Mathematics matters in jigsaw puzzles because every placed piece creates obligations. Neighbouring edges must be compatible, outside borders must face outward, orientations must agree, and local visual matches must combine into one globally consistent arrangement. A piece that looks excellent beside one neighbour can make the remaining puzzle impossible.
For students asking why mathematics is important in problem solving and computing, jigsaws connect geometry, graphs, permutations, constraint propagation, scoring, optimisation and computational complexity. They also teach a careful distinction: proving that a formal family of puzzles is hard is not the same as predicting how long one picture puzzle will take a particular person.
This article uses paper pieces and synthetic compatibility data. It does not claim that physical puzzles reduce perfectly to coloured edges. Shape tolerances, image texture, manufacturing and wear add evidence. For complexity context, Bosboom, Demaine and colleagues show strong hardness results for formal one-row rotating edge-matching and jigsaw variants in Even 1 × n Edge-Matching and Jigsaw Puzzles Are Really Hard. Their theorem supports precise claims about those defined variants, not every casual puzzle-solving situation.
Quick Reading Route
- Count adjacencies before treating every edge as a constraint.
- Represent orientations without double-counting symmetric states.
- Build compatibility scores and keep evidence types separate.
- Propagate constraints before brute-force search.
- Understand complexity claims without exaggeration.
- Use the learning plan to create a transparent toy solver.
Why Mathematics Is Important in Jigsaw Puzzles
A rectangular jigsaw can be modelled as positions, pieces, orientations and neighbour constraints. Each position receives exactly one piece; every piece is used exactly once; compatible edges meet; and boundary edges satisfy the outer-frame rule.
That model is a constraint satisfaction problem. A position is a variable. Its domain contains possible piece–orientation states. Adjacent positions create compatibility constraints. The all-different rule prevents one piece from occupying two positions.
Real solving adds softer evidence. Colour, line continuation, texture, object recognition and image semantics produce degrees of similarity rather than simple yes/no matches. Mathematics can combine these scores, but the weights and feature quality matter.
The puzzle is therefore both discrete and perceptual. Border shape may create hard exclusions; a faint cloud gradient gives probabilistic evidence. Strong solvers know which is which.
A Grid Has Fewer Neighbour Relations Than Piece Edges
For an r-row by c-column rectangular grid, each row has c − 1 horizontal adjacencies, giving r(c − 1). Each column has r − 1 vertical adjacencies, giving (r − 1)c. Total internal neighbour relations are:
r(c − 1) + (r − 1)c = 2rc − r − c.
For a 4×5 puzzle, horizontal adjacencies are 4×4 = 16 and vertical adjacencies are 3×5 = 15, for 31 internal adjacencies. There are 20 pieces and 80 physical piece edges, but internal joins use 62 edge sides because each of 31 joins contains two sides. The remaining 18 edge sides lie on the outside perimeter, matching 2r + 2c = 18.
This reconciliation is a valuable check: 62 internal sides + 18 boundary sides = 80 total sides.
Avoid double-counting a shared border
If piece A touches piece B, that is one adjacency, not two. An adjacency graph has an undirected edge between the two positions. Counting “A touches B” and “B touches A” separately doubles the total.
Directed compatibility may still be useful because A’s right edge must match B’s left edge, not any edge of B. The relationship is one physical join with two directed edge roles.
Corners and Borders Reduce Candidate Positions
In a standard rectangular frame with recognisable flat outer edges, four corner positions require pieces with two adjacent boundary sides. Noncorner boundary positions require one boundary side. Interior positions require none.
For an r×c grid with r,c > 1, the number of boundary positions is 2r + 2c − 4. Of these, four are corners and 2r + 2c − 8 are noncorner edge positions. Interior positions number (r − 2)(c − 2).
For a 4×5 grid, there are four corners, 2×4 + 2×5 − 8 = 10 noncorner boundary positions, and 2×3 = 6 interior positions. The counts sum to 20.
This classification dramatically shrinks domains when flat sides are reliable. Yet a cropped image puzzle, irregular frame or ambiguous manufacturing shape may violate the model. A solver should record the assumption rather than force it.
Orientation follows from boundary geometry
A corner piece may fit four corners under rotation, but once assigned to the top-left corner its two flat edges must face up and left. That often fixes orientation. A noncorner edge piece has one flat side and its orientation is similarly restricted by which boundary it occupies.
Geometry supplies hard information before image analysis begins.
Orientation Multiplies the State Space
A square piece with no symmetry has four rotations: 0°, 90°, 180° and 270°. If n distinct pieces can occupy n positions with four orientations each, a crude upper bound is n! × 4ⁿ states.
For 10 pieces, 10! = 3,628,800 and 4¹⁰ = 1,048,576. Their product is about 3.805×10¹². For 20 pieces, 20! ≈ 2.433×10¹⁸ and 4²⁰ ≈ 1.100×10¹², giving about 2.675×10³⁰ raw states.
These are not counts of genuinely plausible solutions. Boundary types, piece shapes, edge compatibility and image evidence remove most states. The calculation explains why unstructured enumeration is poor, not why a particular puzzle necessarily takes astronomical time.
Symmetry can collapse orientations
A piece with rotationally symmetric appearance and shape may have fewer distinguishable orientation states. A completely uniform square has one visual orientation; a half-turn-symmetric piece has two. Counting all four would duplicate equivalent states.
Symmetry reduction is valid only for the evidence used. A shape might be symmetric while the printed image is not. The state model should include every feature that affects compatibility.
Edge Compatibility Can Be Hard or Scored
In a formal edge-matching puzzle, neighbouring edge colours or labels may have to match exactly. Compatibility is binary: allowed or forbidden. A physical jigsaw adds shape: a tab may pair with a pocket, but two tabs cannot occupy the same join.
Image compatibility is usually scored. Sample colour near each edge, reverse one sequence to align directions, then compute a distance such as mean squared colour difference. Smaller distance suggests a smoother continuation.
If edge profiles are p_i and q_i across k samples, one simplified score is D = (1/k)Σ||p_i − q_{k−1−i}||². The reversal is important because the two facing edges run in opposite local directions.
One score can hide different evidence
Shape, colour gradient, texture and line continuation have different reliability. A combined cost might be C = w_sC_s + w_cC_c + w_tC_t + w_lC_l. The weights express modelling judgement.
If colour dominates, two blue-sky edges may look compatible even when the line continuation fails. If shape dominates, worn pieces may be rejected incorrectly. Report component scores and test sensitivity to weights.
Hard constraints versus soft preferences
A confirmed tab–tab conflict can be a hard exclusion. A slightly imperfect colour match should usually remain a soft penalty. Converting every weak cue into a hard rule risks deleting the true solution.
This distinction appears in scheduling and diagnosis too: some combinations are impossible, while others are merely less likely.
A Compatibility Graph Organises Possibilities
Create a vertex for each oriented piece edge. Draw a directed compatibility edge from state A to state B when A’s right side could meet B’s left side, with a weight representing match quality. Similar graphs can be built for vertical joins.
A complete puzzle seeks a selection of piece–orientation states assigned to positions so that all required graph relationships exist and every piece is used once. Pairwise compatibility is necessary but not sufficient because the same piece cannot satisfy two positions and cycles must close consistently.
For a small puzzle, a compatibility matrix helps. Rows represent right edges, columns represent left edges, and entries are scores or infinity for impossible matches. Sorting each row produces candidate neighbours.
Mutual ranking is useful evidence
If B is A’s best right neighbour and A is B’s best left neighbour, the pair has a mutual-best relationship. It may deserve priority, but it is not proof. Repeated sky texture can create several similar scores, and a globally valid arrangement may require a second-best local pair.
Strong evidence should guide search without freezing it prematurely.
Constraint Propagation Shrinks the Candidate Domains
Represent every position as a domain of allowable piece–orientation states. At first, corner domains contain only corner-compatible states, top-edge domains only states with a boundary side facing up, and so on.
When a state is assigned to one position, remove every orientation of that piece from all other positions. Then remove neighbour states incompatible with its exposed edges. If any neighbouring domain becomes a singleton, that position is forced and propagation continues.
More generally, arc consistency removes a state if it has no supporting compatible state in an adjacent position’s domain. Repeating this process can reveal contradictions before a complete arrangement is attempted.
All-different is a global constraint
Pairwise “these two positions cannot share a piece” rules express uniqueness weakly. An all-different constraint reasons about a whole group. If three positions together have candidate piece IDs {A,B,C}, those pieces are reserved for them and can be removed elsewhere.
This resembles Sudoku subsets. The surface story changes, but the mathematics of domains and exclusive resources transfers.
Worked Example: Four Pieces Around a Cycle
Consider a 2×2 toy puzzle with positions TL, TR, BL and BR. There are four internal joins: TL–TR, BL–BR, TL–BL and TR–BR. Suppose local scores strongly favour A beside B on top, B above C on the right, C beside D on the bottom and D above A on the left.
Those four choices form a consistent cycle if orientations also align and each piece is used once. Now change the final relationship so D’s top edge is incompatible with A’s bottom edge. The first three locally attractive matches cannot all remain; the cycle fails to close.
A greedy solver that commits A–B, then B–C, then C–D reaches a dead end. Backtracking must undo at least one earlier pair and try a lower-ranked alternative.
The example shows why loops are powerful checks. A tree of pairwise relationships can grow without contradiction, while the final closure exposes accumulated inconsistency.
Greedy Placement Can Fall into a Trap
A greedy strategy repeatedly selects the lowest-cost available edge match. It is fast and often useful for generating hypotheses. It can fail because local cost ignores future scarcity and global piece usage.
Suppose one rare edge has only one compatible partner, while a common blue edge has many. If the rare partner is consumed in a slightly better blue match, the rare edge becomes impossible later. A constraint-aware strategy protects scarce resources.
One heuristic selects the position with the smallest remaining domain. Another prioritises the most distinctive exposed edge. A third considers regret: the difference between the best and second-best candidates. Large regret suggests a decision where losing the best option is costly.
No heuristic wins every instance. Compare explored states and solution quality across a test set rather than celebrating one puzzle.
Backtracking Makes Early Choices Reversible
Choose a position, try a candidate state, propagate, and continue. If a domain becomes empty or a join fails, restore the prior domains and try another candidate. This is backtracking.
Correct restoration is essential. Efficient solvers record only the values removed at each decision level, then reinstate them when leaving the branch. Copying the entire state is simpler but can use more memory.
Branch-and-bound adds an objective. If current cost plus an optimistic lower bound already exceeds the best complete solution found, prune the branch. The lower bound must truly be optimistic; an invalid bound can delete the optimum.
Search order changes effort, not correctness
With exhaustive backtracking and correct pruning, variable ordering should not change which solutions exist. It changes how quickly they are found. If two orders produce different feasible sets, the implementation likely has a restoration or propagation bug.
This gives students a valuable testing principle: performance choices should not silently change semantics.
Image Features Add Measurement Uncertainty
Photographed pieces may differ in lighting, scale, perspective and white balance. Pixels near a cardboard edge may include background or shadow. Comparing raw RGB values can therefore punish true neighbours.
Preprocessing might rectify perspective, normalise illumination, separate piece masks and sample just inside the printed region. Each step introduces parameters and possible errors. The solver should retain the original image and transformation metadata.
Texture descriptors or learned embeddings can capture patterns beyond pointwise colour. They may also be less interpretable. A high score from a model is evidence, not an explanation of physical compatibility.
Calibration and validation
Use known neighbouring and non-neighbouring edge pairs to study score distributions. Plot false-match and missed-match rates across thresholds. Do not tune and evaluate on the same pairs if claiming general performance.
This is where puzzle solving meets statistical classification. A threshold trades one error type against another; there is no context-free perfect cutoff.
Complexity Results Need Precise Language
Computational complexity studies families of instances whose size can grow. NP-hardness means, roughly, that a polynomial-time algorithm for every instance of the formal problem would have major consequences in complexity theory. It does not mean every small instance is difficult or that no useful heuristic exists.
Bosboom and co-authors prove hardness results even for formal 1×n rotating edge-matching and jigsaw variants. Ebbesen, Fischer and Witt also analyse edge-matching problems with rotations, showing that complexity changes across settings and parameters.
The exact problem definition matters: board shape, rotations, edge types, blanks, optimisation objective and input encoding. A theorem for labelled square tiles is not automatically a theorem about every photographic jigsaw.
Worst case versus typical case
Complexity results often concern worst-case instances. Commercial puzzles may contain strong image cues designed for humans, making many placements easy. Conversely, repeated textures and nearly identical shapes can create hard practical cases.
A responsible statement is: formal variants are computationally intractable in a precise theoretical sense, while actual solving effort depends on instance structure, evidence and method.
Counting Is a Bound, Not a Runtime Prediction
The raw n!4ⁿ state count is an upper bound before constraints. It does not mean a solver will visit every state. Propagation, symmetry breaking and good heuristics can reduce the explored tree by many orders of magnitude.
Nor does a huge count prove the puzzle has a unique solution. Some pieces may be interchangeable under the available evidence, producing multiple equivalent arrangements. Provenance or a reference image may be needed to identify the intended original.
Logarithms help communicate scale. log₁₀(n!4ⁿ) = log₁₀(n!) + n log₁₀4. Software can compute log-factorials without overflowing, and Stirling’s approximation explains growth for large n.
Symmetry Breaking Avoids Duplicate Search
If rotating the entire completed puzzle gives an equivalent solution and no fixed outer orientation exists, the search may find four rotated copies. Fixing one distinguished corner or orientation can remove this global symmetry.
Likewise, visually identical pieces may be interchangeable. Ordering their assigned positions can select one representative from equivalent permutations. Symmetry-breaking constraints improve efficiency but must not remove genuinely distinct solutions.
Before adding a rule, define the equivalence. Two arrangements that look identical under a simplified colour model may differ in shape or provenance.
Exact-Cover and Integer Models Offer Other Views
A position–piece–orientation assignment can be represented by a binary variable x_{p,k,o}: one when piece k in orientation o occupies position p. Every position needs exactly one state, so Σ_{k,o}x_{p,k,o} = 1. Every physical piece is used exactly once, so Σ_{p,o}x_{p,k,o} = 1.
Neighbour compatibility can forbid pairs of binary variables from both being one. If state a at the left position is incompatible with state b at the right position, add x_a + x_b ≤ 1. A weighted objective can minimise edge mismatch scores.
This is an integer-programming view. A pure exact-cover model handles the exactly-once requirements elegantly, while compatibility may be encoded through extra columns or conflict handling. A SAT model uses Boolean clauses. A CSP model uses domains and adjacency tables.
The puzzle has not changed, but the representation changes which solver tools are natural. Comparing formulations teaches that modelling is a design choice, not mechanical translation.
Model size grows quickly
For n positions, n pieces and four orientations, the assignment layer alone has up to 4n² binary variables. At n = 100, that is 40,000 variables before compatibility encoding. Boundary and symmetry reductions should remove impossible states early.
Pairwise conflict clauses can be numerous. Table constraints or specialised propagators may express compatibility more compactly. Counting model elements helps students understand why a mathematically correct encoding may still be computationally awkward.
Probabilities Need Normalisation and Calibration
Edge scores are not automatically probabilities. A distance of 12 is not “twice as likely” as a distance of 24. To obtain a probability-like distribution over candidates, a model might use a softmax transformation P(j|i) = exp(−C_{ij}/τ) / Σ_k exp(−C_{ik}/τ).
The temperature τ controls sharpness. Small τ concentrates probability on the lowest-cost candidate; large τ spreads it. Changing τ can alter search order without changing raw costs.
These probabilities are conditional on the candidate set and model. If the true neighbour was omitted by a hard filter, the remaining probabilities still sum to one and can look falsely confident.
Calibration experiment
Group predicted joins by reported confidence. Among joins assigned confidence around 0.8, roughly 80% should be correct for a calibrated system on comparable data. Plot observed accuracy against predicted confidence.
If 0.9 predictions are correct only 60% of the time, the model is overconfident. Temperature scaling may improve calibration on held-out data, but it cannot repair missing features or biased samples.
Bayes combines prior and evidence
A boundary position has a prior preference for a border piece. Edge observations supply likelihood. Bayes’ rule combines them: posterior is proportional to likelihood times prior.
This language clarifies why a weak image match may be outweighed by strong geometric impossibility. Priors should reflect genuine structure, not the desired answer.
Assemblies Can Be Built as Verified Components
Human solvers often create small clusters before placing them on the frame. Model a cluster as a connected subgraph with known relative piece positions and orientations. Its exposed boundary edges become the cluster’s interface.
Joining two clusters requires compatible exposed edges, no overlapping coordinates and no repeated pieces. Once joined, their coordinate frames must be aligned consistently. A single rotation or translation maps every piece in one cluster.
This reduces the number of objects but creates larger compound states. A mistaken internal match can contaminate a whole cluster, so confidence and reversibility remain important.
Union-find tracks connectivity, not geometry
A disjoint-set data structure can quickly record which pieces belong to the same cluster. It cannot by itself verify relative coordinates, orientation or loop consistency. Additional transforms and constraints are needed.
This is a useful software lesson: a data structure answers the question it was designed for, not every nearby question.
Loop closure checks a cluster
If a path of relative transforms returns to its starting piece, the combined rotation and translation should be the identity. A nonzero residual signals inconsistent matches or measurement error.
Robotics uses a related idea in pose graphs. A puzzle cluster offers a safe miniature example of local measurements meeting global consistency.
Worked Example: Candidate Domains and Hall’s Condition
Suppose three interior positions P1, P2 and P3 collectively have candidate piece IDs {A,B,C}, while a fourth position P4 has candidates {A,B,C,D}. Because the first three positions must use three distinct pieces drawn from exactly {A,B,C}, those pieces are reserved. P4 must therefore use D, even if D was not its best-scoring local option.
This is an instance of Hall-style reasoning behind the all-different constraint. A set of k variables whose union of candidate values has size k locks those values to that set.
Now suppose P1, P2 and P3 have candidate union only {A,B}. Three positions cannot receive distinct values from two pieces. The model is contradictory before any orientation or edge score is considered.
Global resource reasoning can therefore outperform repeated pairwise checks. It exposes scarcity early, much like a scheduling system notices that three jobs cannot fit into two exclusive slots.
Benchmarking Needs Repeated, Separated Tests
One solved puzzle proves that an implementation can succeed once. It does not establish reliability. Create a collection of synthetic and photographed puzzles with different sizes, repeated textures, border ambiguity and noise.
Separate training, parameter tuning and final testing. If feature weights are adjusted after viewing test failures, the test set has become part of training. Preserve an unseen set or use cross-validation when data is limited.
Report success rate, exact arrangement rate, correct adjacency rate, runtime distribution and number of explored states. A solver may place 95% of adjacencies correctly but fail to produce one complete exact grid.
Timeouts are censored observations
If a run stops after 60 seconds, its true completion time is not 60 seconds; it is at least 60. Averaging timeouts as if completed biases runtime summaries. Report success within the limit and distributions among completed runs.
Hardware, implementation language and parallelism also affect runtime. Algorithm comparisons need controlled environments or carefully normalised claims.
Physical Tolerances Complicate Shape Matching
Real pieces are not ideal polygons. Cutting tools have kerf, cardboard compresses, tabs bend and scans introduce pixel error. Two true neighbours need not have perfectly complementary digitised outlines.
Align edge curves before comparing them. A distance measure may allow small translations, rotations or elastic deviations. Too much flexibility, however, lets unrelated shapes match. The tolerance becomes another model parameter.
Manufacturing patterns can create correlations: several tabs cut by the same tool may resemble one another. Treating edge errors as independent can exaggerate certainty.
Destructive and non-destructive evidence
For ordinary puzzle solving, never alter pieces to improve a fit. In cultural-heritage or forensic contexts, provenance and conservation rules dominate. Mathematics can organise non-destructive measurements, but authority and ethics govern what may be done.
This boundary matters when presenting careers. An algorithmic suggestion is not permission to manipulate an object.
Misconceptions Worth Correcting
“The best edge match must be correct”
It is the best under a chosen score among tested alternatives. Global consistency, other evidence and score uncertainty may overturn it.
“Corner pieces make the puzzle easy”
They reduce candidate positions but do not determine the interior. Repeated borders or irregular frames can also weaken classification.
“NP-hard means impossible”
It describes asymptotic worst-case difficulty for a formal problem. Small or structured instances can be solved effectively.
“More image features always improve accuracy”
Extra features can add noise, redundancy or bias. Validate their contribution and examine sensitivity to weights.
“A complete arrangement proves it is the original”
Multiple arrangements may satisfy the model. Provenance, reference imagery and stronger evidence may be required to identify intent.
Did You Know?
A 4×5 puzzle has 31 internal adjacencies, not 40. The formula separates horizontal and vertical joins and prevents double-counting shared borders.
The hardest-looking part—a giant permutation count—often collapses after a few strong constraints. Mathematics helps by applying structure early rather than searching faster through bad possibilities.
Local similarity and global consistency are different types of evidence. A slightly weaker edge score may belong to the only arrangement in which every loop closes and every piece is used exactly once.
What Students Should Practise
- Derive internal adjacency and perimeter counts for rectangular grids.
- Represent piece–orientation states without double-counting symmetry.
- Build hard and soft compatibility matrices.
- Propagate boundary, neighbour and all-different constraints.
- Create a greedy counterexample and repair it with backtracking.
- Calculate raw search bounds using factorials and logarithms.
- Explain worst-case complexity separately from one puzzle’s difficulty.
- Record intermediate decisions so a failed branch can be reproduced.
A Four-Week Learning Plan
Week 1: geometry and counting
Use a 3×4 paper grid. Count boundary positions, internal joins and edge sides. Rotate labelled pieces and identify equivalent orientation states.
Week 2: compatibility data
Assign synthetic labels or colour sequences to edges. Build horizontal and vertical compatibility tables, including impossible pairs and graded scores.
Week 3: propagation and search
Create position domains, apply boundary rules and all-different removal, then implement backtracking. Compare greedy, random and minimum-domain orders by explored states.
Week 4: evidence audit
Add noisy image scores or ambiguous repeated patterns. Vary feature weights, test solution uniqueness and write a report separating model evidence from claims about an original image.
Guidance for Parents and Teachers
Begin with four to twelve hand-labelled square tiles. A small puzzle lets learners draw every state and understand the algorithm before software hides the logic.
Ask for two explanations after each placement: “Why does this neighbour fit?” and “What does this choice remove elsewhere?” The second question develops global reasoning.
Do not equate speed with mathematical ability. Visual memory, motor skills, experience and image familiarity affect human performance. The learning target is explicit reasoning, not a race.
When introducing complexity theory, keep the quantifiers visible. Say “a defined family of growing instances has this hardness result,” not “jigsaws are mathematically impossible.”
Frequently Asked Questions
How many internal joins does an r×c puzzle have?
r(c−1) horizontal plus (r−1)c vertical, giving 2rc−r−c.
Why multiply by 4ⁿ?
If every square piece has four distinguishable rotations, each of n placed pieces contributes four orientation choices. Symmetry or boundary information can reduce this count.
What is an edge-matching puzzle?
It is a formal placement puzzle in which adjacent edge labels, colours or shapes must be compatible. It abstracts part of a jigsaw’s neighbour condition.
Why can greedy matching fail?
A locally best pair may consume a scarce piece or make a later cycle impossible. Greedy choice does not account for all future constraints.
What does all-different mean?
Every position must use a different physical piece. It is a global resource constraint, not just a set of edge comparisons.
Is a photographic jigsaw exactly an edge-matching problem?
No. It also uses shape, image content, texture, orientation and physical tolerances. Edge matching is a useful abstraction with limits.
Can a puzzle have multiple solutions?
Yes, especially under simplified or ambiguous evidence. Uniqueness must be tested relative to the stated model.
Does NP-hardness tell me how long my 1,000-piece puzzle will take?
No. It is a property of a formal growing problem family, not a personal time forecast. Instance structure and solving method matter greatly.
Useful Next Reading
Compare physical assembly with stained glass, tessellation and panel geometry and compare search structure with Sudoku, constraints and backtracking search. The Mathematics Learning Hub provides broader routes through combinatorics, graphs and algorithms.
For research, read Bosboom et al. on 1×n edge-matching and jigsaw hardness, Ebbesen, Fischer and Witt on rotations, and Kovalsky, Glasner and Basri’s algebraic approach. Each studies a stated formal model; compare definitions before transferring claims.
Conclusion
Jigsaw puzzles answer “why mathematics?” one neighbour at a time. Counting defines the grid. Geometry restricts borders and orientations. Compatibility scores quantify evidence. Constraint propagation removes impossibilities, while backtracking repairs local choices that fail globally.
The puzzle’s delight comes from seeing fragments become a whole. Mathematics explains why that transformation is hard, how structure makes it manageable and why a transparent solver should report both its matches and its uncertainty.
A Practical Investigation Studio
These investigations use paper pieces or synthetic data. Complete two or three for a short project or the sequence as a portfolio. The aim is to expose assumptions, constraints and error signals, not to imitate regulated operations or claim that one classroom model captures every real case.
Investigation 1: Adjacency count
For several rectangular grids, derive horizontal and vertical neighbour counts separately and verify the total by drawing the shared borders. Explain why outside edges are not adjacency constraints.
Investigation 2: Orientation states
List the distinct rotations of square and non-square teaching pieces. Identify cases where visual symmetry makes two nominal rotations equivalent and avoids double-counting.
Investigation 3: Compatibility table
Create synthetic edge labels and build a matrix of pairwise compatibility scores. Keep hard impossibility separate from weak visual similarity so later search decisions remain interpretable.
Investigation 4: Corner classifier
Classify fictional pieces by number and arrangement of flat sides. Compare the reduction in candidate positions with the risk of misclassification from worn or cropped edges.
Investigation 5: Permutation growth
Calculate n! and 4^n n! for small n, then use logarithms for larger n. State that these are raw upper bounds before geometry and evidence remove states.
Investigation 6: Constraint propagation
Place one synthetic corner and remove incompatible states from neighbouring positions. Continue until no domain changes, recording each consequence rather than jumping to a final arrangement.
Investigation 7: Greedy trap
Design a small compatibility table in which the locally best edge match blocks a complete solution. Compare greedy placement with backtracking and explain why reversibility matters.
Investigation 8: Search ordering
Try minimum-domain, strongest-edge and random placement orders on the same toy puzzle. Count explored states and avoid treating one instance as a universal performance proof.
Investigation 9: Image cue fusion
Combine fictional edge-shape, colour and texture scores with declared weights. Vary the weights and observe which placement changes first, revealing sensitivity to modelling choices.
Investigation 10: Loop consistency
Arrange four pieces around a small cycle and show how the final closure constraint can reject pairwise plausible choices. This demonstrates why local matches are not enough.
Investigation 11: Complexity language
Write separate claims for a particular 20-piece puzzle, a formal family of growing instances and a published NP-hard variant. Keep those levels distinct so theory is not overstated.
Investigation 12: Solver record
Store piece IDs, rotations, candidate domains, score definitions, search order, backtracks and stopping rule. A peer should be able to reproduce one branch and understand why it failed.