Constraint

• How constraint satisfaction puzzles encode rules as must-be, cannot-be, and conditional links.
• To formalise clues as logical constraints before searching for solutions.
• To detect contradiction early and prune impossible branches.
• To apply constraint thinking to scheduling, seating, and allocation problems in Class 12 reasoning.

Level 1 — Foundations

Verbal: A constraint is any rule the solution must obey. Collect all constraints before placing values.

Constraint categories:

Propagation: When Ravi in Team A, remove Ravi from Team B column (all-different). When Team A full, others forced to Team B.

Level 2 — Exam depth

Hard constraints first: Apply most restrictive rules early (exact count, fixed position).

Soft constraints: Preferences ("prefer morning slot") — only after hard rules satisfied in optimisation puzzles.

Constraint graph: Nodes = variables; edges = constraints — helps see which variables are tightly coupled.

Infeasibility signal: Empty domain for any variable → backtrack or re-read clues.

Exam parallel: Sudoku is pure constraint satisfaction — same propagation mindset.

Schedule three exams in three slots with constraints

Subjects: Math, Physics, Chemistry. Slots: Mon, Tue, Wed (one each).
(1) Math not Mon. (2) Chemistry before Physics. (3) Physics not Wed.
Step 1 — Order: Chem < Phys in {Mon,Tue,Wed} → Chem Mon, Phys Tue OR Chem Tue, Phys Wed.
Step 2 — Phys not Wed → Chem Mon, Phys Tue, Chem can't be Wed if Phys Tue... Chem Mon, Phys Tue, Math Wed.
Step 3 — Math not Mon ✓ (Math Wed). **Mon=Chem, Tue=Phys, Wed=Math**.

Detect contradiction from constraints

A must be 1 or 2. B must be 1. A ≠ B → A cannot be 1 → A=2. If another clue says A=1, **contradiction** — recheck clue reading.

• Write "Either A in 1 or B in 5" as constraint types.
• What happens if all domains empty for one variable?
• Difference between implication and inequality constraint?
• Stretch: Map seating puzzle clues to constraint table with 5 rows.

Revision tip: Revisit adjacent topics in Advanced Puzzles before mixed practice on Constraint.

Open the Practice tab for graded questions on Constraint.

Translate English clues into symbols on rough paper (A≠B, A→B) before touching the matrix. Apply cardinality constraints (exactly two, at most one) immediately — they shrink domains fastest. When a branch contradicts, mark the branching clue and try the alternate only once. Constraint puzzles reward patience; rushing placements causes cascade errors.

Constraint propagation is the engine behind Sudoku, grouping, and multi-variable grids — one skill, many skins. Scheduling questions encode constraints as precedence (A before B) — draw arrows before filling calendars. In optimisation, constraints define feasible region; objective picks the best point inside. Programming-minded students may note similarity to if-then rules in spreadsheets — same logic, friendlier notation.