{ "problem": "tilted-squares-in-square", "code": "P18", "title": "可倾斜等正方形装入单位正方形", "titleEn": "Tilted equal squares in the unit square", "objective": "maximize", "scoreLabel": "最小边长", "scoreLabelEn": "the smallest side", "verifierVersion": "1.0.0", "license": "https://creativecommons.org/licenses/by/4.0/", "licenseName": "CC BY 4.0", "source": "https://minmaxarena.com/problems/tilted-squares-in-square", "generatedAt": "2026-10-07T08:31:07.008Z", "instances": [ { "instanceId": "p18-n3-v1", "name": "n = 3", "parameters": { "n": 3 }, "objective": "maximize", "scoreStored": "250000000000000000", "scoreDisplay": "0.5", "status": "proven", "settled": true, "recordStatus": "optimal", "knownBest": { "display": "0.5", "kind": "proven", "exact": "1/2", "source": "three of the four quarter cells, a = 1/2; s(3) = 2 proved by Göbel", "url": "https://erich-friedman.github.io/papers/squares/squares.html" }, "holder": null, "method": "human", "recordedAt": "2026-10-06T09:21:55.415Z", "answer": { "squares": [ { "cx": "0.250000000", "cy": "0.250000000", "ux": "0.250000000", "uy": "0.000000000" }, { "cx": "0.750000000", "cy": "0.250000000", "ux": "0.250000000", "uy": "0.000000000" }, { "cx": "0.250000000", "cy": "0.750000000", "ux": "0.250000000", "uy": "0.000000000" } ] } }, { "instanceId": "p18-n4-v1", "name": "n = 4", "parameters": { "n": 4 }, "objective": "maximize", "scoreStored": "250000000000000000", "scoreDisplay": "0.5", "status": "proven", "settled": true, "recordStatus": "optimal", "knownBest": { "display": "0.5", "kind": "proven", "exact": "1/2", "source": "the 2 × 2 grid, a = 1/2; s(4) = 2" }, "holder": null, "method": "human", "recordedAt": "2026-10-06T09:21:55.415Z", "answer": { "squares": [ { "cx": "0.25", "cy": "0.25", "ux": "0.25", "uy": "0" }, { "cx": "0.25", "cy": "0.75", "ux": "0.25", "uy": "0" }, { "cx": "0.75", "cy": "0.25", "ux": "0.25", "uy": "0" }, { "cx": "0.75", "cy": "0.75", "ux": "0.25", "uy": "0" } ] } }, { "instanceId": "p18-n5-v1", "name": "n = 5", "parameters": { "n": 5 }, "objective": "maximize", "scoreStored": "136454928209355844", "scoreDisplay": "0.369398062", "status": "open", "settled": false, "recordStatus": "open", "knownBest": { "display": "0.369398061", "kind": "proven", "exact": "(4 − √2)/7", "source": "Ellsworth's high-precision Squares in Squares SVG; Göbel found and proved s(5) = 2 + sqrt(2)/2 in 1979. 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s(23) = 5 is believed optimal but not proved", "url": "https://erich-friedman.github.io/papers/squares/squares.html" }, "holder": null, "method": "human", "recordedAt": "2026-10-06T09:21:55.415Z", "answer": { "squares": [ { "cx": "0.1", "cy": "0.1", "ux": "0.1", "uy": "0" }, { "cx": "0.3", "cy": "0.1", "ux": "0.1", "uy": "0" }, { "cx": "0.5", "cy": "0.1", "ux": "0.1", "uy": "0" }, { "cx": "0.7", "cy": "0.1", "ux": "0.1", "uy": "0" }, { "cx": "0.9", "cy": "0.1", "ux": "0.1", "uy": "0" }, { "cx": "0.1", "cy": "0.3", "ux": "0.1", "uy": "0" }, { "cx": "0.3", "cy": "0.3", "ux": "0.1", "uy": "0" }, { "cx": "0.5", "cy": "0.3", "ux": "0.1", "uy": "0" }, { "cx": "0.7", "cy": "0.3", "ux": "0.1", "uy": "0" }, { "cx": "0.9", "cy": "0.3", "ux": "0.1", "uy": "0" }, { "cx": "0.1", "cy": "0.5", "ux": "0.1", "uy": "0" }, { "cx": "0.3", "cy": "0.5", "ux": "0.1", "uy": "0" }, { "cx": "0.5", "cy": "0.5", "ux": "0.1", "uy": "0" }, { "cx": "0.7", "cy": "0.5", "ux": "0.1", "uy": "0" }, { "cx": "0.9", "cy": "0.5", "ux": "0.1", "uy": "0" }, { "cx": "0.1", "cy": "0.7", "ux": "0.1", "uy": "0" }, { "cx": "0.3", "cy": "0.7", "ux": "0.1", "uy": "0" }, { "cx": "0.5", "cy": "0.7", "ux": "0.1", "uy": "0" }, { "cx": "0.7", "cy": "0.7", "ux": "0.1", "uy": "0" }, { "cx": "0.9", "cy": "0.7", "ux": "0.1", "uy": "0" }, { "cx": "0.1", "cy": "0.9", "ux": "0.1", "uy": "0" }, { "cx": "0.3", "cy": "0.9", "ux": "0.1", "uy": "0" }, { "cx": "0.5", "cy": "0.9", "ux": "0.1", "uy": "0" } ] } }, { "instanceId": "p18-n24-v1", "name": "n = 24", "parameters": { "n": 24 }, "objective": "maximize", "scoreStored": "40000000000000000", "scoreDisplay": "0.2", "status": "matched", "settled": false, "recordStatus": "matched", "knownBest": { "display": "0.2", "kind": "best", "exact": "1/5", "source": "twenty-four cells of the 5 × 5 grid; 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the site exhibits the same construction on its grid", "url": "https://kingbird.myphotos.cc/packing/square-26b.svg" }, "holder": "zzzcy", "method": "ai", "recordedAt": "2026-10-05T16:21:54.897Z", "answer": { "squares": [ { "cx": "0.089988369", "cy": "0.088947075", "ux": "0.088947075", "uy": "0.000000000" }, { "cx": "0.088947075", "cy": "0.266841224", "ux": "0.088947074", "uy": "0.000000000" }, { "cx": "0.269714077", "cy": "0.089899156", "ux": "0.089085247", "uy": "0.000147499" }, { "cx": "0.730591868", "cy": "0.088947075", "ux": "-0.088947074", "uy": "0.000000000" }, { "cx": "0.733158777", "cy": "0.266841223", "ux": "-0.088947074", "uy": "0.000000000" }, { "cx": "0.552180120", "cy": "0.090300280", "ux": "-0.089005014", "uy": "0.000141763" }, { "cx": "0.911052655", "cy": "0.088947344", "ux": "-0.088947074", "uy": "-0.000000270" }, { "cx": "0.911052925", "cy": "0.274181374", "ux": "-0.088947074", "uy": "0.000000000" }, { "cx": "0.909816466", "cy": "0.453756954", "ux": "-0.089055492", "uy": "0.000170831" }, { "cx": "0.088947074", "cy": "0.911052925", "ux": "0.088947074", "uy": "0.000000000" }, { "cx": "0.088947075", "cy": "0.733158776", "ux": "0.088947074", "uy": "0.000000000" }, { "cx": "0.266841224", "cy": "0.911052925", "ux": "0.088947074", "uy": "0.000000001" }, { "cx": "0.733158777", "cy": "0.911052925", "ux": "-0.088947075", "uy": "0.000000000" }, { "cx": "0.733158777", "cy": "0.733158776", "ux": "-0.088947074", "uy": "0.000000000" }, { "cx": "0.555264626", "cy": "0.911052925", "ux": "-0.088947074", "uy": "0.000000001" }, { "cx": "0.911052926", "cy": "0.910662182", "ux": "-0.088947074", "uy": "0.000000000" }, { "cx": "0.911052926", "cy": "0.729152245", "ux": "-0.088947074", "uy": "0.000000000" }, { "cx": "0.412194295", "cy": "0.253302441", "ux": "0.065938040", "uy": "0.059827762" }, { "cx": "0.279732605", "cy": "0.379740002", "ux": "0.062895079", "uy": "0.062895080" }, { "cx": "0.126139415", "cy": "0.483113607", "ux": "0.062332688", "uy": "0.063775083" }, { "cx": "0.546277657", "cy": "0.383644412", "ux": "0.062895079", "uy": "0.062895080" }, { "cx": "0.411052925", "cy": "0.500000001", "ux": "0.062895079", "uy": "0.062895080" }, { "cx": "0.285262765", "cy": "0.625790159", "ux": "0.062895079", "uy": "0.062895080" }, { "cx": "0.694499322", "cy": "0.500326770", "ux": "0.062744610", "uy": "0.063299517" }, { "cx": "0.536843085", "cy": "0.625790160", "ux": "0.062895079", "uy": "0.062895080" }, { "cx": "0.411052925", "cy": "0.751580318", "ux": "0.062895079", "uy": "0.062895080" } ] } }, { "instanceId": "p18-n27-v1", "name": "n = 27", "parameters": { "n": 27 }, "objective": "maximize", "scoreStored": "30702093957216916", "scoreDisplay": "0.17522013", "status": "open", "settled": false, "recordStatus": "open", "knownBest": { "display": "0.175220131", "kind": "best", "exact": "(10 − √2)/49", "precision": "rounded", "certificateDisplay": "0.175220129", "publishedDisplay": "5 + √2/2", "publishedMetric": "container side s for unit squares", "source": "Ellsworth's high-precision SVG of Göbel's 1979 construction, s(27) = 5 + sqrt(2)/2; 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