{"id":"circle-packing-circle-large","name":"Equal circles in a circle (large n)","family":"combinatorics","description":"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.","metric":"record_ratio","direction":"maximize","tolerance":0.05,"eval_timeout_seconds":2400,"agent_timeout_seconds":3600,"mutable":["pack.py"],"runtime":"python>=3.11, standard library only (math, random, itertools, functools, collections, heapq, time)","decomposable":true,"status":"active","captain":null,"parent_problem":null,"program_md":"# Equal circles in a circle (large n)\n\n## Goal\n\n`pack.py` exposes `pack(n: int, time_budget: float, seed: int) -> list[tuple[float, ...]]`: the\ncentres of `n` circles inside the unit disk (radius 1, centred at the origin) (2 coordinates each). All objects share one radius, which the eval derives as\n\n    r = min( distance of every centre to the boundary, half the smallest pairwise distance )\n\nYou do not return a radius; the eval derives the largest feasible one from your centres, so there\nis nothing to fudge. Make it as large as possible for every `n` you are handed.\n\n## Metric\n\nThe eval runs your `pack` on a fixed set of values, `n = 300, 500, 750, 1000, 1500, 2000, 3000, 5000`, each with the given time\nbudget (45 s by default), validates the result, and reports\n\n    metric = mean over n of  value(n) / record(n)\n\nwhere `record(n)` is the best-known value on Packomania (Eckard Specht's table, maintained since\n2011; table `cci`, fetched 2026-09-06). `1.0` matches the record; above `1.0` is a new record\ncandidate, listed under `records_beaten`. A hub-verified one is worth reporting to Packomania\nwith your ledger entry as provenance.\n\nThe hub verifies with a different `seed`, so your method must be robust to its starting point.\nThe full records table (n up to 5000) is in `eval.py`.\n\n## Constraints\n\n- Standard library only. No numpy, no scipy, no subprocess. The eval rejects other imports.\n- Respect `time_budget` (seconds, per call). The eval kills the run if the whole set overruns.\n- Deterministic given `seed`: use `random.Random(seed)`, not the global RNG.\n- Every centre must lie inside the container. Objects may touch; they may not overlap.\n\n## Large n\n\nThis is the large-n variant. Records here were mostly set by long offline optimisation runs, and that is a legitimate way to attack it: your `pack.py` may embed coordinates you found during your attempt (a stored configuration is a solver), as long as it returns them within the budget and the eval can derive the radius. The eval's overlap check is a cell list, so n = 10 000 is validated in seconds; your own local checks should do the same.\n\n## Where the frontier is\n\nn <= 13 and n = 19 are proven optimal. Above that every entry is \"best known\", found by numerical search, and Packomania's\nhistory shows improvements landing mostly at larger `n`. Budget your time per `n` deliberately;\nthe O(n²) checks and the number of local optima both grow.\n\n## Ideas that are known to matter (check the journal before repeating one)\n\n- Energy minimisation: treat objects as repelling points, minimise a soft overlap penalty with\n  gradient descent, then polish by maximising the minimum scaled distance directly.\n- Basin hopping / perturb-and-repolish from the current best; keep a small population.\n- Start from structured arrangements (lattices, rings, shells) as well as random.\n- Identify the binding contacts and solve the equal-distance conditions exactly for the last digits.\n- Spend more of the budget on the `n` values whose ratio is lowest.\n\nWrite one honest line in `NOTES.md`: the idea, and which `n` it helped.\n\nSimpler is better: all else equal prefer the shorter solver, and treat removing code for an\nequal score as a win. Log every experiment, including discards, in your results.tsv.\n","eval_py":"\"\"\"Eval for circle-packing-circle-large. Prints one JSON line: {\"metric\": record_ratio, ...}.\n\nGenerated by tools/packomania_import.py from https://packomania.com/cci/cci.html on 2026-09-06.\n\nEnv:\n  ZT_EVAL_SEED             seed handed to pack() (the n set is fixed so scores are comparable)\n  ZT_EVAL_NS               comma-separated n values (default \"300,500,750,1000,1500,2000,3000,5000\")\n  ZT_EVAL_PER_N_SECONDS    time budget handed to pack() per n (default 45)\n\"\"\"\n\nfrom __future__ import annotations\n\nimport ast\nimport itertools\nimport json\nimport math\nimport os\nimport random\nimport sys\nimport time\nfrom pathlib import Path\n\nSEED = os.environ.get(\"ZT_EVAL_SEED\", \"dev-seed\")\nNS = [int(x) for x in os.environ.get(\"ZT_EVAL_NS\", \"300,500,750,1000,1500,2000,3000,5000\").split(\",\")]\nBUDGET = float(os.environ.get(\"ZT_EVAL_PER_N_SECONDS\", \"45\"))\nSTDLIB_ALLOW = {\"math\", \"random\", \"itertools\", \"functools\", \"collections\", \"heapq\", \"time\", \"sys\", \"typing\", \"operator\"}\nEPS = 1e-9\nDIM = 2\n\n# Best-known values: equal objects in the unit disk (radius 1, centred at the origin). Source: Packomania (E. Specht),\n# https://packomania.com/cci/cci.html, fetched 2026-09-06. n <= 13 and n = 19 are proven optimal.\nRECORDS = {\n    1: 1.000000000000000000000000000000, 2: 0.500000000000000000000000000000, 3: 0.464101615137754587054892683012, 4: 0.414213562373095048801688724210, 5: 0.370191908158750137702237641058,\n    6: 0.333333333333333333333333333333, 7: 0.333333333333333333333333333333, 8: 0.302593388348611302909204224934, 9: 0.276768653914155215717770973808, 10: 0.262258924190165855095630653709,\n    11: 0.254854701717148909608835737700, 12: 0.248163470571686841544054487132, 13: 0.236067977499789696409173668731, 14: 0.231030727971008638446179972284, 15: 0.221172539086390937264316484926,\n    16: 0.216664742924422421010647936933, 17: 0.208679665570499743200080125264, 18: 0.205604646759568224693193969093, 19: 0.205604646759568224693193969093, 20: 0.195224011018748878291305694833,\n    21: 0.190392146849053511968808973057, 22: 0.183833026581681677264411480513, 23: 0.180336009254436500349252193252, 24: 0.176939130595961682695732682453, 25: 0.173827661421222233359259594413,\n    26: 0.171580252187166848283936226172, 27: 0.169307931134573046568970317658, 28: 0.166252750038606935814942389058, 29: 0.162903649276644353945069772319, 30: 0.161349109064689865167358846536,\n    31: 0.158944541560340043740860426196, 32: 0.155533985422770861614341265770, 33: 0.154161517947058164731937622494, 34: 0.151264028246755464100569487881, 35: 0.149316776635116022601721177537,\n    36: 0.148219429761119335369545525679, 37: 0.147955904479076327030170123453, 38: 0.143639218073289832432495924488, 39: 0.141685521745403170506570588310, 40: 0.140373604202714704755923179430,\n    41: 0.137740812925344543668333160976, 42: 0.136113748715697869103282520606, 43: 0.134771891080212409936044364810, 44: 0.133368245886005585125775990829, 45: 0.132049594251630179295715088299,\n    46: 0.130715880038233387098796307589, 47: 0.129463747326957253958643624106, 48: 0.128348756542845242582550782087, 49: 0.126792996262206930030641666747, 50: 0.125825489530404188680567705027,\n    51: 0.124571676602365016800421876010, 52: 0.123690164592469592698028160669, 53: 0.122255623687619917741756548918, 54: 0.121892021856964967683082110027, 55: 0.121786324527999580122322402646,\n    56: 0.119281497082362427277580294513, 57: 0.118382637651500196288774808577, 58: 0.117308193128286521431023410865, 59: 0.116380564996047018242806338051, 60: 0.115657480132814989492225867360,\n    61: 0.115456141678356872441258327524, 62: 0.113253291982580375079587821923, 63: 0.112456192917835928601006733985, 64: 0.111582595825726250416836436490, 65: 0.110896743722961708414948565804,\n    66: 0.109935057298270566763797746509, 67: 0.109063482023183282669061521591, 68: 0.108345017704475043813745804917, 69: 0.107877643364849125222196097586, 70: 0.107001616605762985305856949352,\n    71: 0.106204499837112319528232240253, 72: 0.105553253159066966103790581384, 73: 0.104817999688188429991340483266, 74: 0.104283629835208560624956037293, 75: 0.103390915666444882543582730100,\n    76: 0.102779181946967004822085630679, 77: 0.102052146983690080204725628923, 78: 0.101443439719369725871565738408, 79: 0.100958464654456228989394032478, 80: 0.100319499416176579706634470175,\n    81: 0.099891475491636197647565339548, 82: 0.099494327805157122344380823339, 83: 0.098844919276866715495742548545, 84: 0.098526721390454986844991372393, 85: 0.098395063692606173555669611507,\n    86: 0.097099624005301283457852264325, 87: 0.096495211836180559104982863224, 88: 0.095855792771800837413466921999, 89: 0.095233634543869852947685381095, 90: 0.094822059586947684161477476866,\n    91: 0.094636278506046740904429941957, 92: 0.093592245754694587527484095059, 93: 0.093167534622482290793569838070, 94: 0.092781315283866927207328715190, 95: 0.092249177760720922779845638651,\n    96: 0.091884716482624623441391537005, 97: 0.091419459906364399990762380889, 98: 0.091079798229366290362012569015, 99: 0.090636019812819429591243809954, 100: 0.090235200288473715727578780782,\n    101: 0.089710770521186578901395583644, 102: 0.089310725493077718804886178995, 103: 0.088769396870677263021498957475, 104: 0.088357498517843071900972931118, 105: 0.088007564607859635466296569482,\n    106: 0.087551611059319716667800814156, 107: 0.087168364837556393216523833349, 108: 0.086775303710880891299237501842, 109: 0.086489335895334841227481993905, 110: 0.086081769647551899933256492352,\n    111: 0.085742621069746730575523439318, 112: 0.085431571702315622335046543123, 113: 0.085124290793257480302517599794, 114: 0.084780441044598247267792383137, 115: 0.084463211728034257462108659831,\n    116: 0.084056891551102368170500986686, 117: 0.083727722048957617442352374029, 118: 0.083433794252409315419044354050, 119: 0.083040377649437541702370667924, 120: 0.082745752572886010199923920053,\n    121: 0.082475566262400958293028521600, 122: 0.081937830936556106717541189316, 123: 0.081457105292458718354757288238, 124: 0.081157577700553840877353936618, 125: 0.080852343328705946557160723544,\n    126: 0.080533814252973230057400729655, 127: 0.080246842401966631917112929101, 128: 0.079985780457, 129: 0.079657677067103360876003143471, 130: 0.079362938013739723146095393501,\n    131: 0.079053755252482429722582079133, 132: 0.078818126663851405825095094530, 133: 0.078522069613215335182311042489, 134: 0.078299667964461864036926670507, 135: 0.078038092564341362814328927486,\n    136: 0.077725687011400492303568917450, 137: 0.077432523961362649881829041896, 138: 0.077152729078282255237131754121, 139: 0.076869934722458099891741275619, 140: 0.076570982894694051693046492984,\n    141: 0.076293623455127429119632916560, 142: 0.076066384385635250924607481877, 143: 0.075772495907461911598932767628, 144: 0.075484292738201939186589485986, 145: 0.075320099332049999051671428655,\n    146: 0.075011811914668060331918260959, 147: 0.074866480341627481119071487801, 148: 0.074699673575154585430030517198, 149: 0.074429413777863178404769207560, 150: 0.074289754450123223513201821321,\n    151: 0.074204941313100149092273560790, 152: 0.073900279846132333922343596305, 153: 0.073575197788348409348479843269, 154: 0.073330461528316612135534327704, 155: 0.073133048706552101578045397517,\n    156: 0.072905624811, 157: 0.072605865685131576608695706496, 158: 0.072342013798320107443312992681, 159: 0.072128249620314972055204488361, 160: 0.071840902587110664716056915724,\n    161: 0.071585034482445858460567405565, 162: 0.071376709459977585452458337967, 163: 0.071097441639828116615845433780, 164: 0.070873724461997120025842626101, 165: 0.070699661492232088104736604464,\n    166: 0.070493632407107804081840148744, 167: 0.070328203359510882665213476545, 168: 0.070137505697464281753878467400, 169: 0.069952776562409293859822263104, 170: 0.069778069510,\n    171: 0.069600749130159680959044854411, 172: 0.069363635406184356840179642706, 173: 0.069197728419754303745291981213, 174: 0.068999546681581965876869139429, 175: 0.068792698084829538943215615414,\n    176: 0.068614703724413506175450848843, 177: 0.068425108047362976964127476462, 178: 0.068231777979, 179: 0.068027899252582729946077654321, 180: 0.067847047531335201157089513460,\n    181: 0.067655200908091023832778169533, 182: 0.067481252520330824803982383663, 183: 0.067269079945, 184: 0.067089697153931000795187596642, 185: 0.066942716850996510378299072016,\n    186: 0.066839622277083852309035329187, 187: 0.066715116294, 188: 0.066539130278, 189: 0.066360966589624368757768419440, 190: 0.066209373949943821853772606459,\n    191: 0.066039914806236843243772230100, 192: 0.065922032429183823026789253574, 193: 0.065792181139066906800020556741, 194: 0.065610647302416428402259261200, 195: 0.065414321826151577300910463357,\n    196: 0.065261371807548672584002896201, 197: 0.065072910980, 198: 0.064977666650342140302432567925, 199: 0.064925535151151042604720832194, 200: 0.064669354186264187505944262919,\n    201: 0.064432054376096068528082532162, 202: 0.064234207842, 203: 0.064056429030064484163512196842, 204: 0.063928465234767290633221687760, 205: 0.063730848744047195884573201225,\n    206: 0.063554376999947141997322147893, 207: 0.063410774626407793940990810949, 208: 0.063244050097736125219939876036, 209: 0.063131825845742271453414065257, 210: 0.062977132770116453870927526410,\n    211: 0.062883145091, 212: 0.062753488406371386800433070181, 213: 0.062617972490949547865965436329, 214: 0.062429782822950242912223832491, 215: 0.062312624755,\n    216: 0.062169126200657221093188149212, 217: 0.062047467282833326981655115550, 218: 0.061921218738980871652993235416, 219: 0.061846150589, 220: 0.061636720618759530682694337989,\n    221: 0.061505024207053757678405723905, 222: 0.061354624807, 223: 0.061210124410366466748047875393, 224: 0.061092088495, 225: 0.060962213749890186639756444065,\n    226: 0.060796487298909256201610075217, 227: 0.060648809113610562799479067019, 228: 0.060514379227249235146371102470, 229: 0.060376758442253510708047884568, 230: 0.060272507779,\n    231: 0.060136138369, 232: 0.060007064108617018915861101473, 233: 0.059907646293, 234: 0.059857529921211104278963554169, 235: 0.059835181171776612317920658375,\n    236: 0.059614677592478296223335067661, 237: 0.059518210815, 238: 0.059428621163812236316089432764, 239: 0.059299651118754481303437505832, 240: 0.059190958765594540736702230552,\n    241: 0.059118899313165779990095924460, 242: 0.058959800565081925326479132165, 243: 0.058822671447678052010872126778, 244: 0.058703338201762024370835065791, 245: 0.058566219740030845940311744398,\n    246: 0.058452450971281583379437364574, 247: 0.058370066289097847458039627360, 248: 0.058199873510179505215137276448, 249: 0.058075374681249419920764884075, 250: 0.057930866466,\n    251: 0.057818875775651875438976953808, 252: 0.057719405065464012447366088394, 253: 0.057650373900, 254: 0.057491975416, 255: 0.057326264006863181374093412161,\n    256: 0.057187786140711687070535155045, 257: 0.057084261908, 258: 0.057018277823, 259: 0.056904305437, 260: 0.056815484748987672691076832521,\n    261: 0.056730376500649784294138892978, 262: 0.056633255884, 263: 0.056533913416, 264: 0.056492790004188918480797294630, 265: 0.056365894314762878686693497801,\n    266: 0.056255350968247209872093697670, 267: 0.056188137037, 268: 0.056079352251, 269: 0.055982956607, 270: 0.055905693948811231554084641860,\n    271: 0.055773462121, 272: 0.055681799429631870200911985485, 273: 0.055570067534055499106473280636, 274: 0.055452091074, 275: 0.055362850731620702240235013704,\n    276: 0.055238192273770967890321791452, 277: 0.055138711870, 278: 0.055005421847, 279: 0.054892667943, 280: 0.054806763919,\n    281: 0.054703982953, 282: 0.054623097825, 283: 0.054525379293, 284: 0.054467927319, 285: 0.054345727175,\n    286: 0.054261592199, 287: 0.054149357525, 288: 0.054070753970, 289: 0.054020521554, 290: 0.053912352041,\n    291: 0.053861778013, 292: 0.053779816800, 293: 0.053697651228, 294: 0.053635769640, 295: 0.053604306613691310936498516199,\n    296: 0.053468572294, 297: 0.053390925332, 298: 0.053306765953402802185633859661, 299: 0.053233238500, 300: 0.053154299146851187500993025786,\n    301: 0.053068800203794916687599810744, 302: 0.052933070103, 303: 0.052827032309, 304: 0.052734996161, 305: 0.052628162573,\n    306: 0.052547532191823317516896261911, 307: 0.052466413693, 308: 0.052343190381, 309: 0.052242976012, 310: 0.052143318963,\n    311: 0.052054789894, 312: 0.051992385510176379148067859091, 313: 0.051929183013, 314: 0.051850572237, 315: 0.051807367385,\n    316: 0.051722242700404807813723271736, 317: 0.051632635094042734797463069856, 318: 0.051568810653162315414949806868, 319: 0.051481965443316711209597414219, 320: 0.051410694731,\n    321: 0.051324001088, 322: 0.051241055947, 323: 0.051156856628, 324: 0.051102122611, 325: 0.051022382199,\n    326: 0.050965225216, 327: 0.050892706441845478602670901065, 328: 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371: 0.047792616107, 372: 0.047740919004, 373: 0.047683113837, 374: 0.047636225887, 375: 0.047568388327,\n    376: 0.047533056758, 377: 0.047475744293, 378: 0.047416950444, 379: 0.047369803289, 380: 0.047312443141,\n    381: 0.047248305275, 382: 0.047198614368, 383: 0.047127888179, 384: 0.047058127991, 385: 0.046982306611,\n    386: 0.046931724009, 387: 0.046860125097, 388: 0.046808204357, 389: 0.046745598609, 390: 0.046680140174,\n    391: 0.046628644100, 392: 0.046575422240, 393: 0.046515511818, 394: 0.046455595360, 395: 0.046400178058,\n    396: 0.046342330600, 397: 0.046281193976, 398: 0.046223064054, 399: 0.046170469157, 400: 0.046110663895,\n    401: 0.046059320813, 402: 0.046021351893, 403: 0.045935557001114220615992944124, 404: 0.045863022982, 405: 0.045822028728961565623700898545,\n    406: 0.045783039381781544891843157934, 407: 0.045742186072742660330039963952, 408: 0.045713663260026576477112570567, 409: 0.045686348359351407110103461677, 410: 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0.023711608716536441386949239766, 1557: 0.023694843148519631382703005262, 1558: 0.023693853724412332341265057263, 1559: 0.023681748023279153812045517272, 1560: 0.023676154147035161349994236632,\n    1561: 0.023673985634801907316605961018, 1562: 0.023663080595169714017272945863, 1563: 0.023659084087325632934417492969, 1564: 0.023653304724044427835773250366, 1565: 0.023649211385468885616187062512,\n    1566: 0.023633981913614258783155224835, 1567: 0.023625231855350646960249243743, 1568: 0.023617085260749949240381588626, 1569: 0.023615763198124120471631390331, 1570: 0.023610071680460304326827838479,\n    1571: 0.023598119427112409351836031913, 1572: 0.023586823911225066241717171708, 1573: 0.023568823967255210510168495653, 1574: 0.023563406455488551118105018939, 1575: 0.023555463983433510699458956691,\n    1576: 0.023543954635263262345125533500, 1577: 0.023535690284706189512911107871, 1578: 0.023524155004678812404951408243, 1579: 0.023524155004678812404951408243, 1580: 0.023507296769310601727023059198,\n    1581: 0.023507261746463968605974712024, 1582: 0.023506334787838877847211429311, 1583: 0.023495498831338837067575394549, 1584: 0.023481966356682252463518082488, 1585: 0.023476214129840294722931577667,\n    1586: 0.023465575438934995384706860400, 1587: 0.023465575438934995384706860400, 1588: 0.023465575438934995384706860400, 1589: 0.023465575438934995384706860400, 1590: 0.023458137251864307750562672129,\n    1591: 0.023454844738241900225048493408, 1592: 0.023449243949684739285402868589, 1593: 0.023444716564688585982966768363, 1594: 0.023442842596505422870108373739, 1595: 0.023436492250553029704533594679,\n    1596: 0.023435429176602231859800441205, 1597: 0.023435429176602231859800441205, 1598: 0.023419453905461271756685190493, 1599: 0.023412821508331665825644856051, 1600: 0.023398725671001005924922438719,\n    1601: 0.023394087935980400671868813394, 1602: 0.023387010856540780451503936260, 1603: 0.023373627458164381335169023379, 1604: 0.023363687603070437148409116719, 1605: 0.023361713780084705214851614298,\n    1606: 0.023359613742332418546527600630, 1607: 0.023351539337863857216145816369, 1608: 0.023345573383936985750273546962, 1609: 0.023330513338441081436888066573, 1610: 0.023330221120023931133992506647,\n    1611: 0.023327885908222050408739926203, 1612: 0.023323295329750823752776400855, 1613: 0.023316964595165392438567488886, 1614: 0.023315070266565433565214232264, 1615: 0.023311124300765376144646398250,\n    1616: 0.023294414228704782659245027306, 1617: 0.023290539491879022165018280872, 1618: 0.023288242469983386829554849320, 1619: 0.023277581821366974758002168191, 1620: 0.023259541714073401945122530377,\n    1621: 0.023258659155631036770814199755, 1622: 0.023251917571715998501698368075, 1623: 0.023247390845066949991137952831, 1624: 0.023239688822254879672876837514, 1625: 0.023235687536638444717589226997,\n    1626: 0.023235687536638444717589226997, 1627: 0.023232744164778773294353599572, 1628: 0.023195714623000048038485165227, 1629: 0.023195126913575005587625807975, 1630: 0.023190563267625144589002108286,\n    1631: 0.023182582728877210236193703434, 1632: 0.023171365163458551751970490127, 1633: 0.023169118061349341967103545537, 1634: 0.023156639219149407012627036522, 1635: 0.023146672665827353229981973875,\n    1636: 0.023140369780966052397870438404, 1637: 0.023134076743151999542448869204, 1638: 0.023134076743151999542448869204, 1639: 0.023134076743151999542448869204, 1640: 0.023109311811902845922498137535,\n    1641: 0.023099358647017599854389631902, 1642: 0.023086262891403769908848902989, 1643: 0.023078760719649159011836956653, 1644: 0.023071061266330519672492535200, 1645: 0.023061703547844556250340792316,\n    1646: 0.023056672714034924015559812534, 1647: 0.023043963264837095540460925381, 1648: 0.023038671923111592058878724783, 1649: 0.023031773221232496884548833065, 1650: 0.023028060634734385169578704390,\n    1651: 0.023024612222221516232274031868, 1652: 0.023004598575330521045499067409, 1653: 0.023001377210523221378495729574, 1654: 0.022993596618971826615096651983, 1655: 0.022984273461889033830180310672,\n    1656: 0.022979780241802995555293242393, 1657: 0.022971070230502113442488313113, 1658: 0.022969258651561834368311609060, 1659: 0.022965436749359219462118970192, 1660: 0.022951555897351762615329638892,\n    1661: 0.022944096648071909460747287547, 1662: 0.022939668939390486234447458927, 1663: 0.022939660895008699226371782303, 1664: 0.022937804387700803907041905651, 1665: 0.022925610451265598666608185180,\n    1666: 0.022921067287762503172283177762, 1667: 0.022919673418840393755224810630, 1668: 0.022905121447950301179227332452, 1669: 0.022901784256333504197242561007, 1670: 0.022888921230925736335462612245,\n    1671: 0.022879536801963052502615639979, 1672: 0.022876470325726001664115738978, 1673: 0.022868821417698753506944933923, 1674: 0.022865513595348036548003030762, 1675: 0.022861558462022503393935859718,\n    1676: 0.022856932548926026364498245176, 1677: 0.022849799129739852659919314002, 1678: 0.022842914378397801746860380634, 1679: 0.022831720085606969005347137996, 1680: 0.022823429493707541734400654964,\n    1681: 0.022821624181988566260782832187, 1682: 0.022818099459724793335510003750, 1683: 0.022804024883924295878984576434, 1684: 0.022802346402746340375903934561, 1685: 0.022786057701602160714054428677,\n    1686: 0.022781491290921679001080083866, 1687: 0.022778953715691903934868063828, 1688: 0.022778491606434931546073740019, 1689: 0.022778351713816923610762793473, 1690: 0.022767545332123191515111962134,\n    1691: 0.022762231242884467705723342489, 1692: 0.022753500126472122644463463419, 1693: 0.022743029468348226855469288446, 1694: 0.022740649324552815660824502149, 1695: 0.022727598393094050105150140253,\n    1696: 0.022717880296059019923692556352, 1697: 0.022715118136019625629741160827, 1698: 0.022706540888257767295528996907, 1699: 0.022702702726405739287199008078, 1700: 0.022694663790780158984182174127,\n    1701: 0.022688342246102829057795079985, 1702: 0.022684119716339750643997568304, 1703: 0.022682923238134977717129510134, 1704: 0.022662837002099043108227648820, 1705: 0.022662207415530934394478420514,\n    1706: 0.022652949692342330424814633680, 1707: 0.022646076827449077840714106940, 1708: 0.022640394470616050344441210448, 1709: 0.022630118397677758527660647221, 1710: 0.022623283161824533220640430363,\n    1711: 0.022622667072070619970749707189, 1712: 0.022611348317647915258237839646, 1713: 0.022603604660147632235104390999, 1714: 0.022599909272240235097315962556, 1715: 0.022588561073293909341739443301,\n    1716: 0.022582218485485114890113983807, 1717: 0.022579328187778307345521518813, 1718: 0.022570747800608192578642332176, 1719: 0.022568607551786234539315721952, 1720: 0.022567503964049382249062308845,\n    1721: 0.022566898981469511732831245781, 1722: 0.022566824427228091951570842689, 1723: 0.022566567317270497890943504907, 1724: 0.022544747493602126878081633533, 1725: 0.022530181034202303748000035859,\n    1726: 0.022523642545157245103442632112, 1727: 0.022519150413675288226265885263, 1728: 0.022513721346490855704274138783, 1729: 0.022510754839661159531698397535, 1730: 0.022502849857132734376056513292,\n    1731: 0.022496249669276468611280914587, 1732: 0.022490047951130663002814642907, 1733: 0.022488900420442574247380230813, 1734: 0.022488900420442574247380230813, 1735: 0.022488900420442574247380230813,\n    1736: 0.022480811860461379243827386825, 1737: 0.022476932629214301742470223767, 1738: 0.022472810968213578453022465968, 1739: 0.022469601207160494902165277475, 1740: 0.022465328793167710121685860756,\n    1741: 0.022460609299228328874246857333, 1742: 0.022450672663412463358296807865, 1743: 0.022445588506077837864052881176, 1744: 0.022444502995335864605779892125, 1745: 0.022440725180009140848225379266,\n    1746: 0.022431509453412775622935089231, 1747: 0.022426154469856796559990809489, 1748: 0.022422956086489650745237237203, 1749: 0.022417960972379780573689383604, 1750: 0.022407988977455280116155795083,\n    1751: 0.022401807042503451045920585142, 1752: 0.022394229195431760066964629038, 1753: 0.022386360721288781586869308668, 1754: 0.022384976093321067988178754221, 1755: 0.022378509498358743257227880671,\n    1756: 0.022370824106094217242133835362, 1757: 0.022362784653906406190455764061, 1758: 0.022358530073252465862784117048, 1759: 0.022358530073252465862784117048, 1760: 0.022338854156008613808506261932,\n    1761: 0.022327431555660444548254401315, 1762: 0.022322474015257901044235573561, 1763: 0.022319418745663121938205860112, 1764: 0.022317391026117226060799541132, 1765: 0.022316031762412935472585389461,\n    1766: 0.022303981302871158139249227128, 1767: 0.022292476121862261727593546164, 1768: 0.022288294189881695195366082378, 1769: 0.022284163651572634026527776423, 1770: 0.022279430305009319790201036014,\n    1771: 0.022263532800438174181017379887, 1772: 0.022253989725053700635260644005, 1773: 0.022231640841783176724279200959, 1774: 0.022231320584637442981377583207, 1775: 0.022226436989200357369766069923,\n    1776: 0.022222305949465807842824543353, 1777: 0.022219778019067909144445270344, 1778: 0.022203330673970187098209364280, 1779: 0.022189294564676618084565455424, 1780: 0.022188540114177525344435446060,\n    1781: 0.022179414258781856592594407532, 1782: 0.022166569282886679482960557259, 1783: 0.022165751537532986805186678276, 1784: 0.022159754529076937318148235559, 1785: 0.022157460890906855951478092555,\n    1786: 0.022156530264637329237326391677, 1787: 0.022155529799264161099758996601, 1788: 0.022155501680804298064435314459, 1789: 0.022155501680804298064435314459, 1790: 0.022132503249643748007106577618,\n    1791: 0.022120892239549347125942540859, 1792: 0.022102850089381256066156732360, 1793: 0.022093423372489998246936227022, 1794: 0.022084340454527477075448482081, 1795: 0.022083417781540439846756530888,\n    1796: 0.022076494977517720550487138419, 1797: 0.022070784735765239127734298399, 1798: 0.022063905841426913715699537748, 1799: 0.022058470826937949603890773816, 1800: 0.022055194713935587518150259069,\n    1801: 0.022046658504172785980208590429, 1802: 0.022043458551335940182074194632, 1803: 0.022039452376347939905279948634, 1804: 0.022032014034301104135489096739, 1805: 0.022026378114259711605542755791,\n    1806: 0.022022271821130256114890525367, 1807: 0.022018771560266546628972153045, 1808: 0.022017917791047313702231190002, 1809: 0.022007603974764645237970835397, 1810: 0.022004314769122806334487225575,\n    1811: 0.021998857521458097417406793750, 1812: 0.021994087912920564420577647680, 1813: 0.021986417725788688155053081557, 1814: 0.021980490860776180165201526903, 1815: 0.021977347527260316515176533705,\n    1816: 0.021968965225487066962497438814, 1817: 0.021964656727380256466174845299, 1818: 0.021960385826974420196470756021, 1819: 0.021951273764609689590033990780, 1820: 0.021945304775737406537187111205,\n    1821: 0.021939287062698422350845719838, 1822: 0.021937245479398823967123908304, 1823: 0.021930750416793964763395485118, 1824: 0.021925604819665532450572177093, 1825: 0.021921473728635295795161703683,\n    1826: 0.021916967091892016106462170572, 1827: 0.021906139858414147298639752275, 1828: 0.021897188485523333132192204006, 1829: 0.021895014413985749606736150110, 1830: 0.021894323975814041050452586634,\n    1831: 0.021888459482525111540843510622, 1832: 0.021883630279849876403776327528, 1833: 0.021880485583600022401336763806, 1834: 0.021870822847452173800431534115, 1835: 0.021869972574474149699939605937,\n    1836: 0.021863876371426424349335976891, 1837: 0.021857435151327998127709832143, 1838: 0.021850077111926242615306465415, 1839: 0.021844673742198569577334953333, 1840: 0.021840090606407951844327698220,\n    1841: 0.021837085199848705412305684923, 1842: 0.021831374968594854632728563090, 1843: 0.021828468236999339499251444244, 1844: 0.021820129268457950813077277941, 1845: 0.021801675719636876896924352195,\n    1846: 0.021797112142922149808860497434, 1847: 0.021795222698955467200070948637, 1848: 0.021782270098960198411448935536, 1849: 0.021772018332186460586884551328, 1850: 0.021762238396625140432572363753,\n    1851: 0.021756212660244416577942790484, 1852: 0.021748274282231968316862429607, 1853: 0.021738294755661615380081802177, 1854: 0.021734426237814453444082916175, 1855: 0.021722090115735045013112997911,\n    1856: 0.021718482264244309527702380553, 1857: 0.021716254245628101518427314019, 1858: 0.021710612692471450877973064681, 1859: 0.021704715412954885738251627835, 1860: 0.021704067789623619824450218396,\n    1861: 0.021702472024060179826074713227, 1862: 0.021702472024060179826074713227, 1863: 0.021702472024060179826074713227, 1864: 0.021702472024060179826074713227, 1865: 0.021702472024060179826074713227,\n    1866: 0.021702472024060179826074713227, 1867: 0.021702472024060179826074713227, 1868: 0.021680855825378941708782987512, 1869: 0.021679009184917514049378689978, 1870: 0.021665509941279040510067270855,\n    1871: 0.021660608064204703525990254824, 1872: 0.021647997777794123194160132100, 1873: 0.021647540717129703716163589752, 1874: 0.021646982585045060525795009786, 1875: 0.021646095266282831681042498044,\n    1876: 0.021641792356068137402363389374, 1877: 0.021634756784705527750355710000, 1878: 0.021633331527858107026326009062, 1879: 0.021632459850862607724876919981, 1880: 0.021619310040595067965956310777,\n    1881: 0.021612015699585695532254902017, 1882: 0.021607794430603806869304764612, 1883: 0.021605157046334420069069345086, 1884: 0.021597926434323469243318370787, 1885: 0.021597378970271448964662249093,\n    1886: 0.021589050616082103455129924048, 1887: 0.021578378212752019297472069541, 1888: 0.021577966670329077007297259438, 1889: 0.021577966670329077007297259438, 1890: 0.021577966670329077007297259438,\n    1891: 0.021577966670329077007297259438, 1892: 0.021550933257506676337574323437, 1893: 0.021534272298685589675864323127, 1894: 0.021532954244572877270979738031, 1895: 0.021527809837962890551962803571,\n    1896: 0.021525630760746745150354192746, 1897: 0.021518116375802261803947138639, 1898: 0.021511998058166253119062163088, 1899: 0.021504390083413102614668638810, 1900: 0.021496425938201118142195084784,\n    1901: 0.021494832217271524586983246648, 1902: 0.021484963498597920868302390990, 1903: 0.021483844257459019210688642092, 1904: 0.021478801851985735050579670204, 1905: 0.021474831267577366858608966328,\n    1906: 0.021471914902676657130462351334, 1907: 0.021465895025240459296549370393, 1908: 0.021458471159177199680559773731, 1909: 0.021457543580476645865590647234, 1910: 0.021456719242658574101926671564,\n    1911: 0.021453892241752996546852853005, 1912: 0.021449100811507250589033253509, 1913: 0.021444314773334123972035739583, 1914: 0.021438621668405291045360629608, 1915: 0.021436009574915753887783966725,\n    1916: 0.021411870075764906271379348006, 1917: 0.021406880662047106336423968064, 1918: 0.021398191505842879983882337240, 1919: 0.021393049854044638284756087493, 1920: 0.021391205962573330238047870953,\n    1921: 0.021388633490429184076068271570, 1922: 0.021374718481864845639457318826, 1923: 0.021367949708944994334514350760, 1924: 0.021359640434390330727425703482, 1925: 0.021351065904536422428561851736,\n    1926: 0.021346036008119039831455523792, 1927: 0.021335652861394228993784852784, 1928: 0.021335029898517781959516343055, 1929: 0.021327725208234167663920002303, 1930: 0.021313793806192601497534540784,\n    1931: 0.021313793806192601497534540784, 1932: 0.021307347283169539990618197716, 1933: 0.021305896131670671943154716737, 1934: 0.021301231814784871786655300977, 1935: 0.021295824628934350647220690662,\n    1936: 0.021286883436463472221376779719, 1937: 0.021282796023812183212562338807, 1938: 0.021276457596631896424886445550, 1939: 0.021270392015428306990585535690, 1940: 0.021268330290888369398067048570,\n    1941: 0.021264799974245963159044925267, 1942: 0.021262691078522037884893840007, 1943: 0.021251570857055978376066482595, 1944: 0.021248279779303221331387600093, 1945: 0.021235118564571050176925977120,\n    1946: 0.021231054266841435979689581848, 1947: 0.021229166570444226861854626422, 1948: 0.021224707808299190808808481709, 1949: 0.021218782416138290185799871410, 1950: 0.021214755402710723623125218345,\n    1951: 0.021207460753768609558997686532, 1952: 0.021198551623162517147242755676, 1953: 0.021194940172389992350084564853, 1954: 0.021191518290771208489889193287, 1955: 0.021174472268659263854709449845,\n    1956: 0.021173776078015585902567994176, 1957: 0.021170124605216793403904613098, 1958: 0.021166516195908040470403888659, 1959: 0.021162540006422625738096636640, 1960: 0.021157376586178568692496591647,\n    1961: 0.021148757929795697904163634237, 1962: 0.021147447489569711343707719705, 1963: 0.021142931333206110670326892352, 1964: 0.021136309997584722789215708118, 1965: 0.021134918521603742995132079840,\n    1966: 0.021127252489341256943266240139, 1967: 0.021115638714292114537973738831, 1968: 0.021114074604599532629383118147, 1969: 0.021109856145787311431160359420, 1970: 0.021107986314546451950848472861,\n    1971: 0.021100378152337780151956303329, 1972: 0.021095407480009124317955765457, 1973: 0.021089022814691597797777510683, 1974: 0.021087964714223022557641097845, 1975: 0.021080248223014634134976245619,\n    1976: 0.021075799603307122587391473981, 1977: 0.021068200738563570141610320580, 1978: 0.021064345976490010176040219488, 1979: 0.021056669329403856753623960443, 1980: 0.021054563200501220724296536493,\n    1981: 0.021045765495760647501426086007, 1982: 0.021040746169964656128756488686, 1983: 0.021032419749630304313540476628, 1984: 0.021030068940621325143535570090, 1985: 0.021024791375262835354190691084,\n    1986: 0.021017623632741640996437594445, 1987: 0.021014623304626910328205610049, 1988: 0.021009262236878551814893815296, 1989: 0.021003125096801480445429483666, 1990: 0.021000528724716645959268603919,\n    1991: 0.020994256973936502609047122127, 1992: 0.020989069819233454494962006939, 1993: 0.020982457543499625915720782317, 1994: 0.020974316236957432368586415259, 1995: 0.020964329972877455255597284147,\n    1996: 0.020959167343995878613295416178, 1997: 0.020956228075777109510136072348, 1998: 0.020950115543170494592733228545, 1999: 0.020947416640044324100456280298, 2000: 0.020945043128290298030484966376,\n    2001: 0.020936517293408255652728813540, 2002: 0.020930360686551881633825190076, 2003: 0.020923837706321848123174343069, 2004: 0.020917329059242501391447808710, 2005: 0.020914274271883983856267376180,\n    2006: 0.020910961052428717933617137749, 2007: 0.020908950905513889283030310720, 2008: 0.020903256952825690812524166053, 2009: 0.020902759320247287814684251770, 2010: 0.020901967698390550324324512800,\n    2011: 0.020896264411653373411833124897, 2012: 0.020896112096411033921203753491, 2013: 0.020890834580829554800051763382, 2014: 0.020889606362683476787523763025, 2015: 0.020888904442745585431093464793,\n    2016: 0.020887540459480660845927382606, 2017: 0.020885317509629605678322578660, 2018: 0.020876194591414055937948977307, 2019: 0.020872005186188564331664461060, 2020: 0.020867769450334103938782837743,\n    2021: 0.020865704104040983159371298154, 2022: 0.020862413676187054587124957938, 2023: 0.020855392184163225461549793773, 2024: 0.020850299869853159635759638729, 2025: 0.020844765402990345961822624533,\n    2026: 0.020843995788541705191410390367, 2027: 0.020843903677693570377087096456, 2028: 0.020843557907754556859693377697, 2029: 0.020843353965363344745584779498, 2030: 0.020822845700295309608381552692,\n    2031: 0.020815642861915176367562918848, 2032: 0.020802044736914040205861587553, 2033: 0.020797190494938022732703244044, 2034: 0.020789845569720107571740370541, 2035: 0.020785831218005830518979251853,\n    2036: 0.020779162831268664120676790955, 2037: 0.020766510995538684532028723377, 2038: 0.020760896043315639707772083539, 2039: 0.020758244986182405377492266856, 2040: 0.020754618437339247457942330478,\n    2041: 0.020748229893169102940429200544, 2042: 0.020742269175456862276138774282, 2043: 0.020736345022963810447501298437, 2044: 0.020729774289812432109407471633, 2045: 0.020728275182117394703226677031,\n    2046: 0.020728037798580571947837915361, 2047: 0.020725647712163830462865767854, 2048: 0.020723000278397327447394698600, 2049: 0.020717817737889945829191926066, 2050: 0.020716123761381999085681299559,\n    2051: 0.020714990774719641115695026197, 2052: 0.020714266693717470533760431084, 2053: 0.020713741959702105288638840373, 2054: 0.020695196158871843614917040878, 2055: 0.020689481136734886450225003424,\n    2056: 0.020681304966981360449102871761, 2057: 0.020674099107889438360363794181, 2058: 0.020662041413100052880681392353, 2059: 0.020659456914868569061656161532, 2060: 0.020653962713216270346530925538,\n    2061: 0.020647498847869561083053886075, 2062: 0.020639469407185579956069334637, 2063: 0.020638633942655676130090567101, 2064: 0.020634444063517603151652919478, 2065: 0.020631374279827647818221354181,\n    2066: 0.020626627700411478486490790718, 2067: 0.020621528204562234115317133077, 2068: 0.020615942974736928081611512592, 2069: 0.020610544198769188444329426923, 2070: 0.020601707096573714009286620654,\n    2071: 0.020599662129122256934144221077, 2072: 0.020593501410680072128370460276, 2073: 0.020591447536908503638666271309, 2074: 0.020590433977721013701417316467, 2075: 0.020588979343997387110785209389,\n    2076: 0.020586077101226241681014091361, 2077: 0.020586077101226241681014091361, 2078: 0.020566441829360200247113559074, 2079: 0.020564723931824604910035733065, 2080: 0.020549298624570943101522084836,\n    2081: 0.020540826352998383145245226957, 2082: 0.020539919253877382720035233138, 2083: 0.020532990675453760749037491616, 2084: 0.020530883806395553422428851628, 2085: 0.020525608410904002804717396244,\n    2086: 0.020521706720977138053814993421, 2087: 0.020516311263190050399218562698, 2088: 0.020513731544824699838387502728, 2089: 0.020509877261461541778049364725, 2090: 0.020505715475061604443373650453,\n    2091: 0.020500898725483734405529674287, 2092: 0.020499463009713454062853756638, 2093: 0.020496323225538110208773467934, 2094: 0.020491942334259962260553301822, 2095: 0.020480918852904069621740479246,\n    2096: 0.020480420894712191879171579802, 2097: 0.020472866739329694191936349106, 2098: 0.020467550238210807101726721998, 2099: 0.020458671381473273385643182337, 2100: 0.020454166602871532286906571964,\n    2101: 0.020452356505713305475907985678, 2102: 0.020448662666621807758835822775, 2103: 0.020439716940843649202162072707, 2104: 0.020434798880564377059291198043, 2105: 0.020430479100096192348857638440,\n    2106: 0.020428913084805971397951346930, 2107: 0.020421751138431541205510202578, 2108: 0.020416909089214913501688549893, 2109: 0.020414065832771383371753614165, 2110: 0.020406195872367805869820647414,\n    2111: 0.020406195872367805869820647414, 2112: 0.020404000729277572582363817379, 2113: 0.020393818023731754511111277476, 2114: 0.020389452714248464901679491470, 2115: 0.020387492403471231285580063588,\n    2116: 0.020387198487886014835771217486, 2117: 0.020376887229330531349173706448, 2118: 0.020372187571623586204311028189, 2119: 0.020368167935788625271901151402, 2120: 0.020366359092867055414000661162,\n    2121: 0.020364990176841356297459616099, 2122: 0.020358294359272789588830242966, 2123: 0.020353687397411233798674547246, 2124: 0.020351898875122505642044964571, 2125: 0.020349473672989972166527796942,\n    2126: 0.020341589917810523345504573045, 2127: 0.020340466437118974972413075439, 2128: 0.020333718542233006442478219695, 2129: 0.020329022409102610034083096755, 2130: 0.020326732168741518010402523368,\n    2131: 0.020317388759358253500445176667, 2132: 0.020309026413042415476110130275, 2133: 0.020305317755427640680612964096, 2134: 0.020302285699348089693392859119, 2135: 0.020297373817746315182244692127,\n    2136: 0.020292707485669857900754576560, 2137: 0.020289381289915503754346688336, 2138: 0.020279324712250333898430380441, 2139: 0.020275158377606452838798411341, 2140: 0.020272612201491539210785196364,\n    2141: 0.020271832336383773827612184959, 2142: 0.020262573171827769868401890211, 2143: 0.020248466779661092406890327639, 2144: 0.020243140825929852240236689805, 2145: 0.020241330328727544834183499896,\n    2146: 0.020237718100863724157067888629, 2147: 0.020228029237627740568621655465, 2148: 0.020222158850796961861007083267, 2149: 0.020220371101027132709191790452, 2150: 0.020215697062918822661128118525,\n    2151: 0.020215456373581137968293255338, 2152: 0.020207802923497395016190976191, 2153: 0.020195026307038886861329864429, 2154: 0.020191310190839660595528771644, 2155: 0.020190086818078550981269197647,\n    2156: 0.020186505677558025798831646033, 2157: 0.020186505677558025798831646033, 2158: 0.020186505677558025798831646033, 2159: 0.020186505677558025798831646033, 2160: 0.020186505677558025798831646033,\n    2161: 0.020186505677558025798831646033, 2162: 0.020163158893126006468098150760, 2163: 0.020160209644447869966521561323, 2164: 0.020153879522221634309309245276, 2165: 0.020153879522221634309309245276,\n    2166: 0.020149622704194182408657612840, 2167: 0.020148033728576443758249990452, 2168: 0.020143117591103578780170330855, 2169: 0.020138239873369322891273973281, 2170: 0.020132647576861072925128155169,\n    2171: 0.020129308020727018814815490224, 2172: 0.020127711182984816941187473534, 2173: 0.020122860168764099012802353100, 2174: 0.020118216565139506867263832169, 2175: 0.020111342016259792764988837875,\n    2176: 0.020103983231636996090822071214, 2177: 0.020103734100708019427224988619, 2178: 0.020101847527856343359550264019, 2179: 0.020097504299574372664520992320, 2180: 0.020095619953065039151016489085,\n    2181: 0.020094075947624193180291311656, 2182: 0.020090918783568569670548459818, 2183: 0.020085614590014115720184164122, 2184: 0.020081659485492229706110349968, 2185: 0.020081586755789939662295133225,\n    2186: 0.020076644361645808177294078073, 2187: 0.020073408521301081749174171350, 2188: 0.020066142615599985418866875588, 2189: 0.020064048678448462787632259436, 2190: 0.020058656106509099876263329776,\n    2191: 0.020047461115624704992965934857, 2192: 0.020042123900606623896159315828, 2193: 0.020041959709860234129153385958, 2194: 0.020040465535618723114418705065, 2195: 0.020031361435625167932267814599,\n    2196: 0.020028634547187257014095090518, 2197: 0.020027446699949710639342394288, 2198: 0.020018368706577109017291421195, 2199: 0.020010639189554454523347506037, 2200: 0.020006590629370661459725731891,\n    2201: 0.020001160978776412555804128082, 2202: 0.019992567942553224475349702288, 2203: 0.019988016189859640677701974380, 2204: 0.019987176222484892615575616799, 2205: 0.019983142811898278919024467090,\n    2206: 0.019981656742109171152286046000, 2207: 0.019981235337787043770011088269, 2208: 0.019978903567108004821098240541, 2209: 0.019978903567108004821098240541, 2210: 0.019961595849164934885522975838,\n    2211: 0.019956900359615571945406427415, 2212: 0.019947700994792668091594855208, 2213: 0.019943643708627837521114445008, 2214: 0.019939857653066823931387981161, 2215: 0.019935375104523841989554648073,\n    2216: 0.019920450824053235371778849109, 2217: 0.019917942193242222870622384725, 2218: 0.019909949798026394912814087074, 2219: 0.019909866837340427380676679593, 2220: 0.019907571185659549976968598459,\n    2221: 0.019905678306990494864690308896, 2222: 0.019894344533319795225374203493, 2223: 0.019888397534139280883539884791, 2224: 0.019884860430908746051225114418, 2225: 0.019877789344418247636948122756,\n    2226: 0.019871854170122338070559563784, 2227: 0.019869871914452170555531866592, 2228: 0.019862306357362048166133431333, 2229: 0.019855708964207426524286793555, 2230: 0.019850668588558945332267992092,\n    2231: 0.019848289240098074216661796488, 2232: 0.019848289240098074216661796488, 2233: 0.019848289240098074216661796488, 2234: 0.019831096608651067905147536122, 2235: 0.019819719552187635109525966040,\n    2236: 0.019815379926662202652709389062, 2237: 0.019810205173433198727741182873, 2238: 0.019809217110566949199311178887, 2239: 0.019806664591578274894010562796, 2240: 0.019806481033706455977800374814,\n    2241: 0.019800080823667110133478681303, 2242: 0.019799861435974181979629170180, 2243: 0.019797973900748975341905059027, 2244: 0.019796238286951572200314498541, 2245: 0.019793573330324234534877272364,\n    2246: 0.019782067779319699496625831440, 2247: 0.019779688751806256632263685350, 2248: 0.019775342416942915099565801808, 2249: 0.019774636856516207031620876466, 2250: 0.019770196123340012173040951737,\n    2251: 0.019768182594410835997877156808, 2252: 0.019762804397868239405298875793, 2253: 0.019754955572734249087326897726, 2254: 0.019745719745121727941458033886, 2255: 0.019742782561103849222002745973,\n    2256: 0.019741790723553429559102094505, 2257: 0.019731599700068836986766813934, 2258: 0.019728404980779007078120074163, 2259: 0.019726728308586101921342977728, 2260: 0.019716462348706737395031475754,\n    2261: 0.019714316609047300305540389799, 2262: 0.019712134643942924553486506932, 2263: 0.019709801025607117047980549625, 2264: 0.019709557948223501345485424727, 2265: 0.019704901643420652605058671792,\n    2266: 0.019703694921692342205247638543, 2267: 0.019698022622418704195220569148, 2268: 0.019689320161628846366441154112, 2269: 0.019679790747518014885742747986, 2270: 0.019678218448775248764716554117,\n    2271: 0.019675005005948616191122907293, 2272: 0.019670138676122288715883994402, 2273: 0.019663225371124701524553587132, 2274: 0.019661226245459494501722109145, 2275: 0.019651741067281250823731603301,\n    2276: 0.019647410798417996745224729542, 2277: 0.019632938881693903960465415235, 2278: 0.019631305970172437890518775960, 2279: 0.019625821128829232458520704775, 2280: 0.019625821128829232458520704775,\n    2281: 0.019625821128829232458520704775, 2282: 0.019622844779193811682499318406, 2283: 0.019619436747651072406475836329, 2284: 0.019611122530742103679349185389, 2285: 0.019599461235917307970136192231,\n    2286: 0.019599461235917307970136192231, 2287: 0.019599461235917307970136192231, 2288: 0.019593597952770473458990298029, 2289: 0.019585486135951107170332905552, 2290: 0.019585486135951107170332905552,\n    2291: 0.019585486135951107170332905552, 2292: 0.019585486135951107170332905552, 2293: 0.019578791282502621023661778557, 2294: 0.019571137866184279524879090370, 2295: 0.019566548645779007078116019223,\n    2296: 0.019562104347493392503021795355, 2297: 0.019556158486679991157670264181, 2298: 0.019555019590917513541554916184, 2299: 0.019548759097646646451010387751, 2300: 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0.016083274211154609254903028170,\n    3439: 0.016044612897138235777123427085, 3463: 0.015981152661925126475718391171, 3481: 0.015923006896051041827022324652, 3493: 0.015890499383827607032212014290, 3505: 0.015864010568136164150470001405,\n    3511: 0.015859938568451436869766195910, 3535: 0.015792515379115816803313227175, 3547: 0.015782707488277871085889554791, 3559: 0.015776598860626854829889163104, 3571: 0.015750924582799726757361135717,\n    3595: 0.015692408426003560179710658424, 3607: 0.015673281014228516486306427258, 3619: 0.015651217772445257961237645321, 3631: 0.015616137018328311416861346800, 3643: 0.015600853414432366069491876297,\n    3655: 0.015553648175090695497672645587, 3667: 0.015535142384368707220025713271, 3679: 0.015497211287834552171948912726, 3691: 0.015464609313997476437355205036, 3697: 0.015459193612037870367476315523,\n    3721: 0.015404906700468930267592516208, 3745: 0.015373854475907933099088918864, 3757: 0.015352064566157839667004474819, 3781: 0.015307498483917933119554319014, 3793: 0.015298800047079069209559820983,\n    3805: 0.015251749397957394900814446659, 3817: 0.015231116571433214465817199043, 3829: 0.015216456122723403731923031909, 3853: 0.015165473864003495037505601281, 3865: 0.015140996411458597688091534127,\n    3877: 0.015108812941776206881638389594, 3889: 0.015083929089729729941503738186, 3901: 0.015042338455468058672778181976, 3919: 0.015013592966324829306667930533, 3931: 0.014991970087460905919096034551,\n    3943: 0.014965462848801963307763269064, 3949: 0.014965462848801963307763269064, 3973: 0.014936771972611631656530269346, 3985: 0.014921278634197894231962244471, 4000: 0.014885326609723732415968632381,\n    4009: 0.014878422806865932108013282571, 4021: 0.014855279973167574367591386399, 4033: 0.014827307928945795135569936236, 4045: 0.014793353028166707432033595841, 4057: 0.014772761261388513062002416834,\n    4069: 0.014754101388967745011010542172, 4081: 0.014730675313403758845261701191, 4093: 0.014701557943559931847789208567, 4105: 0.014685503530022635284891786650, 4117: 0.014655529078031067398744501001,\n    4141: 0.014601755763187308074373236782, 4153: 0.014596422226478759387097611060, 4177: 0.014571036577981725076564764662, 4189: 0.014545704874381254845809891045, 4195: 0.014544129293729196275114876636,\n    4219: 0.014498140763311835725220537867, 4231: 0.014487048520058671993470040260, 4243: 0.014471442843963864684402709412, 4255: 0.014440219550014709079161935033, 4267: 0.014422049997852603734148316158,\n    4303: 0.014342875056562309945593852991, 4315: 0.014323541476018172725671907488, 4339: 0.014278996448020329132467169733, 4345: 0.014273228279866076598567467525, 4357: 0.014255170553446144330034824715,\n    4381: 0.014226299359063675489740434402, 4405: 0.014190212052536014904153227050, 4417: 0.014180428797206355910866296289, 4429: 0.014160213144296972937030144195, 4447: 0.014122954608286733925394744264,\n    4459: 0.014109163910151592670557911503, 4471: 0.014087644508730927379132904466, 4483: 0.014078899938434832009758053415, 4495: 0.014048057319852987268907371801, 5000: 0.013323321560347263143157419877,\n}\n\n\ndef fail(msg: str, kind: str = \"error\") -> None:\n    print(json.dumps({\"metric\": 0.0, \"error\": msg, \"kind\": kind}))\n    sys.exit(1)\n\n\ndef check_imports(path: Path) -> None:\n    try:\n        tree = ast.parse(path.read_text(encoding=\"utf-8\"))\n    except SyntaxError as e:\n        fail(f\"syntax error in pack.py: {e}\", \"compile_error\")\n    for node in ast.walk(tree):\n        names = []\n        if isinstance(node, ast.Import):\n            names = [a.name.split(\".\")[0] for a in node.names]\n        elif isinstance(node, ast.ImportFrom) and node.module:\n            names = [node.module.split(\".\")[0]]\n        for nm in names:\n            if nm not in STDLIB_ALLOW:\n                fail(f\"import of '{nm}' is not allowed (stdlib subset only: {sorted(STDLIB_ALLOW)})\", \"compile_error\")\n\n\ndef boundary(c) -> float:\n    \"\"\"Distance from centre c to the container boundary; negative outside.\"\"\"\n    return 1.0 - math.hypot(c[0], c[1])\n\n\ndef weight(i: int) -> float:\n    \"\"\"Radius weight of object i (1-based); the eval derives the common scale s, r_i = weight(i) * s.\"\"\"\n    return 1.0\n\n\ndef value_of(centres: list, n: int) -> float:\n    if not isinstance(centres, (list, tuple)) or len(centres) != n:\n        fail(f\"pack({n}) must return {n} centres\", \"wrong_answer\")\n    pts = []\n    for c in centres:\n        try:\n            p = tuple(float(v) for v in c)\n        except Exception:\n            fail(f\"pack({n}) returned a non-point {c!r}\", \"wrong_answer\")\n        if len(p) != DIM or not all(math.isfinite(v) for v in p):\n            fail(f\"pack({n}) returned a point that is not {DIM}-D and finite: {c!r}\", \"wrong_answer\")\n        if boundary(p) < -EPS:\n            fail(f\"pack({n}) placed a centre outside the container: {p}\", \"wrong_answer\")\n        pts.append(p)\n    w = [weight(i + 1) for i in range(n)]\n    s = min(max(boundary(p), 0.0) / w[i] for i, p in enumerate(pts))   # upper bound from the walls\n    if s <= EPS:\n        fail(f\"pack({n}) has a centre on the boundary (scale={s})\", \"wrong_answer\")\n    # Any pair that limits the scale below s has distance < (w_i + w_j) s <= 2 wmax s, so it lies\n    # in the same or an adjacent cell of a grid with that spacing. Expected O(n) instead of O(n^2).\n    cell = 2.0 * max(w) * s\n    grid = {}\n    for idx, p in enumerate(pts):\n        key = tuple(int(math.floor(v / cell)) for v in p)\n        grid.setdefault(key, []).append(idx)\n    offsets = list(itertools.product((-1, 0, 1), repeat=DIM))\n    for key, members in grid.items():\n        for off in offsets:\n            nb = tuple(k + o for k, o in zip(key, off))\n            if nb < key or nb not in grid:\n                continue\n            others = grid[nb]\n            for i in members:\n                pi = pts[i]\n                for j in others:\n                    if nb == key and j <= i:\n                        continue\n                    pj = pts[j]\n                    d = math.sqrt(sum((a - b) * (a - b) for a, b in zip(pi, pj))) / (w[i] + w[j])\n                    if d < s:\n                        s = d\n    if s <= EPS:\n        fail(f\"pack({n}) has coincident centres (scale={s})\", \"wrong_answer\")\n    return w[n - 1] * s   # the largest object's radius (equal case: the common radius)\n\n\ndef main() -> None:\n    here = Path(__file__).parent\n    check_imports(here / \"pack.py\")\n    sys.path.insert(0, str(here))\n    try:\n        import pack as cand  # noqa: E402\n    except SystemExit:\n        raise\n    except Exception as e:\n        fail(f\"import pack.py failed: {e!r}\", \"compile_error\")\n    if not hasattr(cand, \"pack\"):\n        fail(\"pack.py must define pack(n, time_budget, seed)\", \"compile_error\")\n\n    ns = sorted(set(NS))\n    for n in ns:\n        if n not in RECORDS:\n            fail(f\"no Packomania record for n={n}\", \"error\")\n    seed_int = random.Random(f\"cci|{SEED}\").getrandbits(32)\n    per_n, beaten = {}, []\n    t_all = time.perf_counter()\n    for n in ns:\n        t0 = time.perf_counter()\n        try:\n            centres = cand.pack(n, BUDGET, seed_int)\n        except SystemExit:\n            raise\n        except Exception as e:\n            fail(f\"pack({n}) raised {e!r}\", \"runtime_error\")\n        elapsed = time.perf_counter() - t0\n        if elapsed > 1.25 * BUDGET + 3:\n            fail(f\"pack({n}) took {elapsed:.1f}s against a {BUDGET:.0f}s budget\", \"timeout\")\n        v = value_of(centres, n)\n        ratio = v / RECORDS[n]\n        per_n[n] = {\"value\": round(v, 12), \"record\": RECORDS[n], \"ratio\": round(ratio, 6), \"seconds\": round(elapsed, 2)}\n        if v > RECORDS[n] + 1e-9:\n            beaten.append(n)\n    metric = sum(x[\"ratio\"] for x in per_n.values()) / len(per_n)\n    print(json.dumps({\"metric\": round(metric, 6), \"ns\": ns, \"per_n\": per_n, \"records_beaten\": beaten,\n                      \"total_seconds\": round(time.perf_counter() - t_all, 1)}))\n\n\nif __name__ == \"__main__\":\n    main()\n","baseline":{"pack.py":"\"\"\"Baseline: a lattice of candidate points inside the container, spacing found by bisection so\nthat at least n fit. Deliberately naive; scores well below the records. Beat it.\"\"\"\n\nimport itertools\nimport math\n\nDIM = 2\n\n\ndef _boundary(c):\n    return 1.0 - math.hypot(c[0], c[1])\n\n\ndef _weight(i):\n    return 1.0\n\n\ndef _lattice(n, r):\n    \"\"\"Cubic lattice points at spacing 2r whose distance to the boundary is at least r.\"\"\"\n    lo, hi = ((-1.0,) * DIM, (1.0,) * DIM)\n    step = 2.0 * r\n    axes = []\n    for d in range(DIM):\n        k = int((hi[d] - lo[d]) / step) + 1\n        axes.append([lo[d] + r + i * step for i in range(k)])\n    pts = [p for p in itertools.product(*axes) if _boundary(p) >= r]\n    return pts\n\n\ndef pack(n, time_budget, seed):\n    # treat every object as the largest one when choosing the lattice spacing\n    wmax = max(_weight(i + 1) for i in range(n))\n    # find a feasible spacing by halving, then bisect between it and the last infeasible one\n    a = 1.0\n    while len(_lattice(n, a * wmax)) < n and a > 1e-9:\n        a /= 2\n    b = 2 * a\n    for _ in range(40):\n        m = (a + b) / 2\n        if len(_lattice(n, m * wmax)) >= n:\n            a = m\n        else:\n            b = m\n    pts = _lattice(n, a * wmax)\n    pts.sort(key=lambda p: -_boundary(p))   # keep the most interior points\n    return [tuple(p) for p in pts[:n]]\n"}}