. . "\uB9C8\uC721\uAC01\uC9C4(\u9B54\u516D\u89D2\u9663, magic hexagon) \uB610\uB294 \uC721\uAC01\uC9C4(\u516D\u89D2\u9663)\uC740 \uC0AC\uAC01\uD615\uC73C\uB85C \uC774\uB8E8\uC5B4\uC9C4 \uB9C8\uBC29\uC9C4\uACFC\uB294 \uB2EC\uB9AC \uC721\uAC01\uD615\uC73C\uB85C \uC774\uB8E8\uC5B4\uC838 \uC788\uB294 \uB9C8\uBC95\uC9C4\uC774\uB2E4. 1\uBD80\uD130 \uAE4C\uC9C0\uC758 \uC815\uC218\uB97C \uD3EC\uD568\uD55C\uB2E4. \uC5EC\uAE30\uC11C \uC740 n\uBC88\uC9F8 \uC721\uAC01\uC218\uC774\uB2E4. \uC774\uB7EC\uD55C \uC721\uAC01\uC9C0\uC740 1\uCC28\uC640 3\uCC28\uB9CC \uAC00\uB2A5\uD558\uBA70, \uC65C\uB0D0\uD558\uBA74 \uD569\uC774 \uC815\uC218\uC5EC\uC57C\uD558\uAE30 \uB54C\uBB38\uC774\uB2E4. 4\uCC28 \uC774\uC0C1\uC740 1\uC774 \uC544\uB2CC \uC22B\uC790\uC5D0\uC11C \uC2DC\uC791\uD574\uC57C \uAC00\uB2A5\uD558\uB2E4."@ko . "Magisches Sechseck"@de . . . "\u041C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A"@ru . "\u041C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u0438\u043B\u0438 \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0433\u0435\u043A\u0441\u0430\u0433\u043E\u043D \u043F\u043E\u0440\u044F\u0434\u043A\u0430 \u2014 \u043D\u0430\u0431\u043E\u0440 \u0447\u0438\u0441\u0435\u043B, \u0440\u0430\u0441\u043F\u043E\u043B\u043E\u0436\u0435\u043D\u043D\u044B\u0439 \u0432 \u0446\u0435\u043D\u0442\u0440\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u043E\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u043E\u0439 \u0440\u0435\u0448\u0451\u0442\u043A\u0435 \u0441\u043E \u0441\u0442\u043E\u0440\u043E\u043D\u043E\u0439 \u0442\u0430\u043A\u0438\u043C \u043E\u0431\u0440\u0430\u0437\u043E\u043C, \u0447\u0442\u043E \u0441\u0443\u043C\u043C\u0430 \u0447\u0438\u0441\u0435\u043B \u0432 \u043A\u0430\u0436\u0434\u043E\u0439 \u0441\u0442\u0440\u043E\u043A\u0435 \u0432\u043E \u0432\u0441\u0435\u0445 \u043D\u0430\u043F\u0440\u0430\u0432\u043B\u0435\u043D\u0438\u044F\u0445 \u0440\u0430\u0432\u043D\u0430 \u043D\u0435\u043A\u043E\u0435\u0439 \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u043E\u043D\u0441\u0442\u0430\u043D\u0442\u0435 \u041E\u0431\u044B\u0447\u043D\u044B\u0439 \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0442\u043E\u043B\u044C\u043A\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430 (\u0441\u043B\u0443\u0447\u0430\u0439 \u0442\u0440\u0438\u0432\u0438\u0430\u043B\u0435\u043D, \u0438 \u0437\u0434\u0435\u0441\u044C \u0440\u0435\u0447\u044C \u043E \u043D\u0451\u043C \u0438\u0434\u0442\u0438 \u043D\u0435 \u0431\u0443\u0434\u0435\u0442) \u0438\u043B\u0438 \u0438 \u043C\u043E\u0436\u0435\u0442 \u0441\u043E\u0434\u0435\u0440\u0436\u0430\u0442\u044C \u0447\u0438\u0441\u043B\u0430 \u043E\u0442 \u0435\u0434\u0438\u043D\u0438\u0446\u044B \u0434\u043E \u0411\u043E\u043B\u0435\u0435 \u0442\u043E\u0433\u043E, \u0435\u0441\u043B\u0438 \u043D\u0435 \u0441\u0447\u0438\u0442\u0430\u0442\u044C \u0437\u0435\u0440\u043A\u0430\u043B\u044C\u043D\u044B\u0445, \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0442\u043E\u043B\u044C\u043A\u043E \u043E\u0434\u0438\u043D \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u043F\u043E\u0440\u044F\u0434\u043A\u0430 \u041C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u043F\u0443\u0431\u043B\u0438\u043A\u043E\u0432\u0430\u043B\u0441\u044F \u043C\u043D\u043E\u0433\u043E \u0440\u0430\u0437 \u043A\u0430\u043A \u043D\u043E\u0432\u043E\u0435 \u044F\u0432\u043B\u0435\u043D\u0438\u0435. \u041F\u0435\u0440\u0432\u043E\u043E\u0442\u043A\u0440\u044B\u0432\u0430\u0442\u0435\u043B\u0435\u043C, \u0432\u0435\u0440\u043E\u044F\u0442\u043D\u043E, \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u042D\u0440\u043D\u0441\u0442 \u0444\u043E\u043D \u0425\u0430\u0441\u0435\u043B\u044C\u0431\u0435\u0440\u0433 (\u043D\u0435\u043C. Ernst von Haselberg) \u0432 1887 \u0433\u043E\u0434\u0443."@ru . . . . . . . . . "\uB9C8\uC721\uAC01\uC9C4(\u9B54\u516D\u89D2\u9663, magic hexagon) \uB610\uB294 \uC721\uAC01\uC9C4(\u516D\u89D2\u9663)\uC740 \uC0AC\uAC01\uD615\uC73C\uB85C \uC774\uB8E8\uC5B4\uC9C4 \uB9C8\uBC29\uC9C4\uACFC\uB294 \uB2EC\uB9AC \uC721\uAC01\uD615\uC73C\uB85C \uC774\uB8E8\uC5B4\uC838 \uC788\uB294 \uB9C8\uBC95\uC9C4\uC774\uB2E4. 1\uBD80\uD130 \uAE4C\uC9C0\uC758 \uC815\uC218\uB97C \uD3EC\uD568\uD55C\uB2E4. \uC5EC\uAE30\uC11C \uC740 n\uBC88\uC9F8 \uC721\uAC01\uC218\uC774\uB2E4. \uC774\uB7EC\uD55C \uC721\uAC01\uC9C0\uC740 1\uCC28\uC640 3\uCC28\uB9CC \uAC00\uB2A5\uD558\uBA70, \uC65C\uB0D0\uD558\uBA74 \uD569\uC774 \uC815\uC218\uC5EC\uC57C\uD558\uAE30 \uB54C\uBB38\uC774\uB2E4. 4\uCC28 \uC774\uC0C1\uC740 1\uC774 \uC544\uB2CC \uC22B\uC790\uC5D0\uC11C \uC2DC\uC791\uD574\uC57C \uAC00\uB2A5\uD558\uB2E4."@ko . "Ein magisches Sechseck ist eine sechseckige Anordnung von Zahlen, bei der die Summen aller Reihen in den drei Richtungen jeweils den gleichen Wert ergeben. Insbesondere geht es darum, analog zum magischen Quadrat die ganzen Zahlen, beginnend ab 1, so in dem Sechseck anzuordnen, dass die Summen aller Reihen gleich sind. Abgesehen vom trivialen Fall , in dem das Sechseck nur aus einer Zahl besteht, ist dies nur bei der Seitenl\u00E4nge m\u00F6glich."@de . . . . . . . "7285"^^ . . "Hexagone magique"@fr . . . . . "Esagono magico"@it . "\u9B54\u516D\u89D2\u9663"@ja . "En math\u00E9matiques, un hexagone magique d'ordre n est un arrangement de nombres formant un gabarit hexagonal centr\u00E9 avec n cellules sur chaque c\u00F4t\u00E9. La somme des nombres dans chaque rang\u00E9e ou dans les trois directions font la m\u00EAme somme. Un hexagone magique normal contient tous les entiers allant de 1 \u00E0 3n2 \u2212 3n + 1. Il existe seulement deux arrangements respectant ces conditions, celui d'ordre 1 et celui d'ordre 3. De plus, la solution d'ordre 3 est unique. Meng en donne une preuve constructive ."@fr . . "\u9B54\u516D\u89D2\u9663\uFF08\u307E\u308D\u3063\u304B\u304F\u3058\u3093\uFF09\u306F\u3001\u9B54\u65B9\u9663\u306E\u516D\u89D2\u5F62\u7248\u3067\u3001\u5DE6\u659C\u3081\u30FB\u53F3\u659C\u3081\u30FB\u6A2A\u306E\u3044\u305A\u308C\u306E\u65B9\u5411\u306E\u548C\u3082\u7B49\u3057\u304F\u306A\u308B\u3088\u3046\u306B 1 \u304B\u3089\u59CB\u307E\u308B\u9023\u7D9A\u3057\u305F\u6570\u5B57\u3092\u3042\u3066\u306F\u3081\u305F\u3082\u306E\u3067\u3042\u308B\u3002 \n* \u5927\u304D\u30551\u548C = 1 \n* \u5927\u304D\u30553\u548C = 38 \u9B54\u516D\u89D2\u9663\u306F\u3001\u5927\u304D\u30551\u306E\u3082\u306E\u3068\u5927\u304D\u30553\u306E\u3082\u306E\u306E\u4E8C\u3064\u3057\u304B\u5B58\u5728\u3057\u306A\u3044\u3002\u307E\u305F\u3001\u93E1\u50CF\u30FB\u5BFE\u79F0\u306A\u3082\u306E\u3092\u9664\u304F\u3068\u5171\u306B1\u7A2E\u985E\u3057\u304B\u6570\u5B57\u306E\u5F53\u3066\u306F\u3081\u65B9\u304C\u5B58\u5728\u3057\u306A\u3044\u3053\u3068\u304C\u77E5\u3089\u308C\u3066\u3044\u308B\u3002 \u9B54\u516D\u89D2\u9663\u306F\u591A\u304F\u306E\u4EBA\u306B\u3088\u3063\u3066\u518D\u767A\u898B\u3055\u308C\u3066\u3044\u308B\u3002\u73FE\u5728\u5224\u660E\u3057\u3066\u3044\u308B\u6700\u3082\u53E4\u3044\u767A\u898B\u8005\u306F\u30A8\u30EB\u30F3\u30B9\u30C8\u30FB\u30D5\u30A9\u30F3\u30FB\u30CF\u30C3\u30BB\u30EB\u30D9\u30EB\u30B0\u3067\u30011887\u5E74\u306B\u767A\u8868\u3057\u3066\u3044\u308B\u3002"@ja . . . . . . "1106755460"^^ . . "\u9B54\u516D\u89D2\u9663\uFF08\u307E\u308D\u3063\u304B\u304F\u3058\u3093\uFF09\u306F\u3001\u9B54\u65B9\u9663\u306E\u516D\u89D2\u5F62\u7248\u3067\u3001\u5DE6\u659C\u3081\u30FB\u53F3\u659C\u3081\u30FB\u6A2A\u306E\u3044\u305A\u308C\u306E\u65B9\u5411\u306E\u548C\u3082\u7B49\u3057\u304F\u306A\u308B\u3088\u3046\u306B 1 \u304B\u3089\u59CB\u307E\u308B\u9023\u7D9A\u3057\u305F\u6570\u5B57\u3092\u3042\u3066\u306F\u3081\u305F\u3082\u306E\u3067\u3042\u308B\u3002 \n* \u5927\u304D\u30551\u548C = 1 \n* \u5927\u304D\u30553\u548C = 38 \u9B54\u516D\u89D2\u9663\u306F\u3001\u5927\u304D\u30551\u306E\u3082\u306E\u3068\u5927\u304D\u30553\u306E\u3082\u306E\u306E\u4E8C\u3064\u3057\u304B\u5B58\u5728\u3057\u306A\u3044\u3002\u307E\u305F\u3001\u93E1\u50CF\u30FB\u5BFE\u79F0\u306A\u3082\u306E\u3092\u9664\u304F\u3068\u5171\u306B1\u7A2E\u985E\u3057\u304B\u6570\u5B57\u306E\u5F53\u3066\u306F\u3081\u65B9\u304C\u5B58\u5728\u3057\u306A\u3044\u3053\u3068\u304C\u77E5\u3089\u308C\u3066\u3044\u308B\u3002 \u9B54\u516D\u89D2\u9663\u306F\u591A\u304F\u306E\u4EBA\u306B\u3088\u3063\u3066\u518D\u767A\u898B\u3055\u308C\u3066\u3044\u308B\u3002\u73FE\u5728\u5224\u660E\u3057\u3066\u3044\u308B\u6700\u3082\u53E4\u3044\u767A\u898B\u8005\u306F\u30A8\u30EB\u30F3\u30B9\u30C8\u30FB\u30D5\u30A9\u30F3\u30FB\u30CF\u30C3\u30BB\u30EB\u30D9\u30EB\u30B0\u3067\u30011887\u5E74\u306B\u767A\u8868\u3057\u3066\u3044\u308B\u3002"@ja . . . . . . . . . "A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant M. A normal magic hexagon contains the consecutive integers from 1 to 3n2 \u2212 3n + 1. It turns out that normal magic hexagons exist only for n = 1 (which is trivial, as it is composed of only 1 cell) and n = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof."@en . . "Magic hexagon"@en . "Un esagono magico di ordine n \u00E8 una disposizione di numeri tra loro distinti in una tabella esagonale composta da n celle per ogni lato, in modo che la somma dei numeri in ogni riga, in ciascuna delle tre direzioni possibili, abbia come somma la stessa costante magica. Un esagono magico normale ha il vincolo ulteriore di usare gli interi consecutivi da 1 a 3n\u00B2 \u2212 3n + 1. Si pu\u00F2 dimostrare che esistono esagoni magici normali solo per n = 1 (banale) e n = 3; inoltre, la soluzione di ordine 3 \u00E8 essenzialmente unica, a meno di rotazioni e riflessioni."@it . . . "\u041C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u0438\u043B\u0438 \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0433\u0435\u043A\u0441\u0430\u0433\u043E\u043D \u043F\u043E\u0440\u044F\u0434\u043A\u0430 \u2014 \u043D\u0430\u0431\u043E\u0440 \u0447\u0438\u0441\u0435\u043B, \u0440\u0430\u0441\u043F\u043E\u043B\u043E\u0436\u0435\u043D\u043D\u044B\u0439 \u0432 \u0446\u0435\u043D\u0442\u0440\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u043E\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u043E\u0439 \u0440\u0435\u0448\u0451\u0442\u043A\u0435 \u0441\u043E \u0441\u0442\u043E\u0440\u043E\u043D\u043E\u0439 \u0442\u0430\u043A\u0438\u043C \u043E\u0431\u0440\u0430\u0437\u043E\u043C, \u0447\u0442\u043E \u0441\u0443\u043C\u043C\u0430 \u0447\u0438\u0441\u0435\u043B \u0432 \u043A\u0430\u0436\u0434\u043E\u0439 \u0441\u0442\u0440\u043E\u043A\u0435 \u0432\u043E \u0432\u0441\u0435\u0445 \u043D\u0430\u043F\u0440\u0430\u0432\u043B\u0435\u043D\u0438\u044F\u0445 \u0440\u0430\u0432\u043D\u0430 \u043D\u0435\u043A\u043E\u0435\u0439 \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u043A\u043E\u043D\u0441\u0442\u0430\u043D\u0442\u0435 \u041E\u0431\u044B\u0447\u043D\u044B\u0439 \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0442\u043E\u043B\u044C\u043A\u043E \u043F\u043E\u0440\u044F\u0434\u043A\u0430 (\u0441\u043B\u0443\u0447\u0430\u0439 \u0442\u0440\u0438\u0432\u0438\u0430\u043B\u0435\u043D, \u0438 \u0437\u0434\u0435\u0441\u044C \u0440\u0435\u0447\u044C \u043E \u043D\u0451\u043C \u0438\u0434\u0442\u0438 \u043D\u0435 \u0431\u0443\u0434\u0435\u0442) \u0438\u043B\u0438 \u0438 \u043C\u043E\u0436\u0435\u0442 \u0441\u043E\u0434\u0435\u0440\u0436\u0430\u0442\u044C \u0447\u0438\u0441\u043B\u0430 \u043E\u0442 \u0435\u0434\u0438\u043D\u0438\u0446\u044B \u0434\u043E \u0411\u043E\u043B\u0435\u0435 \u0442\u043E\u0433\u043E, \u0435\u0441\u043B\u0438 \u043D\u0435 \u0441\u0447\u0438\u0442\u0430\u0442\u044C \u0437\u0435\u0440\u043A\u0430\u043B\u044C\u043D\u044B\u0445, \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u0435\u0442 \u0442\u043E\u043B\u044C\u043A\u043E \u043E\u0434\u0438\u043D \u043C\u0430\u0433\u0438\u0447\u0435\u0441\u043A\u0438\u0439 \u0448\u0435\u0441\u0442\u0438\u0443\u0433\u043E\u043B\u044C\u043D\u0438\u043A \u043F\u043E\u0440\u044F\u0434\u043A\u0430"@ru . "1629074"^^ . . . . "En math\u00E9matiques, un hexagone magique d'ordre n est un arrangement de nombres formant un gabarit hexagonal centr\u00E9 avec n cellules sur chaque c\u00F4t\u00E9. La somme des nombres dans chaque rang\u00E9e ou dans les trois directions font la m\u00EAme somme. Un hexagone magique normal contient tous les entiers allant de 1 \u00E0 3n2 \u2212 3n + 1. Il existe seulement deux arrangements respectant ces conditions, celui d'ordre 1 et celui d'ordre 3. De plus, la solution d'ordre 3 est unique. Meng en donne une preuve constructive ."@fr . . . . . . . "\uC721\uAC01\uC9C4"@ko . "A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant M. A normal magic hexagon contains the consecutive integers from 1 to 3n2 \u2212 3n + 1. It turns out that normal magic hexagons exist only for n = 1 (which is trivial, as it is composed of only 1 cell) and n = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof. The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is (1887)."@en . . . . . . "Un esagono magico di ordine n \u00E8 una disposizione di numeri tra loro distinti in una tabella esagonale composta da n celle per ogni lato, in modo che la somma dei numeri in ogni riga, in ciascuna delle tre direzioni possibili, abbia come somma la stessa costante magica. Un esagono magico normale ha il vincolo ulteriore di usare gli interi consecutivi da 1 a 3n\u00B2 \u2212 3n + 1. Si pu\u00F2 dimostrare che esistono esagoni magici normali solo per n = 1 (banale) e n = 3; inoltre, la soluzione di ordine 3 \u00E8 essenzialmente unica, a meno di rotazioni e riflessioni. L'esagono magico di ordine 3 \u00E8 stato pubblicato molte volte come una \"nuova\" scoperta. Il riferimento pi\u00F9 antico noto \u00E8 quello di Ernst von Haselberg, nel 1887. Anche se non esistono esagoni magici normali di ordine maggiore di 3, \u00E8 possibile trovare esagoni leggermente \"anormali\", che cio\u00E8 contengano s\u00EC tutte cifre consecutive, ma non inizino con 1. Ne sono un esempio gli esagoni di ordine 4 e 5 scoperti da Zahray Arsen: L'esagono di ordine 4 inizia con 3 e termina con 38; la costante magica \u00E8 111. Quello di ordine 5 inizia con 6 e termina con 66; la sua costante magica \u00E8 244. Al momento il pi\u00F9 grande esagono magico conosciuto \u00E8 stato trovato da Zahray Arsen il 22 marzo 2006: inizia con 2 e termina con 128, con una costante magica di 635. Tuttavia, un esagono magico pi\u00F9 grande, ma \"diverso (essendo formato da interi opposti)\", di ordine 8, \u00E8 stato creato da Louis K. Hoelbling il 5 febbraio 2006: Inizia con -84 e termina con 84, e la sua costante magica \u00E8 0."@it . "Ein magisches Sechseck ist eine sechseckige Anordnung von Zahlen, bei der die Summen aller Reihen in den drei Richtungen jeweils den gleichen Wert ergeben. Insbesondere geht es darum, analog zum magischen Quadrat die ganzen Zahlen, beginnend ab 1, so in dem Sechseck anzuordnen, dass die Summen aller Reihen gleich sind. Abgesehen vom trivialen Fall , in dem das Sechseck nur aus einer Zahl besteht, ist dies nur bei der Seitenl\u00E4nge m\u00F6glich."@de .