PaperlayerView exact revision on arXiv

Source-linked AI summary

Mathematical exploration and discovery at scale

Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao, Adam Zsolt Wagner

arXiv:2511.02864v3cs.NEcs.AImath.CAmath.COmath.MG

TL;DR

The paper asks how AI-guided computational search can discover mathematical constructions across challenging problems. It evaluates AlphaEvolve, an evolutionary system that combines LLM-generated programs with automated evaluation, across a broad mathematical portfolio. The system often matches known bounds, improves several results, and can support generalization and formal proof workflows, but remains limited to problems with suitable optimization structure and can produce only candidate solutions in some settings.

  • Problem

    The paper investigates whether computational tools can autonomously discover and advance mathematical constructions across a broad range of long-standing problems.

  • Method

    AlphaEvolve maintains and evolves a population of programs, using LLM-generated mutations, automated scoring, search heuristics, and, in some cases, generalizer programs.

  • Results

    Across many problems, AlphaEvolve matched or nearly matched existing bounds and improved the state of the art in several cases.

  • Takeaways & Limitations

    The results support using LLM-guided evolutionary search as a scalable complement to mathematical techniques for discovering constructions.

  • Takeaways & Limitations

    AlphaEvolve struggles on problems lacking clearly optimizable smooth score functions, and some outputs remain high-precision solution candidates requiring certification.

Abstract

from arXiv · show

AlphaEvolve (Novikov et al., 2025) is a generic evolutionary coding agent that combines the generative capabilities of LLMs with automated evaluation in an iterative evolutionary framework that proposes, tests, and refines algorithmic solutions to challenging scientific and practical problems. In this paper we showcase AlphaEvolve as a tool for autonomously discovering novel mathematical constructions and advancing our understanding of long-standing open problems. To demonstrate its breadth, we considered a list of 67 problems spanning mathematical analysis, combinatorics, geometry, and number theory. The system rediscovered the best known solutions in most of the cases and discovered improved solutions in several. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. These results demonstrate that large language model-guided evolutionary search can autonomously discover mathematical constructions that complement human intuition, at times matching or even improving the best known results, highlighting the potential for significant new ways of interaction between mathematicians and AI systems. We present AlphaEvolve as a powerful new tool for mathematical discovery, capable of exploring vast search spaces to solve complex optimization problems at scale, often with significantly reduced requirements on preparation and computation time.

1. INTRODUCTION

AlphaEvolve combines LLM-generated programs with evolutionary search and automated evaluation to explore mathematical constructions at scale. Across varied problems, it often matches or improves existing results, while its effectiveness depends on suitable objectives, prompting, and problem structure.

  • Practical findings: The study reports low setup overhead in many cases, with usual preparation taking only up to a few hours, motivating the term “constructive mathematics at scale.”The paper presents this scalability as a contrast with the expected time required for equivalent traditional setup without prior knowledge, information, or code.
  • Method: The system can use search heuristics and generalizer programs to discover fixed-size constructions or infer constructions working across input sizes.Generalizer mode evaluates performance over multiple n values and seeks patterns that extend beyond small cases.
  • Results: Across mathematical problems, AlphaEvolve often matched or nearly matched existing bounds and improved the state of the art in several cases.Some discovered extremizers also had interpretable descriptions, and certain programmatic constructions led to further mathematical work.
  • Results: AlphaEvolve can be combined with Deep Think and AlphaProof to turn discovered constructions into proofs and formalized mathematical results.For the finite field Kakeya problem, Deep Think derived a correctness proof and closed-form size formula, which AlphaProof formalized in Lean.
  • Limitations: The approach is not a general-purpose solver: it is most effective for large portfolios of problems expressible as optimization over constructions and can fail when deeper insight or suitable smooth objectives are absent.The authors also report cases where AlphaEvolve did not match previous best-known constructions.
  • Method: AlphaEvolve evolves programs through iterative population improvement, using automated evaluation to select and refine potential solutions.Its search can evolve improver functions that progressively enhance the best construction found so far.
  • Practical findings: More parallel threads accelerate discovery but increase total LLM-query cost, making resource allocation dependent on whether speed or direct cost is prioritized.Higher-capability models can yield better suggestions, while cheaper models may be more economical and add useful diversity.

1. Finite field Kakeya and Nikodym sets.

AlphaEvolve searched for finite-field Kakeya and Nikodym constructions, finding lower-order improvements in several dimensions while relying on human guidance and manual verification in harder cases.

  • Search setup: AlphaEvolve searched for Kakeya and Nikodym sets across finite fields and dimensions, using generalizer mode to favor constructions working across many prime powers.Constructions were evaluated over many large inputs, with scores averaged across those inputs.
  • Kakeya sets: In dimension 3, AlphaEvolve found a Kakeya construction with C_K(3,p) ≤ 1/4p^3 + 7/8p^2 − 1/8 for primes p ≡ 1 mod 4.This slightly refines the previously best bound 1/4p^3 + 7/8p^2 + O(p).
  • Kakeya sets: In dimensions 4 and 5, the discovered Kakeya constructions matched leading coefficients from prior work while refining lower-order terms.The dimension-4 result was verified manually because the proofs were too difficult for AlphaProof; the dimension-5 proof was also checked by hand.
  • Nikodym sets: For Nikodym sets in fixed dimensions d ≥ 3, a human-simplified construction established C_N(d,p) ≤ p^d−(((d−2)/log 2)+1+o(1))p^(d−1) log p, improving the best known construction.AlphaEvolve’s original construction was not optimal but served as a starting point for human intuition.
  • Expert guidance: For square-order fields in two dimensions, informing AlphaEvolve that a q^2 − q^(3/2) + O(q log q) construction was possible substantially improved its search outcome.Without the hint it found q^2−O(q log q); with the hint it found q^2 − c q^(3/2) for a small constant c > 0.

2. Autocorrelation inequalities.

AlphaEvolve was applied to autocorrelation inequalities by evolving searches over piecewise-constant functions, producing improved bounds while revealing irregular constructions and practical search trade-offs.

  • Search method: AlphaEvolve used standard search mode to evolve heuristic functions for autocorrelation problems, with piecewise-constant functions serving as the main search space.The method takes the best construction so far as input and generates a good function within a fixed time budget.
  • Problem 6.2: 1.5032 reduced the Problem 6.2 upper bound from π/2 = 1.57079 … through numerical search and evolved cubic-backtracking heuristics.The search was initialized with a Newton-type optimization method.
  • Problem 6.2: A meta-analysis using different optimization methods did not improve the Problem 6.2 result, which was associated with highly irregular numerical optimizers.The authors present this as an example of reduced optimization effort.
  • Problem 6.3: 0.961 improved the Problem 6.3 lower bound using a 50,000-part step function after combining AlphaEvolve heuristics with gradient-based methods.An earlier quick experiment reached 0.8962, while independent gradient-based work reached 0.901564.
  • Problem 6.4: The non-negativity relaxation in Problem 6.4 produced improved bounds of C_6.4 ≤ 1.4688 and C′_6.4 ≤ 1.45810.These improved prior bounds of 1.4993 and 1.45810, respectively, as reported in the supplied passage.
  • Problem 6.5: 0.380924 slightly improved the prior Problem 6.5 upper bound of 0.3809268534330870.The constant controls the asymptotics of Erdős’s minimum overlap problem.

3. Difference bases.

The paper applies AlphaEvolve to difference bases and Kakeya needle constructions, improving some known bounds and finding lower-area or higher-score configurations.

  • Difference bases: Difference bases seek the smallest integer set whose pairwise differences cover every integer from 1 to n.
  • Difference bases: Using Singer difference-set generation code, AlphaEvolve improved the upper bound from 2.6571 to 2.6390.
  • Kissing numbers: For kissing numbers, AlphaEvolve raised the lower bound for C6.8(11) from 592 to 593.
  • Kakeya needle problem: The Kakeya experiments optimize union area and scores for triangle and parallelogram configurations across fixed input sizes.
  • Kakeya needle problem: AlphaEvolve found better union-area constructions within 5 to 10 evolution steps, approximately 1 to 2 hours of wall-clock time.
  • Kakeya needle problem: When optimizing the score S_T directly, AlphaEvolve showed improved performance relative to Keich’s construction.

6. Sphere packing and uncertainty principles.

AlphaEvolve is tested on sphere-packing and uncertainty-principle problems using programmatic searches over mathematical test functions and constructions.

  • Uncertainty principles: AlphaEvolve refined a three-Hermite-polynomial configuration, yielding the bound C6.11 ≤ 0.3521.
  • Sphere packing: The sphere-packing problem asks for the strongest possible upper and lower bounds on maximal packing density in each dimension.
  • Sphere packing: The linear-programming quantity C6.13(n) provides upper bounds for sphere packing because C6.12(n) ≤ C6.13(n).
  • Linear programming bounds: AlphaEvolve constructs spherically symmetric test functions as Laguerre-polynomial combinations and optimizes their roots and coefficients under verifier constraints.
  • Uncertainty principles: The resulting estimates were competitive with prior optimization setups but did not produce qualitatively new state-of-the-art results.
  • Classical inequalities: For classical inequalities, AlphaEvolve almost instantly recovered or guessed the correct structure of extremizers in most runs.

8. The Ovals problem.

The section applies AlphaEvolve to the Ovals problem and several polynomial critical-point conjectures, recovering known constructions and identifying improvements in one analysis.

  • Ovals problem: The Ovals problem asks for the strongest possible upper and lower bounds on C6.19 for normalized simple closed convex curves.C6.19 is defined through the least eigenvalue of a Schrödinger operator associated with the curve.
  • Ovals problem: AlphaEvolve usually found the circle within a few iterations, matching the conjectured lower bound of 1 for λ0(γ), but did not find the ovals.The search represented x1, x2, and ϕ numerically with spline interpolation points and evaluated the variational objective.
  • Sendov’s conjecture and variants: AlphaEvolve did not find counterexamples to Schmeisser’s, Borcea’s, or Sendov’s conjectures in the tested searches.For Schmeisser’s conjecture, it searched over root sets and evaluated distances from critical points; for Borcea’s conjecture, it focused on p = 1.
  • Sendov’s conjecture and variants: AlphaEvolve matched Smale’s best known lower bound by finding the optimizer p(z) = z^n − nz, while also producing similar-score constructions.It did not find a counterexample to Smale’s conjecture.
  • Sendov’s conjecture and variants: AlphaEvolve exposed an oversight in the human analysis by identifying improved conjectured inequalities tied to the zero-sum condition, which Tang later established for all n ≥ 4.The observed extremizers included real-rooted polynomials and a polynomial of the form (z − a)^(n−1)(z + (n−1)a).

10. Crouzeix’s conjecture.

The Crouzeix conjecture asks whether the operator norm of p(A) is bounded by twice the maximum of |p(z)| over the numerical range W(A), and AlphaEvolve tested potential improvements.

  • Conjecture: Crouzeix’s conjecture concerns the smallest constant C6.25 satisfying the numerical-range bound for every square matrix A and complex polynomial p.The numerical range is W(A) = {⟨Ax, x⟩ : ‖x‖ ≤ 1}.
  • Background: The literature bound is 2, with sharpness conjectured and only a few special cases solved.Known cases include monomials, matrices with a quadratic minimum polynomial, and matrices whose numerical range is a disk.
  • Method: AlphaEvolve searched over matrix constructions and polynomial choices using the Kippenhahn–Johnson characterization of the numerical range.The search allowed variable matrix sizes and applied specified scaling, shifting, and similarity transformations.
  • Results: AlphaEvolve found no examples exceeding the literature bound of 2 across matrices of variable sizes.The experiment therefore did not improve the known lower-bound example.

11. Sidorenko’s conjecture.

The Sidorenko section defines a graphon inequality for finite bipartite graphs and tests AlphaEvolve on the smallest graph whose Sidorenko property remains unresolved.

  • Problem: For a finite bipartite graph H, C6.26(H) is the least constant in an inequality relating t(H, W) to the edge density t(K2, W) for all graphons W.A graphon is a symmetric measurable function from [0, 1]^2 to [0, 1].
  • Background: The Sidorenko conjecture asserts that every bipartite graph has C6.26(H) = |E(H)|, a property already known for several graph families.These include complete bipartite graphs, even cycles, trees, hypercubes, and graphs with at most four vertices on one side.
  • Experiment: The smallest bipartite graph whose Sidorenko property is unknown is obtained by removing a 10-cycle from K5,5.This graph was selected as the target H for the computational search.
  • Experiment: AlphaEvolve searched for a graphon violating Sidorenko’s inequality and penalized candidates close to constant graphons, but found no counterexample.Constant graphons trivially give equality, motivating the additional penalty.

13. Flat polynomials and Golay’s merit factor conjecture.

The section studies flat ±1-coefficient polynomials through Golay’s merit factor and uses AlphaEvolve to search numerically for improved bounds and constructions.

  • Problem: Golay’s merit-factor problem asks about the behavior of C−6.28(n), with C6.28(n) defined from degree-n polynomials having coefficients ±1.The associated quantities concern flatness on the unit circle and related normalized extrema.
  • Background: Ultraflat Littlewood polynomials remain an open possibility, while Erdős conjectured that they do not exist.The ultraflat question is expressed through the asymptotic behavior of Cw6.28(n).
  • Background: Prior numerical work estimated lim C+6.28(n) ≈ 1.27, lim C−6.28(n) ≈ 0.64, and lim Cw6.28(n) ≈ 0.79.These estimates came from extensive calculations reaching n = 100.
  • Method: AlphaEvolve searched for lower bounds on C−6.28 and upper bounds on C+6.28 by sampling polynomial values on a dense unit-circle mesh.For n ≤ 100, the experiments used mesh sizes balancing evaluator accuracy and evolutionary search speed, followed by validation at larger mesh sizes.
  • Results: For small n up to 40, AlphaEvolve’s constructions appeared comparable in magnitude to some prior results.The constructions and comparisons were summarized in Figure 16.

14. Blocks Stacking.

AlphaEvolve generalized the block-stacking problem from finite cases to the correct solution form after one or two iterations, yielding a harmonic-number construction.

  • The underlying task maximizes the horizontal displacement of the nth unit-width block while preserving stack stability.The formal definition uses nondecreasing displacements subject to stability constraints.
  • AlphaEvolve produced the correct general solution form after one or two generalizer-mode iterations.The system generated an argument for the recursive relation underlying the construction.
  • The recursive construction sets x_i to the average of later offsets minus 0.5, working backward from x_{n-1}=0.With P = positions[n-1] and x_k = positions[k] - P, the recurrence is x_i = (sum_{j=i+1}^{n-1} x_j)/(n-i-1) - 0.5.
  • The construction selects P just below its upper-bound target, using a small tolerance adjustment to satisfy the strict stability inequality.The chosen adjustment is δ = 2·FLOAT_TOLERANCE.
  • The refined program expresses positions[k] using harmonic numbers and a tolerance correction while satisfying the scoring constraints.The reported form is positions[k] = 0.5·(H_n − H_{n-k-1}) − 2·FLOAT_TOLERANCE.

15. The arithmetic Kakeya conjecture.

For the arithmetic Kakeya conjecture, AlphaEvolve searched explicit discrete distributions and improved a bound for one slope configuration, while inspiring a rigorous asymptotic result for another.

  • The arithmetic Kakeya conjecture asserts that the constants C6.30({r_1, …, r_k}; r_∞) can be made arbitrarily close to 1.These constants are defined through entropy inequalities for discrete random variables and their projections.
  • 1.668 ≤ C6.30({0, 1, 2, ∞}; −1) was obtained by AlphaEvolve, improving the corresponding lower bound.The system generated joint distributions resembling discrete Gaussians while solving many slope sets simultaneously.
  • AlphaEvolve was asked to produce parametrized code handling hundreds of slope sets simultaneously to probe the general solution.The resulting distributions provided patterns that informed subsequent mathematical analysis.
  • The AlphaEvolve-inspired lower bound led the authors to establish rigorously an asymptotic for C6.30({0, 1, ∞}; s) for rational s ≠ 0, 1, ∞.The supplied passage also states bounds for coprime numerator and denominator parameters.
  • For the related Furstenberg–Sárközy problem, AlphaEvolve reproduced known lower bounds for square- and cube-difference avoidance but found no improvements.It used the same moduli, 205 and 91, as the known constructions.
  • For spherical designs, the system optimized a nonnegative error that equals zero exactly when the points form a t-design.The optimization therefore targets finite point configurations satisfying the design conditions.
  • AlphaEvolve found improved spherical-design constructions for C6.32(2, 19) and C6.32(2, 21), while also producing new sizes for C6.32(2, 15).The reported sizes were 198, 200, 202, 204; 234, 236; and 122, 124, 126, 128, 130, respectively.

18. The Thomson and Tammes problems.

AlphaEvolve generated competitive spherical configurations for the Thomson and Tammes problems and improved selected packing bounds, including a cube-packing result for n=11.

  • Thomson problem: The Thomson problem seeks minimum-energy configurations of N electrons on the unit sphere.The paper restricts attention to the classical Coulomb potential for comparison with prior work.
  • Thomson problem: For Thomson configurations with N < 50, AlphaEvolve matched reported results to 10^-8 within the first hour.Larger values of N required O(10) hours, and the search evaluated point positions using the Coulomb-energy functional.
  • Generalization: AlphaEvolve evaluated generalized Thomson programs on N values up to 500 by averaging energies across a prescribed set of inputs.The evolved programs were often based on heuristics from small configurations.
  • Tammes problem: For the Tammes problem, AlphaEvolve matched theoretically known best configurations for small N using its energy-based search mode.The problem can also be interpreted as packing N disks on the unit sphere.
  • Packing problems: The packing search penalized intersecting copies, used binary search for the bounding scale, and assumed fixed orientation.The score combined the scale with an intersection penalty.
  • Packing problems: For hexagonal containers, AlphaEvolve improved the best results for n=11 and n=12, although the final construction was not completely symmetric.The constructions were subsequently analyzed and refined by Johann Schellhorn.
  • Packing problems: 2.894531 improved the cube-packing upper bound for C6.35(11, P), replacing 2 + 3/5 ≈ 2.912096.The new optimum was identified for packing 11 copies of the cube P = [0,1]^3.

20. The Turán number of the tetrahedron.

Across several combinatorial and number-theoretic problems, AlphaEvolve usually reproduced known constructions and sometimes improved lower bounds, while its performance depended on guidance and available computation.

  • Turán hypergraph problem: The Turán hypergraph problem asks for the asymptotic maximum edge density of a 3-uniform hypergraph avoiding the tetrahedron K_4^(3).The paper denotes the target density by C6.37.
  • Turán hypergraph problem: AlphaEvolve quickly discovered the known 5/9 construction for the Turán problem but found no better construction across the tested values of n.The experiments used n = 10, 15, 20, 25, 30.
  • Factoring factorials: For factorial factorizations, AlphaEvolve improved several lower bounds over 80 ≤ N ≤ 600 and matched the benchmark in the remaining tested cases.The setup compared vanilla search with access to Sutherland’s code as a black box or modifiable component.
  • Factoring factorials: Later integer-programming results certified exact values, and AlphaEvolve’s constructions often approached those optima.The integer-programming comparisons were reported in Table 6.
  • Beat the average game: AlphaEvolve obtained the lower bound C6.39 ≥ 0.389 for the Beat the average game within a few hours using a straightforward setup without expert hints.The prior reported lower bound was 0.367, while the contemporaneous interval was 0.400695 ≤ C6.39 ≤ 0.417.
  • Erdős discrepancy problem: For the Erdős discrepancy problem, human guidance increased the discovered sequence length from 200 to 380, still below the optimal value of 1160.The guidance suggested trying a multiplicative or approximately multiplicative function.
  • Sums and differences problems: A longer run produced a further lower bound of 1.1584 using a related set U_2 of 54265 integers.An earlier set U_1 of 2003 integers gave the bound 1.1479.

26. Sum-product problems.

AlphaEvolve was tested on sum-product and graph-density problems, finding constructions that sometimes improved finite-size bounds while exposing pitfalls in asymptotic interpretation and nonconvex scoring.

  • Sum-product problems: AlphaEvolve searched sum-product constructions over finite fields by evaluating programs across multiple primes to encourage generalization.The score averaged normalized log-size quantities across primes.
  • Sum-product problems: For primes near 10^9, AlphaEvolve found examples with max(|A + A|, |A · A|) slightly below N^3/2.Deep Think attributed the construction to Gaussian-integer sets projected to finite fields.
  • Sum-product problems: Finite-size improvements can be misleading when asymptotic convergence is very slow, especially with double-logarithmic corrections.The paper cautions that Gaussian-integer features may matter at small sizes but become negligible asymptotically.
  • Triangle density in graphs: AlphaEvolve represented graph-density solutions as 100 lists of real numbers defining rank-1 step-function graphons, evaluated across values of ρ.The setup withheld the exact target problem to make discovering the structure more challenging.
  • Triangle density in graphs: A naive straight-line area score produced an invalid solution because the feasible region was nonconvex.Replacing straight segments with Lipschitz-type bounds led to convergence to the correct solution.

28. Matrix multiplications and AM-GM inequalities.

The paper applies AlphaEvolve to matrix norm inequalities and geometric extremal problems, recovering known bounds and improving several finite configurations.

  • Matrix multiplications and AM-GM inequalities: Matrix multiplication experiments used Schatten and Ky Fan norms, finding further constructions attaining the upper bound C6.47(n,m,d) ≤ 1 but no counterexample to Duchi’s conjecture.The conjecture concerns positive-semidefinite matrices and unitarily invariant norms.
  • Heilbronn problems: For an equilateral triangle of unit area, AlphaEvolve improved the reported construction for n = 11.The unit-area equilateral-triangle variant differs from the broader arbitrary-triangle formulation.
  • Heilbronn problems: AlphaEvolve matched numerical candidates for the fixed-bounding-box Heilbronn problem through n ≤ 12 and improved constructions for n = 13 and n = 14.The reported improvements concern general convex bounding regions.
  • Max-to-min ratios: For max-to-min distance ratios, AlphaEvolve matched best-known results for most tested parameters and improved C6.50(3,4.165849767) in a few-hour experiment.The search optimized several correlated dimension-and-point-count pairs simultaneously.

31. Erdős–Gyárfás conjecture.

This group covers graph, number-theoretic, geometric, packing, incidence, and cylinder problems, illustrating both successful construction search and limitations outside smooth optimization settings.

  • Erdős–Gyárfás conjecture: The Erdős–Gyárfás question asks whether every finite graph of minimum degree at least 3 contains an even cycle of length at least 4.The question remains open, though stronger minimum-degree assumptions imply many even cycle lengths.
  • Erdős squarefree problem: AlphaEvolve easily found the known residue-class construction for the Erdős squarefree problem but did not find a better one.A later result established that the lower bound is sharp for all sufficiently large N.
  • Touching cylinders: For touching cylinders, AlphaEvolve reached loss O(10^-23) for seven unit cylinders in two hours and found the optimal nine-cylinder configuration when radii differed.The loss is zero exactly when all required tangencies are achieved.

38. Erdős–Szekeres Happy Ending problem.

The paper extends AlphaEvolve experiments to classical geometric, routing, and sofa problems, recovering known constructions and improving several finite lower bounds.

  • Erdős–Szekeres Happy Ending problem: For the Happy Ending problem, AlphaEvolve searched configurations of 2^k−2 + 1 points without convex k-gons for k ≤ 8.The loss counted convex k-gons while penalizing near-collisions and nearly collinear triples.
  • Three-dimensional configurations: AlphaEvolve improved known lower bounds for three-dimensional grid configurations, obtaining C6.60(7) ≥ 21 through C6.60(12) ≥ 33.The constructions avoid five points lying on a sphere or plane.
  • Ring Loading Problem: For the Ring Loading Problem discrepancy, AlphaEvolve found a 15-number construction proving 1.119 ≤ C6.61.The score was computed by checking all 2^m assignments for given input sequences.
  • Moving sofa problems: AlphaEvolve recovered the Gerver and Romik sofa constructions within 0.02% and 1.5%, respectively, when experiments stopped.For the 3D sofa, it produced a rigorous lower bound of 1.81 and an estimated volume of 1.84.
  • Overall findings: The experiments show that AlphaEvolve can match or improve finite constructions across diverse problems, but its success depends strongly on the formulation of the search and score.The paper contrasts successful optimization-based tasks with cases where the system failed to attain optimal or near-optimal results.

43. International Mathematical Olympiad (IMO) 2025: Problem 6.

The paper presents AlphaEvolve’s constructions for IMO 2025 Problem 6.65 and extends the discussion to experiments where evolved programs call language models. These experiments show both successful strategy discovery and important reliability limitations.

  • Problem 6.65: Problem 6.65 asks for the minimum number of non-overlapping rectangular tiles leaving exactly one uncovered unit square in every row and column of an n×n grid.
  • Problem 6.65: The constructive search exceeded current tools such as Deep Think, but AlphaEvolve could not prove that its construction was optimal.
  • Problem 6.65: AlphaEvolve found the optimal solution for every evaluated perfect-square n, although its performance was inconsistent on other integer values.The system was instructed to find a construction valid for all n but evaluated only on perfect squares.
  • Function guessing game: In the function-guessing game, AlphaEvolve evolved programs that queried an oracle with simple questions and used a final language-model call to infer hidden functions.
  • Smullyan-type logic puzzles: In the logic-puzzle experiment, AlphaEvolve found a strategy guaranteeing $110, then discovered prompt-injection-style formulations after cheap guard models mishandled convoluted truth-forcing questions.The strategy initially averaged $83 because the guard language models could not reliably execute the logical questions.
Loading 2511.02864v3…