Logic Grid Puzzles: From Clues to a Unique Solution
A grid is an error-control tool: it turns prose into explicit matches and exclusions that can be checked.
How to solve a logic grid puzzle
Use a separate grid for every pair of categories. Mark confirmed matches and exclusions consistently, propagate every confirmed match across the other grids, and scan each row and column whenever only one candidate remains. Finish with a swap test to confirm that the solution is unique.
A logic grid stores what can and cannot match. Each confirmed match changes the choices left in the same row and column.
Key relationships
- A grid cell records a possible or impossible match.
- A confirmed match eliminates other matches in its row and column.
- A complete grid represents one arrangement that satisfies every clue.
What a grid actually represents
Each cell represents one possible relationship between two categories. A blank means unknown, a cross means impossible, and a dot means confirmed. Rows and columns usually permit exactly one confirmed match. The marks matter less than applying them consistently.
Use a separate sub-grid for each category pair. Do not combine two deductions in your head. If Rowan owns the red bicycle and the red bicycle was purchased Tuesday, explicitly transfer Tuesday to Rowan in the person-day grid.
A miniature worked example
Suppose Ada, Ben, and Chen each choose tea, coffee, or water. Ada does not choose tea. Ben does not choose water. The coffee drinker is not Chen. If Chen chose tea, coffee is ruled out by the third clue and water remains. Then Ada cannot take tea and must take coffee, leaving tea for Ben. But this conflicts with the Chen assumption. Instead, Chen takes water, Ben takes tea, and Ada takes coffee.
The example shows propagation across exclusions. No clue alone states Ada drinks coffee. The answer appears only after candidate removal.
Detecting an ambiguous grid
After filling a grid, test whether two rows can exchange their remaining matches. If a swap preserves every clue, the puzzle has multiple solutions. A well-formed puzzle needs another independent constraint. This test is useful when creating puzzles as well as solving them.
Translate clue language precisely
Direct clues create a mark immediately. Relational clues such as earlier than, next to, or two places after need a small side diagram because a standard matching grid does not preserve order. Conditional clues need both the stated rule and its contrapositive, but not the reversed rule.
Words such as either, only, exactly, and at least change the candidate set in different ways. Rewrite the clue in plain alternatives before marking the grid. If the clue says either Ada or Ben chose tea, confirm whether the puzzle means one or possibly both before assuming exclusivity.
Propagate a confirmed match completely
When a cell is confirmed, eliminate every other cell in its row and column. Then carry the relationship into connected category grids. If Ava is paired with Tuesday and Tuesday is paired with the red bicycle, Ava must be paired with the red bicycle even if no clue states that sentence directly.
Many stalled grids contain a confirmed relationship that was transferred in one direction but not the other. Use a short propagation checklist after every dot: clear the row, clear the column, transfer connected matches, and rescan for single candidates.
Handle ordering clues outside the grid
For arrival times, seats, ranks, or dates, create numbered slots beside the matching grids. Draw blocks for fixed distances and list the legal placements. A clue saying Noor arrived exactly two places before Imani becomes a movable two-person block rather than several disconnected crosses.
Combine the slot diagram with the grids only after a placement becomes forced. This avoids filling the grid with marks that depend on an unrecorded mental picture.
How one confirmed match travels across a grid
Suppose three people, three drinks, and three days must be matched. A confirmed person-to-drink match is incomplete until its consequences are transferred to the day grids.
| New fact | Immediate marks | Cross-grid consequence |
|---|---|---|
| Ben chose tea | Cross coffee and water from Ben's row | Any day matched to tea also belongs to Ben |
| Tea was chosen Tuesday | Cross Monday and Wednesday from tea's row | Confirm Ben with Tuesday |
| Ada was not Monday | Cross Ada and Monday | If Tuesday is occupied by Ben, Ada can only be Wednesday |
| Chen chose coffee | Cross tea and water from Chen's row | The remaining water match belongs to Ada |
Four checks after every confirmed cell
- Clear every other candidate in the same row.
- Clear every other candidate in the same column.
- Transfer the match through any already confirmed third category.
- Rescan all affected rows and columns for a single remaining candidate.
A grid often appears stalled because a solver made the first two marks but skipped the transfer. When order or distance matters, keep a separate numbered slot diagram. A matching grid can store who belongs with what, but it cannot show that one event occurred two positions before another without extra notation.
Build stronger constraint models
A grid records exclusions, while the constraint satisfaction guide explains how those exclusions propagate through the remaining candidates.
Use the logic puzzle generator to test the method on another assignment or ordering problem.
Apply all-different assignments and false labels in the mislabelled fruit boxes logic puzzle.