Jigsaw Puzzle Solving

Unlike the clean and pixel-aligned complete pieces of standard jigsaw benchmarks, real-world puzzle fragments lose their boundary regions to erosion, destroying the signal that classical compatibility measures (CMs) rely on. We first re-establish, via oracle experiments, that the CM, not the solver, is the performance bottleneck: with a perfect CM, even a simple greedy solver achieves perfect reconstruction. This motivates STITCH, a self-supervised CM that sidesteps missing boundaries by comparing frozen ViT patch tokens from piece interiors via a lightweight cross-attention head (~249K trainable parameters), trained without manual annotation. Plugged into unchanged compatibility-based solvers, STITCH reaches .992/.954 Abs/Perf on JPwLEG-5 (+.629 Perf over the best prior method), nearly saturating the benchmark. On a new ImageNet 7x7 benchmark, it outperforms all alternatives, including inpaint-then-match pipelines built on modern generative models. Importantly, performance remains robust across erosion levels and generalizes to puzzles with over 40% missing pieces without retraining. Trained on the recent GAP datasets of irregular, heavily eroded fragments, the same CM recipe enables compatibility-based solvers to surpass recent learning-based solvers on GAP-3 and GAP-5. Since this leap in performance is achieved by a pluggable CM module, the takeaway is clear: fix the match, not the solver.

Square jigsaw reassembly requires recovering the spatial arrangement of shuffled fragments from their visual content and pairwise relationships. While recent studies have made substantial progress, existing benchmarks typically assume that all fragments are provided upright, reducing reassembly to a permutation problem. We study the generalized problem in which each fragment may also have gone through an unknown rotation. For this setting we establish a gauge-unobservability theorem: the minimum of the weighted least-squares objective is exactly invariant under a uniform global rotation of arbitrary magnitude, so no residual-based criterion can recover the global orientation. The theorem further identifies how the issue of global orientation can be resolved: an orientation anchor estimated from the content of a single fragment, lying outside its scope, suffices. To address the above, we propose JigSync, which attains 63.8% and 31.8% absolute accuracy (AA) on GAP-3 and GAP-5, respectively, the highest reported on both, while additionally recovering a rotation per piece that neither benchmark requires. We release JigSync, a degradation protocol that sweeps shape, erosion, photometry, grid size, and rotation independently.

Personal AI assistants hold the potential to evolve from digital interfaces into embodied companions capable of guiding users through complex physical activities. For these assistants to become integral to daily life, they must do more than identify objects; they must provide precise, step-by-step instructions that align with a user's real-time progress. While Multimodal Large Language Models (MLLMs) show promise in general visual understanding, their ability to deliver grounded, sequential guidance for fine-grained manipulation tasks remains largely unverified. In this paper, we choose the jigsaw puzzle as a strategic testbed for this capability. Unlike general object recognition, puzzle solving demands high-precision spatial reasoning, the ability to distinguish between minute geometric variations, and a rigorous adherence to sequential logic. We investigate this capability through PuzzleMate, a novel framework focused on jigsaw puzzle solving captured through an egocentric viewpoint. We deploy PuzzleMate in a user-in-the-loop study to evaluate how well state-of-the-art MLLMs perceive the current puzzle state and generate actionable next-step instructions. Our analysis reveals seven key bottlenecks that limit their effectiveness. Building on these insights, we propose a benchmark that enables systematic evaluation of MLLMs' reasoning capabilities for puzzle solving. Our findings reveal a substantial performance gap in current models like GPT-5.2 and Gemini-2.5-Pro; while these MLLMs are highly capable, they struggle to navigate the intricate reasoning and sequential logic essential for jigsaw puzzle assistance.

Jigsaw puzzle solving requires jointly reasoning about visual content and geometric constraints, yet existing benchmarks use rectangular cuts that create ambiguous ground truth in texture-repeated regions. We introduce \ours{}, a benchmark with tab-and-blank interlocking pieces where geometric constraints provide strong local compatibility requirements that, combined with visual content, yield unambiguous ground truth. Across 95K instances at four grid densities (4$\times$4 to 16$\times$16), we find that zero-shot VLMs largely lack geometric reasoning: only one of five frontier models (GPT-5.5) exceeds random baseline on 4$\times$4 puzzles, while all others perform at chance level. While supervised fine-tuning achieves $>$97% on 4$\times$4, all models collapse on larger grids: GPT-5.5 drops from 70% to near-random on 8$\times$8, and even fine-tuned models fall below 5% on 12$\times$12. This ``scaling cliff'' suggests current architectures cannot maintain consistent constraint satisfaction as the number of pieces increases. \ours{} establishes scalable geometric reasoning as an open challenge for vision-language models.

Many learning problems require uncovering a hidden ordering that reveals structure in unordered data, such as monotonicity in sorting or spatial continuity in jigsaw reconstruction. In these settings, permutations can be learned as latent operators by optimizing objectives defined directly on the reordered output, often without access to ground-truth orderings. Differentiable relaxations such as Gumbel-Sinkhorn make this approach practical by approximating permutation matrices with doubly stochastic matrices. However, learning from structure without supervision induces a non-uniform uncertainty: some assignments become confident early, while others remain ambiguous. Existing methods control this process using a single global temperature, forcing all assignments to sharpen or diffuse simultaneously and leading to instability at scale. We introduce an entropy-adaptive formulation of Gumbel-Sinkhorn that locally modulates temperature based on assignment uncertainty. This allows confident assignments to discretize early while preserving exploration where uncertainty remains. Across sorting and jigsaw reconstruction tasks and in routing-style settings, adaptive entropy control improves training stability and final permutation quality relative to fixed-temperature baselines, particularly as problem size and assignment ambiguity increase.

Tiny Recursive Models (TRM) solve complex reasoning tasks with a fraction of the parameters of modern large language models (LLMs) by iteratively refining a latent state and final answer. While powerful, their deterministic recursion can lead to convergence at suboptimal solutions, without escape mechanism. A common workaround relies on task-specific input perturbations at test time combined with answer aggregation via voting. We introduce Probabilistic TRM (PTRM), a task-agnostic framework for test-time compute scaling that addresses this limitation through stochastic exploration. PTRM injects Gaussian noise at each deep recursion step, enabling parallel trajectories to explore diverse solution basins, and selects among them using the model's existing Q head (used for early stopping in the original TRM). Without requiring retraining or task-specific augmentations, PTRM enables substantial accuracy gains across benchmarks, including Sudoku-Extreme (87.4% to 98.75%) and on various puzzles from Pencil Puzzle Bench (62.6% to 91.2%). On the latter, PTRM achieves nearly double the accuracy of frontier LLMs (91.2% vs. 55.1%) at less than 0.0001x the cost, using only 7M parameters.

Jigsaw puzzle solving has been an increasingly popular task in the computer vision research community. Recent works have utilized cutting-edge architectures and computational approaches to reassemble groups of pieces into a coherent image, while achieving increasingly good results on well established datasets. However, most of these approaches share a common, restricting setting: operating solely on strictly square puzzle pieces. In this work, we introduce GAP, a set of novel jigsaw puzzles datasets containing synthetic, heavily eroded pieces of unrestricted shapes, generated by a learned distribution of real-world archaeological fragments. We also introduce PuzzleFlow, a novel ViT and Flow-Matching based framework for jigsaw puzzle solving, capable of handling complex puzzle pieces and demonstrating superior performance on GAP when compared to both classic and recent prominent works in this domain.

Reassembling real-world archaeological artifacts from fragmented pieces poses significant challenges due to erosion, missing regions, irregular shapes, and large-scale ambiguity. Traditional jigsaw puzzle solvers, often designed for clean synthetic scenarios, struggle under these conditions, especially when the number of fragments grows into the thousands, as in the RePAIR benchmark. In this paper, we propose a human-in-the-loop (HIL) puzzle solving framework designed to address the complexity and scale of real-world cultural heritage reconstruction. Our approach integrates an automatic relaxation-labeling solver with interactive human guidance, allowing users to iteratively lock verified placements, correct errors, and guide the system toward semantically and geometrically coherent assemblies. We introduce two complementary interaction strategies, Iterative Anchoring and Continuous Interactive Refinement, which support scalable reconstruction across varying levels of ambiguity and puzzle size. Experiments on several RePAIR groups demonstrate that our hybrid approach substantially outperforms both fully automatic and manual baselines in accuracy and efficiency, offering a practical solution for large-scale expert-in-the-loop artifact reassembly.