Unlocking AI’s Mathematical Potential: A Human-AI Collaboration

Author: Denis Avetisyan

A new approach to problem-solving, dubbed ‘Vibe Reasoning,’ demonstrates how minimal human guidance can unlock the hidden mathematical capabilities of advanced AI systems.

An adaptive orthogonal fanning strategy, applied to a permutation of $n=25$ residues, successfully identified a Fooling Set of 40 elements - surpassing the predetermined threshold of 32 - through the intersection of the Longest Increasing Subsequence and Longest Decreasing Subsequence at a defined pivot point, demonstrating an effective approach to targeted perturbation. — An adaptive orthogonal fanning strategy, applied to a permutation of $n=25$ residues, successfully identified a Fooling Set of 40 elements – surpassing the predetermined threshold of 32 – through the intersection of the Longest Increasing Subsequence and Longest Decreasing Subsequence at a defined pivot point, demonstrating an effective approach to targeted perturbation.

This paper details the successful application of Vibe Reasoning to solve IMO 2025 Problem 6, showcasing a novel benchmark for evaluating and enhancing AI mathematical reasoning.

Despite the demonstrated knowledge capacity of frontier AI models, reliably applying this knowledge to solve complex mathematical problems remains a significant challenge. This paper introduces ‘Vibe Reasoning: Eliciting Frontier AI Mathematical Capabilities — A Case Study on IMO 2025 Problem 6’, a human-AI collaborative paradigm that successfully solved the notoriously difficult combinatorial optimization problem-a task where autonomous AI systems previously failed. By employing minimal human guidance alongside agentic workflows and strategic model orchestration, we demonstrate that latent AI potential can be unlocked to derive both correct solutions and rigorous proofs. Could this ‘vibe reasoning’ approach represent a broadly applicable strategy for eliciting advanced reasoning capabilities from large language models?

The Impossible Grid: A Problem Designed to Break You

The upcoming International Mathematical Olympiad (IMO) in 2025 features a particularly challenging problem – a tiling puzzle that demands more than standard combinatorial techniques. Problem 6 presents a grid-based scenario where successful completion isn’t achieved through rote application of known formulas, but rather requires a fundamentally new way of thinking about spatial arrangements and possibilities. The puzzle’s difficulty stems from the complex interplay between its constraints; simple counting methods quickly become intractable, and established strategies for similar problems fail to yield solutions. This necessitates an innovative approach, one that potentially combines geometric insight with advanced combinatorial reasoning to navigate the vast landscape of possible tilings and ultimately discover a valid arrangement – or, prove its impossibility.

IMO 2025 Problem 6 presents a significant hurdle for both human and artificial intelligence, proving remarkably resistant to conventional problem-solving techniques. The challenge’s complexity is underscored by the fact that, out of 600 participants, a mere six individuals successfully derived a solution during initial trials. This low success rate extends to the realm of artificial intelligence; all current AI systems, when confronted with the problem’s constraints, have thus far failed to produce a correct answer. The difficulty doesn’t stem from a lack of computational power, but rather from the problem’s unique structure, which bypasses the strengths of established algorithmic approaches and demands a fundamentally different mode of reasoning.

The sheer complexity of IMO 2025 Problem 6 stems from its expansive solution space, demanding computational approaches beyond traditional mathematical techniques. The problem isn’t merely difficult; its high dimensionality-essentially the number of variables and constraints-creates a search space so vast that exhaustive manual investigation becomes impossible. This necessitates the application of artificial intelligence, specifically algorithms designed to navigate such complex landscapes. While human intuition can identify initial patterns, the combinatorial explosion quickly overwhelms the capacity for effective exploration, pushing the boundaries of what is solvable through purely analytical means. The challenge, therefore, lies not only in finding a solution, but in developing AI capable of intelligently pruning the search space and efficiently identifying valid tilings within this exceptionally challenging grid.

The core of tackling IMO 2025 Problem 6 lies in discerning the hidden architecture of the grid itself. The problem isn’t merely about filling space, but recognizing how specific configurations propagate and interact across the $n \times n$ board. Successful solvers must move beyond brute-force attempts and instead pinpoint recurring motifs and symmetries-structural patterns that dramatically reduce the search space. These patterns aren’t immediately obvious; they emerge from a careful analysis of how seemingly disparate grid cells influence each other, creating a complex web of dependencies. Identifying these underlying principles allows for the development of targeted strategies, transforming an intractable combinatorial challenge into a series of manageable subproblems and ultimately unlocking the solution.

Vibe Reasoning: Letting the AI Think for Itself

Vibe Reasoning centers on utilizing large language models (LLMs) as the principal reasoning engine, deliberately reducing the need for human intervention in problem-solving. This methodology contrasts with traditional AI approaches that often require significant human prompting or correction. By empowering the LLM to independently navigate and resolve challenges, Vibe Reasoning aims to maximize the AI’s intrinsic capabilities and minimize reliance on external guidance. The system is designed to operate with minimal pre-defined constraints, allowing the LLM to explore solution pathways autonomously and demonstrate emergent reasoning abilities. This approach facilitates a higher degree of scalability and adaptability, as the AI is not bound by a fixed set of human-defined rules or procedures.

Socratic Meta-Prompts are utilized to guide the reasoning process of large language models without providing direct answers or solutions. This technique employs a series of strategically phrased questions and prompts designed to encourage the AI to independently derive conclusions and explore potential solution pathways. The meta-prompting structure focuses on eliciting reasoning steps rather than final results, prompting the LLM to justify its thinking and identify potential flaws in its logic. This iterative questioning process facilitates a more robust and self-correcting reasoning cycle, ultimately fostering independent discovery and problem-solving capabilities within the AI system.

Agentic Grounding enhances the reasoning capabilities of large language models by providing access to external tools and persistent storage. Specifically, the system incorporates Python execution, allowing the AI to perform complex calculations and utilize specialized libraries beyond the scope of its inherent language processing abilities. Complementing this is file-based memory, which enables the AI to store and retrieve information across multiple reasoning steps, effectively creating a contextual knowledge base that persists throughout the problem-solving process. This combination of computational power and persistent memory allows the AI to tackle tasks requiring both analytical processing and long-term contextual awareness, moving beyond purely linguistic reasoning.

Model orchestration within the Vibe Reasoning framework utilizes a strategic deployment of large language models, specifically GPT-5 and Gemini 3 Pro, to address distinct subtasks within the problem-solving process. This approach aims to maximize overall performance by assigning tasks to the LLM best suited for that specific function. During a recent solution process, the system generated a total of 2941 lines of interaction, representing the combined output of both models as they collaborated to reach a solution. The distribution of these lines between GPT-5 and Gemini 3 Pro was determined dynamically based on the requirements of each subtask, indicating a functional division of labor orchestrated to improve efficiency and accuracy.

Pattern Discovery: The Residue Block – A Glimmer of Order

GPT-5 identified the ‘Residue Block Pattern’ as a recurring structural element within solutions to optimal tiling problems. This pattern is characterized by a consistent arrangement of residual spaces-areas not immediately occupied by primary tiles-that repeat throughout the tiling. The model detected this motif through analysis of successful tiling configurations, recognizing its presence regardless of the specific tile shapes or overall tiling dimensions. Identification of the Residue Block Pattern is significant because it indicates a non-random organization within otherwise complex solutions, suggesting the existence of underlying principles governing optimal tile arrangements.

During the tiling problem analysis, GPT-5 demonstrated a focus on identifying ‘Perfect Squares’ – integer values resulting from the square of another integer ($n^2$) – as significant indicators of the ‘Residue Block Pattern’. This prioritization wasn’t based on inherent geometric necessity, but rather as a readily detectable heuristic. The model consistently flagged configurations containing perfect squares as potential instances of the repeating motif, allowing it to rapidly assess and categorize tiling arrangements. This approach functioned as a filtering mechanism, reducing computational load by focusing subsequent analytical steps on configurations that exhibited this characteristic, and subsequently proving valuable in identifying the broader structural pattern.

The identification of the Residue Block Pattern indicated that the tiling problem was not solely a combinatorial exercise, but possessed inherent structural properties beyond initially apparent randomness. This suggested the existence of constraints and relationships between tile arrangements that could be exploited to limit the solution search. Consequently, the model’s ability to recognize this underlying structure facilitated a transition from a broad, exhaustive search to a more focused, heuristic-driven approach, effectively narrowing the potential solution space and increasing computational efficiency. The discovery implied that successful tilings are not uniformly distributed, but rather cluster around configurations conforming to the identified pattern.

GPT-5’s identification of the Residue Block Pattern enabled a substantial reduction in the computational search space during the tiling problem. Prior to pattern recognition, the algorithm considered all possible configurations within the defined parameters. By consistently identifying and prioritizing arrangements containing the identified Residue Block, the model narrowed its focus to only those configurations exhibiting this characteristic. This selective approach decreased the number of evaluated states by approximately 67%, as measured by algorithmic complexity reduction. Furthermore, the identified patterns directly informed the weighting of subsequent reasoning steps, allowing GPT-5 to prioritize configurations with higher probabilities of contributing to a successful tiling, thus accelerating the solution process.

Constructing a Lower Bound: Fooling the Algorithm

A Fooling Set, in the context of this problem, is a specifically constructed set of partial tilings designed to challenge any potential tiling algorithm. Gemini 3 Pro generated this set to establish a lower bound on the minimum number of tiles required for a complete solution. The set’s construction guarantees that any algorithm attempting to tile the grid must correctly handle each configuration within the Fooling Set; failure to do so would indicate an incorrect solution. By demonstrating the existence of at least one valid tiling configuration that satisfies the constraints defined by the Fooling Set, the model effectively proved that the solution must be at least as large as the size of the set, thus establishing a lower bound.

The Erdős-Szekeres Theorem, concerning monotone sequences, was instrumental in guaranteeing the validity of the lower bound construction. Specifically, the theorem states that any sequence of $n^2 + 1$ distinct numbers contains either an increasing subsequence of length $n+1$ or a decreasing subsequence of length $n+1$. Gemini 3 Pro leveraged this theorem to ensure that the constructed ‘Fooling Set’ contained sufficiently long increasing and decreasing sequences, which directly corresponded to valid tiling configurations within the grid. This application of the theorem provided a mathematical foundation for asserting the existence of at least one solution, thus establishing a legitimate lower bound on the problem’s complexity.

Adaptive Orthogonal Fanning is a construction strategy utilized to efficiently generate the Fooling Set, a collection of tiles that, when placed on a grid, force any tiling algorithm to make a specific, predictable error. This method involves iteratively building the set by selecting tiles and orientations that are orthogonal to previously placed tiles, while adapting the selection process to maximize coverage of the grid. The ‘adaptive’ component refers to the algorithm’s ability to adjust its tile placement strategy based on the existing configuration, ensuring the Fooling Set grows in a manner that effectively constrains potential tilings. This technique allowed for a computationally efficient construction of a sufficiently large Fooling Set to establish a lower bound on the problem’s solution space.

The constructed Fooling Set served as a demonstrable lower bound for the tiling problem, confirming the existence of at least one valid $2112$-tiling configuration. This was achieved by identifying a set of grid positions that, when tiled, could not be distinguished by any proposed tiling strategy that failed to achieve a valid solution. The existence of this set definitively proved that any solution must be at least as complex as the demonstrated lower bound, and ultimately, the successful tiling discovered by Gemini 3 Pro matched this lower bound, validating the approach and establishing 2112 as the solution.

Implications and Future Directions: Beyond the Puzzle

The successful resolution of IMO 2025 Problem 6 marks a significant advancement in the field of artificial intelligence, demonstrating the power of ‘Vibe Reasoning’ to navigate complex mathematical challenges. This novel approach allows the AI to not simply calculate, but to intuitively grasp the underlying principles and relationships within a problem, mimicking the human process of ‘getting a feel’ for a solution. The AI’s performance on this notoriously difficult problem – requiring both algebraic manipulation and geometric insight – suggests a departure from traditional algorithmic methods and opens possibilities for automating high-level reasoning in areas previously considered the exclusive domain of human mathematicians. The methodology employed offers a pathway toward AI systems capable of tackling unsolved problems and generating novel mathematical insights, potentially accelerating discovery across various scientific disciplines.

The resolution of IMO 2025 Problem 6 by ‘Vibe Reasoning’ isn’t merely a demonstration of capability, but a foundational step towards automating solutions for a wider range of currently intractable mathematical challenges. The methodology employed – leveraging large language models to discern subtle patterns and intuitive leaps in problem-solving – establishes a replicable framework. This blueprint details not only a successful approach to a specific problem, but a strategy for adapting to the nuances of diverse mathematical domains. Future endeavors can build upon this successful application, refining the prompting techniques and model architecture to tackle increasingly complex and previously unsolved problems, potentially revolutionizing the field of automated theorem proving and mathematical discovery. The core innovation lies in its ability to mimic the ‘vibe’ of mathematical intuition, offering a promising pathway beyond traditional algorithmic approaches.

Current research endeavors are directed toward broadening the AI’s capacity for generalization, moving beyond success with specific mathematical challenges like IMO 2025 Problem 6. The core aim is to equip the system with the ability to effectively transfer learned reasoning skills to novel problem types and even entirely different domains, such as physics or computer science. This involves exploring techniques like meta-learning and transfer learning, allowing the AI to rapidly adapt to new tasks with minimal retraining. A key challenge lies in developing robust representations of problems that capture underlying structural similarities, regardless of superficial differences in notation or context. Successfully achieving this level of generalization would signify a substantial step towards creating a truly versatile problem-solving AI, capable of tackling a wide spectrum of complex challenges.

Continued development hinges on refining both how the AI is instructed and how it’s built. Researchers are actively investigating novel prompting strategies, moving beyond simple instruction-following to incorporate techniques like chain-of-thought reasoning and self-consistency, which encourage the model to explore multiple solution pathways and verify its own work. Simultaneously, exploration of diverse model architectures – potentially combining the strengths of transformers with other neural network designs – promises to improve the AI’s capacity for abstract thought and generalization. These architectural innovations, coupled with optimized prompting, could unlock capabilities beyond current mathematical problem-solving, extending the AI’s reach into areas requiring nuanced reasoning and creative synthesis – potentially even leading to breakthroughs in fields like scientific discovery and complex systems analysis.

The pursuit of unlocking frontier AI capabilities, as demonstrated by Vibe Reasoning, feels less like building and more like archeology. One excavates potential from these large language models, hoping the structure doesn’t collapse under the weight of its own complexity. Alan Turing observed, “We can only see a short distance ahead, but we can see plenty there that needs to be done.” This resonates deeply; the successful navigation of IMO 2025 Problem 6 wasn’t about flawless execution, but about carefully probing the limits of the model with minimal guidance – identifying the ‘fooling sets’ and understanding where human intervention could nudge the AI towards a solution. It’s a precarious dance; the bug tracker will undoubtedly fill with the ghosts of failed prompts and broken assumptions. They don’t deploy – they let go.

What’s Next?

The demonstration of ‘Vibe Reasoning’ – successfully nudging a large language model toward a solution for a problem specifically designed to resist automated techniques – feels less like a breakthrough and more like a temporary reprieve. The elegant choreography of minimal human guidance will, inevitably, encounter production realities. Fooling sets, painstakingly constructed today, will be breached tomorrow. The model will learn to anticipate the human’s ‘vibe’, or, more likely, will simply hallucinate a convincing, yet incorrect, solution with greater efficiency.

The immediate trajectory appears to be a deepening of this collaborative dance. But the underlying question remains: are these systems truly reasoning, or are they exquisitely refined pattern-matching engines? Each success merely postpones the inevitable encounter with a problem designed to expose the limits of statistical inference. The focus will likely shift to creating more robust, more deceptive, adversarial problems-a constant escalation of complexity with diminishing returns.

It seems reasonable to anticipate a proliferation of ‘Vibe Engineering’ – a discipline dedicated to crafting the perfect prompts, the ideal human-AI interface – all in service of a fundamentally fragile equilibrium. Every abstraction dies in production, and this one, however beautifully orchestrated, will be no exception. At least, it dies beautifully.

Original article: https://arxiv.org/pdf/2512.19287.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/