| C11 | C21 | C31 | C41 | C51 | |
|---|---|---|---|---|---|
| R11 | |||||
| R21 | |||||
| R31 | |||||
| R41 | |||||
| R51 |
Does this board force its centre square to be filled?
Make a prediction before opening the reasoning.
Compare the predictions
- Yes: the diagonal forces it.
Not supported. No diagonal rule exists. A shifted assignment satisfies every clue with an empty centre.
- No: valid completions disagree.
Supported. The diagonal and shifted-column constructions are complete counterexamples to either forced value.
- The clues cannot be satisfied.
Not supported. The main diagonal alone supplies a valid completion.
Predictions stay on this page; they are not saved or sent.
The position and the question
This is a deliberately ambiguous construction, not a daily puzzle advertised as having a unique answer. Each row must contain exactly one filled square, and each column must also contain exactly one. No other givens are available.
Before drawing a diagonal or guessing a picture, separate two questions: can you complete the board, and do all legal completions agree on a particular square? The first asks for one example. The second asks for evidence about every example.
Show the reasoning
Build two witnesses
Fill R1C1, R2C2, R3C3, R4C4 and R5C5. Every clue is satisfied and the centre is filled. Now fill R1C2, R2C3, R3C4, R4C5 and R5C1. Every clue is still satisfied, but the centre is empty. These two complete boards already disprove either forced value for the centre.
Count the whole family
Row 1 can choose any of five columns. Row 2 must use one of the four unused columns, then row 3 has three choices, row 4 two and row 5 one. There are 5 × 4 × 3 × 2 × 1 = 120 legal completions. Choosing a column already used would violate that column’s clue.
Test an additional given
Suppose R3C3 is supplied as a confirmed filled square. Row 3 and column 3 are now settled, leaving four rows to match with four columns. There are 4 × 3 × 2 × 1 = 24 completions. One given reduces the uncertainty without eliminating it.
Row and column clues C11 C21 C31 C41 C51 R11 R21 R31 R41 R51 Test an additional given Make the right stopping decision
If your deductions reach this original unmarked board, the lack of a forced square is not a failure to spot overlap. The supplied information permits different answers. A solver should report ambiguity or request a further given, rather than invent a certainty from a symmetrical-looking picture.
The tempting shortcut
The diagonal looks natural, so the centre must be filled.
Where the shortcut breaks
The shifted-column completion is an explicit counterexample with exactly the same row and column clues. Picture appearance adds no rule.
A better decision
Keep the centre undecided unless a further clue or established mark excludes every centre-empty completion.
Changed conditions
If R1C1, R2C2, R3C3 and R4C4 are confirmed filled, how many completions remain? What if only the first three are confirmed?
Check the changed-condition answer
Four confirmed diagonal squares leave only column 5 available for row 5: exactly one completion. Three confirmed diagonal squares leave rows 4–5 and columns 4–5: two completions, because those last two assignments can be swapped.
Take this back to your next game
A valid completed picture proves existence. Two disagreeing valid pictures prove that a particular square is not forced.
How the claim was checked
Exhaustive permutation enumeration of one filled cell per row, with distinct columns.
- 120 unmarked completions
- 24 completions with centre filled
- 2 completions with first three diagonal givens
- 1 completion with first four diagonal givens
Read the fixed positions and supporting proof data. Computational checks establish only the claim and position stated here. They do not establish a player difficulty rating or make an unproved strategy optimal.
Playfield Arcade · Updated 4 October 2026 · Original case design, AI-assisted writing and code, with reproducible position checks. Our method · Report a correction