297 lines
13 KiB
Markdown
297 lines
13 KiB
Markdown
---
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id: 5900f3cc1000cf542c50fedf
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title: '问题 96：数独'
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challengeType: 5
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forumTopicId: 302213
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dashedName: problem-96-su-doku
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---
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# --description--
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数独（日语含义为*数字位置*）是一个非常流行的解密游戏。 它的起源尚不清楚，但必须归功于莱昂哈德·欧拉（Leonhard Euler），他发明了一种类似的，但更加困难的解密游戏，名叫拉丁方块（Latin Squares）。 数独的目标是用数字替换 9X9 网格中的空白（或零），使得每行，每列和每个 3X3 小网格中都只包含 1 到 9 这 9 个数字。 下面是一个示例，包含一个经典的谜题及其对应解。
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<div style="margin: auto; background-color: white; padding: 10px; width: 80%; text-align: center;">
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<table border="0" cellpadding="0" cellspacing="0" align="center">
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<tbody>
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<tr>
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<td>
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<table cellpadding="5" cellspacing="0" border="1">
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<tbody>
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<tr>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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0 0 3<br />9 0 0<br />0 0 1
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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0 2 0<br />3 0 5<br />8 0 6
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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6 0 0<br />0 0 1<br />4 0 0
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</td>
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</tr>
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<tr>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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0 0 8<br />7 0 0<br />0 0 6
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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1 0 2<br />0 0 0<br />7 0 8
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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9 0 0<br />0 0 8<br />2 0 0
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</td>
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</tr>
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<tr>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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0 0 2<br />8 0 0<br />0 0 5
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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6 0 9<br />2 0 3<br />0 1 0
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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5 0 0<br />0 0 9<br />3 0 0
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</td>
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</tr>
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</tbody>
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</table>
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</td>
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<td width="50">
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<img src="images/spacer.gif" width="50" height="1" alt="" /><br />
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</td>
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<td>
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<table cellpadding="5" cellspacing="0" border="1">
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<tbody>
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<tr>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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4 8 3<br />9 6 7<br />2 5 1
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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9 2 1<br />3 4 5<br />8 7 6
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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6 5 7<br />8 2 1<br />4 9 3
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</td>
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</tr>
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<tr>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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5 4 8<br />7 2 9<br />1 3 6
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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1 3 2<br />5 6 4<br />7 9 8
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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9 7 6<br />1 3 8<br />2 4 5
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</td>
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</tr>
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<tr>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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3 7 2<br />8 1 4<br />6 9 5
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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6 8 9<br />2 5 3<br />4 1 7
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</td>
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<td style="font-family:'courier new';font-size:14pt; color: black; padding: 5px; border: 2px solid black;">
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5 1 4<br />7 6 9<br />3 8 2
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</td>
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</tr>
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</tbody>
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</table>
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</td>
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</tr>
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</tbody>
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</table>
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</div>
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一个构造良好的数独谜题应该只有一个唯一的解，可以通过逻辑解出，虽然可能需要采用“猜测和测试”方法来排除选项（对此有很多争议的意见）。 搜索的复杂性决定了谜题的难度；上面的示例很简单，因此可以通过直接的逻辑推理来解出答案。
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`puzzlesArr` 数组里有若干个数独谜题字符串，难度不一，但是每个谜题的解都是唯一的。
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通过解出 `puzzlesArr` 数组中所有谜题，返回所有解左上角三位数字之和；举个例子，483 就是上述例子中左上角的三位数字。
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# --hints--
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`suDoku(testPuzzles1)` 应该返回一个数字。
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```js
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assert(typeof suDoku(_testPuzzles1) === 'number');
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```
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`suDoku(testPuzzles1)` 应该返回 `1190`。
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```js
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assert.strictEqual(suDoku(_testPuzzles1), 1190);
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```
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`suDoku(testPuzzles2)` 应该返回 `24702`。
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```js
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assert.strictEqual(suDoku(_testPuzzles2), 24702);
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```
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# --seed--
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## --after-user-code--
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```js
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const _testPuzzles1 = [
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'003020600900305001001806400008102900700000008006708200002609500800203009005010300',
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'200080300060070084030500209000105408000000000402706000301007040720040060004010003',
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'000000907000420180000705026100904000050000040000507009920108000034059000507000000'
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];
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const _testPuzzles2 = [
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'003020600900305001001806400008102900700000008006708200002609500800203009005010300',
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'200080300060070084030500209000105408000000000402706000301007040720040060004010003',
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'000000907000420180000705026100904000050000040000507009920108000034059000507000000',
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'030050040008010500460000012070502080000603000040109030250000098001020600080060020',
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'020810740700003100090002805009040087400208003160030200302700060005600008076051090',
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'100920000524010000000000070050008102000000000402700090060000000000030945000071006',
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'043080250600000000000001094900004070000608000010200003820500000000000005034090710',
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'480006902002008001900370060840010200003704100001060049020085007700900600609200018',
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'000900002050123400030000160908000000070000090000000205091000050007439020400007000',
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'001900003900700160030005007050000009004302600200000070600100030042007006500006800',
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'000125400008400000420800000030000095060902010510000060000003049000007200001298000',
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'062340750100005600570000040000094800400000006005830000030000091006400007059083260',
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'300000000005009000200504000020000700160000058704310600000890100000067080000005437',
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'630000000000500008005674000000020000003401020000000345000007004080300902947100080',
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'000020040008035000000070602031046970200000000000501203049000730000000010800004000',
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'361025900080960010400000057008000471000603000259000800740000005020018060005470329',
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'050807020600010090702540006070020301504000908103080070900076205060090003080103040',
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'080005000000003457000070809060400903007010500408007020901020000842300000000100080',
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'003502900000040000106000305900251008070408030800763001308000104000020000005104800',
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'000000000009805100051907420290401065000000000140508093026709580005103600000000000',
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'020030090000907000900208005004806500607000208003102900800605007000309000030020050',
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'005000006070009020000500107804150000000803000000092805907006000030400010200000600',
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'040000050001943600009000300600050002103000506800020007005000200002436700030000040',
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'004000000000030002390700080400009001209801307600200008010008053900040000000000800',
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'360020089000361000000000000803000602400603007607000108000000000000418000970030014',
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'500400060009000800640020000000001008208000501700500000000090084003000600060003002',
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'007256400400000005010030060000508000008060200000107000030070090200000004006312700',
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'000000000079050180800000007007306800450708096003502700700000005016030420000000000',
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'030000080009000500007509200700105008020090030900402001004207100002000800070000090',
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'200170603050000100000006079000040700000801000009050000310400000005000060906037002',
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'000000080800701040040020030374000900000030000005000321010060050050802006080000000',
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'000000085000210009960080100500800016000000000890006007009070052300054000480000000',
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'608070502050608070002000300500090006040302050800050003005000200010704090409060701',
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'050010040107000602000905000208030501040070020901080406000401000304000709020060010',
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'053000790009753400100000002090080010000907000080030070500000003007641200061000940',
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'006080300049070250000405000600317004007000800100826009000702000075040190003090600',
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'005080700700204005320000084060105040008000500070803010450000091600508007003010600',
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'000900800128006400070800060800430007500000009600079008090004010003600284001007000',
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'000080000270000054095000810009806400020403060006905100017000620460000038000090000',
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'000602000400050001085010620038206710000000000019407350026040530900020007000809000',
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'000900002050123400030000160908000000070000090000000205091000050007439020400007000',
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'380000000000400785009020300060090000800302009000040070001070500495006000000000092',
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'000158000002060800030000040027030510000000000046080790050000080004070100000325000',
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'010500200900001000002008030500030007008000500600080004040100700000700006003004050',
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'080000040000469000400000007005904600070608030008502100900000005000781000060000010',
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'904200007010000000000706500000800090020904060040002000001607000000000030300005702',
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'000700800006000031040002000024070000010030080000060290000800070860000500002006000',
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'001007090590080001030000080000005800050060020004100000080000030100020079020700400',
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'000003017015009008060000000100007000009000200000500004000000020500600340340200000',
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'300200000000107000706030500070009080900020004010800050009040301000702000000008006'
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];
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```
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## --seed-contents--
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```js
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function suDoku(puzzlesArr) {
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return true;
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}
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// Only change code above this line
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const testPuzzles1 = [
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'003020600900305001001806400008102900700000008006708200002609500800203009005010300',
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'200080300060070084030500209000105408000000000402706000301007040720040060004010003',
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'000000907000420180000705026100904000050000040000507009920108000034059000507000000'
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];
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suDoku(testPuzzles1);
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```
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# --solutions--
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```js
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function suDoku(puzzlesArr) {
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function solve(puzzle) {
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for (let row = 0; row < 9; row++) {
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for (let col = 0; col < 9; col++) {
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if (puzzle[row][col] > 0) {
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continue;
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}
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const allowedNumbers = getAllowedNumbers(puzzle, row, col);
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for (let number = 1; number <= 9; number++) {
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if (allowedNumbers[number]) {
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puzzle[row][col] = number;
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if (solve(puzzle)) {
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return true;
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}
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}
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}
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puzzle[row][col] = 0;
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return false;
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}
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}
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return true;
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}
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function getAllowedNumbers(puzzle, row, col) {
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const allowed = new Array(10).fill(true);
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allowed[0] = false;
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for (let i = 0; i < 9; i++) {
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const numberInSameRow = puzzle[row][i];
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if (numberInSameRow > 0) {
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allowed[numberInSameRow] = false;
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}
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const numberInSameCol = puzzle[i][col];
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if (numberInSameCol > 0) {
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allowed[numberInSameCol] = false;
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}
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}
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const rowOfSubGrid = Math.floor(row / 3) * 3;
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const colOfSubGrid = Math.floor(col / 3) * 3;
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for (let rowInSubGrid = 0; rowInSubGrid < 3; rowInSubGrid++) {
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for (let colInSubGrid = 0; colInSubGrid < 3; colInSubGrid++) {
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const numberInSameSubGrid =
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puzzle[rowOfSubGrid + rowInSubGrid][colOfSubGrid + colInSubGrid];
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if (numberInSameSubGrid > 0) {
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allowed[numberInSameSubGrid] = false;
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}
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}
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}
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return allowed;
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}
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function parsePuzzle(string) {
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const puzzle = [];
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for (let row = 0; row < 9; row++) {
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puzzle.push(
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string
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.slice(row * 9, 9 + row * 9)
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.split('')
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.map(x => parseInt(x, 10))
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);
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}
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return puzzle;
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}
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let sum = 0;
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for (let i = 0; i < puzzlesArr.length; i++) {
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const puzzle = parsePuzzle(puzzlesArr[i]);
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if (solve(puzzle)) {
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sum += 100 * puzzle[0][0] + 10 * puzzle[0][1] + puzzle[0][2];
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}
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}
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return sum;
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}
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```
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