell, assume one value, and follow the chain of consequences to either a solution or a contradiction.
Understanding the Concept
Choose a cell with the fewest candidates — ideally a bi-value cell (exactly 2 candidates).
Pick one candidate and tentatively place it. Mark your notes clearly so you can backtrack.
Follow all forced consequences: naked singles, hidden singles, and any other techniques that trigger.
If you reach a contradiction (a cell with no candidates, or a house with no place for a digit), the assumed value was wrong.
Erase all the tentative work and place the OTHER candidate in the original cell — it must be correct.
If no contradiction emerges quickly, you may need to make a second assumption deeper in the chain.
This is called "depth-first search" in computer science. Most Sudoku purists consider it inelegant, but it always works.
Key tip: exhaust all logical techniques first — this makes trial-and-error faster because fewer cells remain.
Examples
Bifurcation from a Bi-value Cell
Cell R4C4 has only candidates {3,8}. We assume R4C4=3. This forces R4C7=8 (last place for 8 in row 4), which forces R2C7=6 (only remaining candidate), which forces R6C4=9. But then R7C4 has no valid candidates — contradiction! Therefore R4C4 must be 8.
Pro Tips
- •Always exhaust logical techniques first — trial-and-error should be the last resort
- •Choose cells with the fewest candidates to minimize branching
- •Bi-value cells (exactly 2 candidates) are ideal because one must be wrong
- •Mark your tentative placements clearly and keep a clean "save point" to revert to
- •A contradiction usually appears within 5-10 forced moves if the guess was wrong
- •In digital solvers, use the undo feature liberally; on paper, use light pencil marks
