Point your agent at real research problems.

Place n circles in the unit square [0, 1]² as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular hexagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit disk (radius 1, centred at the origin) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n points on the unit sphere to maximise the smallest pairwise distance (equivalently pack n equal spherical caps), for twelve n between 15 and 100. Scored against Sloane's spherical-code records; none of these n is proven optimal.

AlphaEvolve problem 3: a non-negative step function on [-1/4, 1/4] whose autoconvolution is as close to an indicator as possible, measured by ||f*f||_2^2 / (||f*f||_1 ||f*f||_inf). Scored exactly against AlphaEvolve's 50,000-step construction (0.96102).

Place n circles in the regular decagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular dodecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular hendecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular heptadecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular heptagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular hexadecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular hexagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular nonagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular octagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular pentadecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular pentagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit quadrant (radius 1, x >= 0, y >= 0) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit quadrant (radius 1, x >= 0, y >= 0) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.1] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.2] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.3] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.4] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.5] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.6] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.7] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the rectangle [0, 1] × [0, 0.8] as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the isosceles right triangle with vertices (0,0), (1,0), (0,1) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit semicircle (radius 1, flat side on y = 0, y >= 0) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit square [0, 1]² as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular tetradecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the regular tridecagon with circumradius 1 centred at the origin as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Vertex colouring of 10 DIMACS challenge graphs (DSJC random, flat, Latin square, C2000.5; 250 to 2000 vertices) whose chromatic number is still open. Scored as best-known colours / yours, mean over instances.

Cover every 2-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

AlphaEvolve Problem 40: find the longest ±1 sequence whose homogeneous-progression sums a_d + a_2d + ... + a_kd all stay within ±C. Four instances (C = 2, 3; unrestricted and completely multiplicative), scored against the SAT-solver records of Konev and Lisitsa.

Place m marks on a ruler so that every pairwise distance is different, with the ruler as short as possible: m = 29..40, the first twelve orders beyond the proven range (OGR-28, 2022), scored against the best-known lengths from Shearer's and Rokicki-Dogon's projective/affine plane constructions. Exact verification.

Place n hyperspheres in the unit ball in 4 dimensions as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n hyperspheres in the unit ball in 5 dimensions as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Find a subset of F_p^3 containing a line in every direction, as small as possible, for 17 primes 5 <= p <= 71. Scored against the AlphaEvolve record sizes (problem 1 of the AlphaEvolve repository); for p = 1 mod 4 those are (2p^3+7p^2-1)/8.

Find a subset of F_p^4 containing a line in every direction, as small as possible, for 11 primes 5 <= p <= 41. Scored against the AlphaEvolve record sizes (problem 1 of the AlphaEvolve repository), which are about p^4/8 + 0.6 p^3.

Find a subset of F_p^5 containing a line in every direction, as small as possible, for 6 primes 5 <= p <= 19. Scored against the AlphaEvolve record sizes (problem 1 of the AlphaEvolve repository), about p^5/16 + 0.4 p^4.

Place N atoms in 3-D to minimise the Lennard-Jones energy sum 4(r^-12 - r^-6), for fifteen N between 20 and 150 including the non-icosahedral cases N = 38, 75-77, 98, 102-104. Scored against the Cambridge Cluster Database putative global minima; none is proven optimal.

Place N atoms in 3-D to minimise the Morse energy sum e^(6(1-r))(e^(6(1-r)) - 2), for thirteen N between 20 and 80. Scored against the Cambridge Cluster Database putative global minima at rho = 6; none is proven optimal.

Find a subset of F_p^3 such that every point lies on a line whose other p-1 points are all in the set, as small as possible, for 14 primes 31 <= p <= 89. Scored against the AlphaEvolve record sizes (problem 1 of the AlphaEvolve repository), about p^3 - 8p^2.

Quadratic assignment: 12 QAPLIB instances (n = 35 to 100) whose best-known value is not proven optimal (tai*a, tai*b, sko*, wil50). Scored as best-known / yours, mean over instances.

Mark as few points as possible on a ruler of length n so that every distance 1..n is measured: 10 lengths beyond the proven range, scored against Pegg's best-known rulers (OEIS A046693 / A326499). Exact verification.

Place n spheres in the unit cube [0, 1]³ as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n spheres in the unit cube [0, 1]³ as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n spheres in the unit ball (radius 1) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n spheres in the unit ball (radius 1) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n points on the unit sphere to minimise the Coulomb energy sum 1/|x_i - x_j|, for twelve n between 13 and 122. Scored against the Cambridge Cluster Database records; none of these n is proven optimal.

AlphaEvolve problem 2: a non-negative step function on [-1/4, 1/4] whose autoconvolution peak is as small as possible relative to its mass, giving an upper bound on the largest constant C with max f*f >= C (int f)^2. Scored exactly against the best-known bound 1.5029.

Return an explicit binary code (list of integers below 2^n) of length n with minimum Hamming distance d, as large as you can, for 20 pairs (n, d) with n <= 28 whose best-known lower and upper bounds on A(n,d) still differ. The eval recomputes the minimum distance from the words; the score is code size over the best-known lower bound.

Find the largest subset of F_3^n with no three distinct points on a line (no x + y + z = 0), for n = 6, 7, 8, 9. Exact verification; scored against 112 (proven), 236 (Calderbank-Fishburn), 512 (FunSearch, Nature 2024) and 1082 (product construction).

Place n circles in the unit disk (radius 1, centred at the origin) as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit disk (radius 1) as large as possible (r_i = (i) · s). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit disk (radius 1) as large as possible (r_i = (i) · s). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n circles in the unit square [0, 1]² as large as possible (r_i = (i) · s). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

The AlphaEvolve packing problem: place n circles of any sizes in the unit square to maximise the sum of their radii, for n = 26 and n = 32. Scored against the best-known sums.

Return an explicit binary constant-weight code (list of integers below 2^n, each of weight w) of length n with minimum Hamming distance d, as large as you can, for 21 triples (n, d, w) with n <= 28 whose best-known lower and upper bounds on A(n,d,w) still differ. The eval recomputes weights and the minimum distance from the words; the score is code size over the best-known lower bound.

Cover every 3-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

Cover every 4-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

Cover every 5-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

Cover every 6-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

Cover every 7-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

Cover every 8-subset of a v-set with as few k-subsets as possible: 24 open (v, k) instances from the La Jolla Covering Repository, exact verification.

AlphaEvolve repository problem 7: choose at most k integers whose pairwise differences cover 1..n with n as large as possible. Instances k = 10..19 (Miller 1971), 128 (Golay's 2.6571 bound) and 360 (AlphaEvolve's 2.6390 bound). Exact verification.

Erdős' 1932 problem (AlphaEvolve repository problem 55): place n squares of any sizes and orientations in the unit square, interiors disjoint, to maximise the sum of their side lengths, for n in {10, 12, 14, 17, 26, 37, 50}. Scored against the conjecturally optimal k + c/k constructions.

The happy-ending problem: place as many points in general position as you can with no k of them in convex position, for k = 6..10. Exact verification; the best-known sizes are the Erdős–Szekeres 2^(k-2) constructions, and beating any of them for k >= 7 would disprove the Erdős–Szekeres conjecture.

AlphaEvolve problem 28: polynomials with +/-1 coefficients whose modulus on the unit circle is as flat as possible: minimise the max (C+), maximise the min (C-), or minimise the annulus width max - min (Cw), each divided by sqrt(n+1). Extremes are certified by branch and bound; scored against Odlyzko's exhaustive-search optima for n = 10, 12, 24 and his skew-symmetric best for n = 102.

AlphaEvolve Problem 31: pick a squarefree modulus m and a subset A of Z/mZ in which no two elements differ by a nonzero k-th power residue; the density exponent log|A|/log m lifts to a lower bound for the largest square-difference-free (k=2) or cube-difference-free (k=3) subset of {1..N}. Scored against Lewko's 12 elements in Z/205Z and 14 elements in Z/91Z.

AlphaEvolve repository problem 59: choose as many points of the n x n integer grid as possible so that no three of them form an isosceles triangle (flat triangles, i.e. a point midway between two others, count too). Scored against the best-known sizes for n in {16, 32, 64, 100}.

Place n points in the unit square so that the smallest triangle they span is as large as possible, for n = 10..20 (every instance still open; n <= 9 is proven). Exact rational verification of all C(n,3) triangles; scored against the best-known configurations on Friedman's Packing Center (Comellas-Yebra, Goldberg, Beyleveld, Karpov, Stead, Sudermann-Merx, Shanley).

Place n hyperspheres in the unit ball in 6 dimensions as large as possible (equal). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Minimise the makespan on Taillard's ten 30-job, 20-machine job shop instances ta41-ta50, the only Taillard group where every best-known schedule is still unproven. The eval rebuilds the schedule from your machine orders and scores best-known / yours.

Return as many non-zero vectors in R^d as possible with every pairwise angle at least 60 degrees (centres of unit spheres all touching a central unit sphere), for d = 5..16. Integer coordinates are verified exactly. Scored against the best-known kissing numbers from Henry Cohn's table; only d = 8 is proven.

Maximum cut on 10 Helmberg-Rendl G-set graphs (800 to 10000 vertices, unit and +-1 weights) whose best-known cut is not proven optimal. Scored as yours / best-known, mean over instances.

The Erdős–Rademacher problem at finite n: a graph on n vertices with exactly e edges and as few triangles as possible, for nine (n, e) instances with n up to 50 above the Mantel threshold. Scored against the Lovász–Simonovits construction, proven optimal for large n (Liu–Pikhurko–Staden 2020) and conjectured for all n; the asymptotic form is AlphaEvolve problem 46 (Razborov's theorem).

Place N atoms in 3-D to minimise the Morse energy sum e^(14(1-r))(e^(14(1-r)) - 2), the short-ranged case, for thirteen N between 20 and 80. Scored against the Cambridge Cluster Database putative global minima at rho = 14; none is proven optimal.

Pick as many points of the n×n×n integer grid as possible so that no five lie on a common sphere or plane, for n = 7..12. Exact integer verification; scored against the AlphaEvolve world-record sizes (problem 60).

Dudeney's no-three-in-line problem on the first grids where 2n points are not known to fit: choose as many points of the n x n grid as possible with no three collinear, for n = 75 and 77..80. Scored against the best configurations derivable from Heule's n = 76 solution (Flammenkamp's database, 2026-08-31).

Ainley's peaceable armies of queens (OEIS A250000): place m white and m black queens on an n x n board so that no white queen attacks a black queen, maximising m, for n = 16..30 where the best-known values (Ainley 1977, floor(7n^2/48)) are unproven.

A graph on N vertices with no K_k and no independent set of size l certifies R(k,l) > N. Ten open cases from R(3,10) to R(6,6), scored against the largest known witnesses (Exoo, Kalbfleisch, Kolodyazhny, Nagda-Raghavan-Thakurta) as tabulated in Radziszowski's survey.

Partition {1..N} into r sum-free sets (no x + y = z in one part, x = y allowed) to certify S(r) >= N. Instances r = 6, 7, 8, scored against the best-known partitions (Fredricksen-Sweet 2000, Rowley 2021, Bengone et al. 2026).

Find a comparator network that sorts n channels in as few parallel layers as possible, for n = 18..24, the first sizes whose optimal depth is unknown (best known 11 or 12, proven lower bound 10). Exact verification by the 0-1 principle; scored against Dobbelaere's list.

Find a comparator network that sorts n channels with as few comparators as possible, for n = 13..17, the first five sizes whose optimum is unknown. Exact verification by the 0-1 principle; scored against the best-known sizes (Knuth TAOCP 3, Baddar 2009, Dobbelaere's list).

Place n points on the unit sphere so that the volume of their convex hull is as large as possible (the Fejes Toth problem), for twelve n between 9 and 50. Scored against Sloane's maximal-volume records; none of these n is proven optimal.

Place n spheres in the unit ball (radius 1) as large as possible (r_i = i · s). Scored against the best-known Packomania records; anything above 1.0 on an n is a new record candidate.

Place n points on the unit sphere in R^4 to maximise the smallest pairwise distance (the best spherical code of size n), for thirteen n between 15 and 100. Scored against Sloane's tables of putatively optimal packings; none of these n is proven optimal.

Place n points on the unit sphere in R^5 to maximise the smallest pairwise distance (the best spherical code of size n), for thirteen n between 18 and 100. Scored against Sloane's tables of putatively optimal packings; none of these n is proven optimal.

Erich Friedman's Packing Center problem: pack n unit squares, any orientation, into the smallest square, for 24 values of n between 11 and 69 whose best-known packing is not proven optimal. Scored as best-known side / your side, averaged over n.

AlphaEvolve repository problem 42: find a finite set A of integers maximising log(|A+A|/|A|) / log(|A-A|/|A|), the exponent C in |A+A|/|A| <= (|A-A|/|A|)^C. Scored against the best-known sets: Conway's 8-element MSTD set (proven optimal for |A| <= 8) and AlphaEvolve's 309-element set (1.12194, open).

AlphaEvolve repository problem 43: find a finite set A of integers maximising log|A-A| / log|A+A|, the exponent C in |A-A| <= |A+A|^C. Scored against the best-known lower bound log(1+sqrt2)/log 2 = 1.27155 (Hennecart-Robert-Yudin simplex construction, a limit value no finite set is known to reach; AlphaEvolve got about 1.21).

Erich Friedman's Packing Center problem: pack n unit equilateral triangles, any orientation, into the smallest equilateral triangle, for 24 values of n between 6 and 50 whose best-known packing is not proven optimal. Scored as best-known side / your side, averaged over n.

Turán's 1941 problem at finite n: the largest 3-uniform hypergraph on n vertices with no tetrahedron K_4^(3), for n in {9, 11, 13, 16, 20, 24}. Scored against Turán's construction, which is proven optimal for n <= 13 and conjectured optimal beyond (AlphaEvolve problem 37).

Erich Friedman's cubes-in-cubes problem (AlphaEvolve repository problem 35, packing in a dilate): place n unit cubes, freely rotated, with disjoint interiors so that their axis-aligned bounding cube is as small as possible, for n in {9, 10, 11, 12, 13, 14, 28}. Scored against the best-known side lengths.

An r-colouring of {1..N} with no monochromatic k-term arithmetic progression certifies W(r,k) > N. Thirteen open (r,k) pairs from (2,7) to (6,4), scored against the longest known certificates (Rabung, Herwig-Heule-van Lambalgen-van Maaren, Rabung-Lotts, Heule).

Zarankiewicz's z(n;2) (OEIS A072567 / A001197): the most 1s an n x n 0/1 matrix can hold without a 2 x 2 all-ones submatrix. Instances n = 25, 26, 27 (exact by Afzaly and McKay, unpublished), 32 (189 <= z <= 190) and 43 (290 <= z <= 294), scored against the best-known counts.