Lesson:Bowman's Bingo
This is a hypothesis testing technique. You assume a candidate is true and follow the forced chain of consequences.
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Bowman's Bingo is a systematic hypothesis-testing technique for solving extremely difficult Sudoku puzzles that resist all other logical methods. Unlike analytical techniques that derive placements purely from existing constraints, Bowman's Bingo involves assuming that a specific candidate is TRUE, following the chain of forced consequences, and checking whether a contradiction arises. If it does, the initial assumption was wrong, and the opposite placement is confirmed.
The technique is named after its originator and is closely related to concepts like Nishio and Trial and Error, but it is more structured. Bowman's Bingo always starts with a bi-value cell (a cell with exactly two candidates) and uses only forced placements — Naked Singles — to propagate the hypothesis. This disciplined propagation makes it more principled than random trial and error, and ensures that any contradiction discovered is logically valid.
Bowman's Bingo is a technique of last resort. If you find yourself needing it on a standard published puzzle, it typically means you have missed a more elegant analytical technique elsewhere on the board. However, for computer-generated extreme puzzles and certain diabolical compositions, it may be the only path forward without a computer solver.
How It Works — Step by Step
Step 1 — Select a bi-value cell to start
Find a cell with exactly two remaining candidates. This is your hypothesis starting point. Choose one of the two candidates to assume as TRUE. Mark this assumption clearly — using a different pencil color or notation — so you can easily undo it if needed.
Step 2 — Propagate forced placements
With your assumed placement made, look for Naked Singles — cells in the same row, column, or box that now have only one candidate left. Place each Naked Single and mark it as a hypothetical placement. Continue propagating: each new placement may create more Naked Singles.
Step 3 — Watch for a contradiction
A contradiction occurs when: (a) a cell has no remaining legal candidates (an empty cell with no candidates), or (b) a digit is required in two cells of the same unit simultaneously. If either happens, your initial assumption was FALSE.
Step 4 — Conclude and place
If a contradiction is found, erase all hypothetical placements and place the OPPOSITE candidate in your starting cell. If no contradiction is found after propagating all Naked Singles, the hypothesis may be correct — but this only proves it if the puzzle has a unique solution and the result is complete. If needed, begin the process again from another bi-value cell.
When to Use This Technique
Use Bowman's Bingo only after exhausting every analytical technique you know — Naked and Hidden Singles, Pointing Pairs, Box-Line Reduction, Naked and Hidden Pairs, X-Wing, Swordfish, Coloring, XY-Wing, and AICs. If the puzzle still has not yielded, Bowman's Bingo is a reliable fallback. It is also excellent for verifying a unique solution: if both assumptions from a bi-value cell lead to valid (non-contradictory) completions, the puzzle has multiple solutions.
Worked Examples
Suppose cell R4C6 has candidates {2, 8}. Assume it is 2. Propagate: R4 now has only one empty cell left — it must be 8. Column 6 now has 2 in row 4, which forces Row 2 Column 6 to be 5. But wait — Row 2 already has a 5. Contradiction! Therefore R4C6 cannot be 2; it must be 8. Place 8 in R4C6 and continue solving from there.
A longer chain: assume R1C1 = 3. This forces R1C7 = 7 (Naked Single in Row 1). Then R9C7 = 4 (Naked Single in Column 7). Then R9C3 = 6 (Naked Single in Row 9). Then R5C3 = 3 (Naked Single in Column 3). But R5C3 = 3 is in the same box as R1C1 = 3 — contradiction! Therefore R1C1 ≠ 3, so R1C1 = the other candidate.
Frequently Asked Questions
Is Bowman's Bingo the same as trial and error?
They are related but not identical. Trial and error (or bifurcation) may place any candidate anywhere and follow all logical consequences. Bowman's Bingo is more restricted: it always starts from a bi-value cell and propagates only Naked Singles (forced placements), making it more structured and easier to track. True trial and error is generally considered inelegant in Sudoku culture; Bowman's Bingo occupies a middle ground.
What is a contradiction exactly?
A contradiction is any board state that violates Sudoku's rules: a cell with no candidates remaining (the digit has nowhere to go), or the same digit being required in two different cells within the same unit. Either state proves the current hypothesis is false.
How deep can Bowman's Bingo chains get?
It depends on the puzzle. Some contradictions appear after just two or three forced placements. Others require propagating 20 or more Naked Singles across many units before the board breaks. The depth is unpredictable, which is why careful notation of hypothetical placements is essential.
What if neither candidate leads to a contradiction?
If both candidates in your starting bi-value cell produce valid, fully-solved boards, the puzzle has two solutions and is therefore invalid as a standard Sudoku. If neither produces a contradiction but neither fully solves the board either, you need to apply further analytical techniques to the resulting board state.
Related Lessons
Ready to test your knowledge? Try applying this technique in our Evil Sudoku challenges, Expert level course or explore Expert Sudoku puzzles. Keep training to improve your solve times and master the grid!
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