P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 158

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆,使共同半径尽可能大。

子题n = 158
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.0723420137983201074433129926805, 0.0723420137983201074433129926815]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆,圆心在 (1, 1),两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆:一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内;两两内部不重叠,相切允许
  • 目标让共同半径尽可能大
挑战这个纪录 ↗

当前展示:起步布局

2y0
0x2
已验证构造r = 0.072342013

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构:圆彼此顶住、顶住边界,常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1..14 与 19 已证明;其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型,大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 158 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路,审核采纳后可获得证明分。
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆,圆心在 (1, 1),所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位,直接写成小数,例如 "0.5",最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。

起步布局答案

{
  "centers": [
    [
      "1.103483594",
      "0.078132067"
    ],
    [
      "0.878235416",
      "0.080368158"
    ],
    [
      "1.245568045",
      "0.105435483"
    ],
    [
      "0.621993447",
      "0.152851615"
    ],
    [
      "0.991761038",
      "0.170064316"
    ],
    [
      "0.76552394",
      "0.171085251"
    ],
    [
      "1.370203847",
      "0.178915989"
    ],
    [
      "1.127139246",
      "0.221251783"
    ],
    [
      "0.494866272",
      "0.221932358"
    ],
    [
      "1.507793583",
      "0.223665654"
    ],
    [
      "0.879049563",
      "0.26078141"
    ],
    [
      "1.26166027",
      "0.274580818"
    ],
    [
      "0.627934941",
      "0.297413597"
    ],
    [
      "0.380026829",
      "0.309940143"
    ],
    [
      "1.62233116",
      "0.312065944"
    ],
    [
      "1.013968608",
      "0.313033865"
    ],
    [
      "1.404268437",
      "0.325318639"
    ],
    [
      "0.763527832",
      "0.347891671"
    ],
    [
      "1.148904487",
      "0.365242835"
    ],
    [
      "0.507733508",
      "0.377943612"
    ],
    [
      "0.901257132",
      "0.403750958"
    ],
    [
      "0.280268669",
      "0.414734119"
    ],
    [
      "1.721730095",
      "0.417200713"
    ],
    [
      "1.283797988",
      "0.417561199"
    ],
    [
      "0.652016092",
      "0.440079521"
    ],
    [
      "1.560738808",
      "0.442985193"
    ],
    [
      "1.036266196",
      "0.456050782"
    ],
    [
      "1.418614149",
      "0.470078534"
    ],
    [
      "0.407975349",
      "0.482737588"
    ],
    [
      "0.786909841",
      "0.492397248"
    ],
    [
      "1.171294017",
      "0.508535505"
    ],
    [
      "0.198018481",
      "0.5337651"
    ],
    [
      "0.542869097",
      "0.535055315"
    ],
    [
      "1.803572437",
      "0.536512484"
    ],
    [
      "0.923510842",
      "0.546713332"
    ],
    [
      "1.659411297",
      "0.548802053"
    ],
    [
      "1.306110178",
      "0.56105284"
    ],
    [
      "0.677762846",
      "0.587373041"
    ],
    [
      "1.05840459",
      "0.599031058"
    ],
    [
      "1.441038711",
      "0.613404421"
    ],
    [
      "0.430113743",
      "0.625717865"
    ],
    [
      "0.812656594",
      "0.639690768"
    ],
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      "0.287118475",
      "0.647759217"
    ],
    [
      "1.193220751",
      "0.651548393"
    ],
    [
      "0.13541753",
      "0.664884546"
    ],
    [
      "1.865867315",
      "0.667098915"
    ],
    [
      "1.577071952",
      "0.667771377"
    ],
    [
      "0.565007491",
      "0.678035591"
    ],
    [
      "1.721448372",
      "0.680848213"
    ],
    [
      "0.947550342",
      "0.692008494"
    ],
    [
      "1.328036913",
      "0.704065728"
    ],
    [
      "0.69990124",
      "0.730353318"
    ],
    [
      "1.082366504",
      "0.74452583"
    ],
    [
      "1.462853074",
      "0.756583064"
    ],
    [
      "0.446146226",
      "0.769533902"
    ],
    [
      "0.225055604",
      "0.778456072"
    ],
    [
      "0.834794988",
      "0.782671044"
    ],
    [
      "1.217182665",
      "0.797043165"
    ],
    [
      "1.637751681",
      "0.799116108"
    ],
    [
      "1.907099361",
      "0.8057834"
    ],
    [
      "0.586934226",
      "0.821048479"
    ],
    [
      "0.088665759",
      "0.826739383"
    ],
    [
      "0.969611149",
      "0.835188379"
    ],
    [
      "1.351998826",
      "0.8495605"
    ],
    [
      "1.771119531",
      "0.855209642"
    ],
    [
      "0.334595202",
      "0.872978791"
    ],
    [
      "0.721827974",
      "0.873366206"
    ],
    [
      "1.104427311",
      "0.887705714"
    ],
    [
      "1.521565293",
      "0.888819001"
    ],
    [
      "0.474044799",
      "0.911544032"
    ],
    [
      "0.856721722",
      "0.925683932"
    ],
    [
      "1.239243472",
      "0.940223049"
    ],
    [
      "1.65585718",
      "0.942662821"
    ],
    [
      "1.926265578",
      "0.949192339"
    ],
    [
      "0.217559552",
      "0.96036394"
    ],
    [
      "0.60886096",
      "0.964061367"
    ],
    [
      "0.072809539",
      "0.970551928"
    ],
    [
      "0.991537884",
      "0.978201267"
    ],
    [
      "1.375410096",
      "0.99233788"
    ],
    [
      "1.790285747",
      "0.998618582"
    ],
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      "0.356521936",
      "1.015991679"
    ],
    [
      "0.743754708",
      "1.016379094"
    ],
    [
      "1.126354045",
      "1.030718602"
    ],
    [
      "1.514170434",
      "1.033313928"
    ],
    [
      "0.495971533",
      "1.05455692"
    ],
    [
      "0.878648457",
      "1.06869682"
    ],
    [
      "1.261170206",
      "1.083235937"
    ],
    [
      "1.648628733",
      "1.087166168"
    ],
    [
      "1.922899733",
      "1.093837211"
    ],
    [
      "0.224340309",
      "1.104904758"
    ],
    [
      "0.630787694",
      "1.107074255"
    ],
    [
      "0.079507879",
      "1.115080818"
    ],
    [
      "1.013464618",
      "1.121214155"
    ],
    [
      "1.396069512",
      "1.135539334"
    ],
    [
      "1.78621",
      "1.143245191"
    ],
    [
      "0.765681443",
      "1.159391982"
    ],
    [
      "0.361159826",
      "1.160601353"
    ],
    [
      "1.148280779",
      "1.17373149"
    ],
    [
      "1.534829849",
      "1.176515382"
    ],
    [
      "0.517898267",
      "1.197569808"
    ],
    [
      "0.900575191",
      "1.211709708"
    ],
    [
      "1.283096941",
      "1.226248825"
    ],
    [
      "1.670555467",
      "1.230179056"
    ],
    [
      "1.897083703",
      "1.236199427"
    ],
    [
      "0.652714429",
      "1.250087143"
    ],
    [
      "0.108597837",
      "1.256810287"
    ],
    [
      "0.253281718",
      "1.257015987"
    ],
    [
      "1.035391352",
      "1.264227043"
    ],
    [
      "1.417996246",
      "1.278552222"
    ],
    [
      "0.411933768",
      "1.29608373"
    ],
    [
      "0.787608177",
      "1.30240487"
    ],
    [
      "1.170207514",
      "1.316744379"
    ],
    [
      "1.556756584",
      "1.31952827"
    ],
    [
      "1.78142917",
      "1.323133292"
    ],
    [
      "0.54674993",
      "1.348601065"
    ],
    [
      "0.922501925",
      "1.354722596"
    ],
    [
      "1.305023675",
      "1.369261714"
    ],
    [
      "0.15937178",
      "1.392292664"
    ],
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      "0.304055661",
      "1.392498365"
    ],
    [
      "0.681643678",
      "1.400918792"
    ],
    [
      "1.057318087",
      "1.407239932"
    ],
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      "1.667630286",
      "1.412482505"
    ],
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      "1.43992298",
      "1.42156511"
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      "0.438871822",
      "1.4450157"
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      "1.192134248",
      "1.459757267"
    ],
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      "1.801908417",
      "1.466360622"
    ],
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      "0.797580286",
      "1.487476119"
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      "0.573765571",
      "1.497333426"
    ],
    [
      "0.941873449",
      "1.498103948"
    ],
    [
      "1.557330303",
      "1.506116801"
    ],
    [
      "1.328480312",
      "1.514905588"
    ],
    [
      "0.230594595",
      "1.518232247"
    ],
    [
      "1.076689611",
      "1.550621283"
    ],
    [
      "1.691608433",
      "1.559994919"
    ],
    [
      "0.458612017",
      "1.588346759"
    ],
    [
      "0.678787463",
      "1.596851616"
    ],
    [
      "1.211505772",
      "1.603138618"
    ],
    [
      "0.81950702",
      "1.630489007"
    ],
    [
      "0.320533737",
      "1.631565465"
    ],
    [
      "1.48732987",
      "1.632739893"
    ],
    [
      "0.963800184",
      "1.641116836"
    ],
    [
      "1.345025088",
      "1.658870656"
    ],
    [
      "0.565079654",
      "1.686316704"
    ],
    [
      "1.622965951",
      "1.687359268"
    ],
    [
      "1.098616345",
      "1.693634171"
    ],
    [
      "0.427001375",
      "1.72953541"
    ],
    [
      "0.699128569",
      "1.740762607"
    ],
    [
      "1.232135661",
      "1.749366209"
    ],
    [
      "0.840010608",
      "1.773712855"
    ],
    [
      "1.508510028",
      "1.775865253"
    ],
    [
      "0.984303772",
      "1.784340684"
    ],
    [
      "1.36643159",
      "1.803199947"
    ],
    [
      "0.584952576",
      "1.829629421"
    ],
    [
      "1.118366312",
      "1.838753026"
    ],
    [
      "0.719034578",
      "1.884077086"
    ],
    [
      "1.252662241",
      "1.892586764"
    ],
    [
      "0.859919408",
      "1.917020592"
    ],
    [
      "1.004212571",
      "1.927648421"
    ]
  ],
  "radius": "0.072342013"
}
提交格式与技术细节需要编写程序或准备 JSON 答案时再查看+

子题参数

{
  "n": 158
}

起步布局答案

{
  "centers": [
    [
      "1.103483594",
      "0.078132067"
    ],
    [
      "0.878235416",
      "0.080368158"
    ],
    [
      "1.245568045",
      "0.105435483"
    ],
    [
      "0.621993447",
      "0.152851615"
    ],
    [
      "0.991761038",
      "0.170064316"
    ],
    [
      "0.76552394",
      "0.171085251"
    ],
    [
      "1.370203847",
      "0.178915989"
    ],
    [
      "1.127139246",
      "0.221251783"
    ],
    [
      "0.494866272",
      "0.221932358"
    ],
    [
      "1.507793583",
      "0.223665654"
    ],
    [
      "0.879049563",
      "0.26078141"
    ],
    [
      "1.26166027",
      "0.274580818"
    ],
    [
      "0.627934941",
      "0.297413597"
    ],
    [
      "0.380026829",
      "0.309940143"
    ],
    [
      "1.62233116",
      "0.312065944"
    ],
    [
      "1.013968608",
      "0.313033865"
    ],
    [
      "1.404268437",
      "0.325318639"
    ],
    [
      "0.763527832",
      "0.347891671"
    ],
    [
      "1.148904487",
      "0.365242835"
    ],
    [
      "0.507733508",
      "0.377943612"
    ],
    [
      "0.901257132",
      "0.403750958"
    ],
    [
      "0.280268669",
      "0.414734119"
    ],
    [
      "1.721730095",
      "0.417200713"
    ],
    [
      "1.283797988",
      "0.417561199"
    ],
    [
      "0.652016092",
      "0.440079521"
    ],
    [
      "1.560738808",
      "0.442985193"
    ],
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      "0.456050782"
    ],
    [
      "1.418614149",
      "0.470078534"
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      "0.704065728"
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      "0.69990124",
      "0.730353318"
    ],
    [
      "1.082366504",
      "0.74452583"
    ],
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      "1.462853074",
      "0.756583064"
    ],
    [
      "0.446146226",
      "0.769533902"
    ],
    [
      "0.225055604",
      "0.778456072"
    ],
    [
      "0.834794988",
      "0.782671044"
    ],
    [
      "1.217182665",
      "0.797043165"
    ],
    [
      "1.637751681",
      "0.799116108"
    ],
    [
      "1.907099361",
      "0.8057834"
    ],
    [
      "0.586934226",
      "0.821048479"
    ],
    [
      "0.088665759",
      "0.826739383"
    ],
    [
      "0.969611149",
      "0.835188379"
    ],
    [
      "1.351998826",
      "0.8495605"
    ],
    [
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      "0.855209642"
    ],
    [
      "0.334595202",
      "0.872978791"
    ],
    [
      "0.721827974",
      "0.873366206"
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    [
      "1.104427311",
      "0.887705714"
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    [
      "1.521565293",
      "0.888819001"
    ],
    [
      "0.474044799",
      "0.911544032"
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    [
      "0.856721722",
      "0.925683932"
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    [
      "1.239243472",
      "0.940223049"
    ],
    [
      "1.65585718",
      "0.942662821"
    ],
    [
      "1.926265578",
      "0.949192339"
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    [
      "0.217559552",
      "0.96036394"
    ],
    [
      "0.60886096",
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    ],
    [
      "0.072809539",
      "0.970551928"
    ],
    [
      "0.991537884",
      "0.978201267"
    ],
    [
      "1.375410096",
      "0.99233788"
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    [
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      "0.998618582"
    ],
    [
      "0.356521936",
      "1.015991679"
    ],
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  "radius": "0.072342013"
}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。 · 验证器 v1.0.0

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