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 A min problem of the year 2010
03-08-2010, 01:19 PM
Post: #1
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
A min problem of the year 2010
Let $$0 < a,b,c \in \mathbf{N}^+$$ such that $$a+b+c=2010$$
Find the smallest possible $$n \in \mathbf{N}^+$$ so that $$a! \ast b! \ast c! = 10^n M$$ where $$M$$ is not divisible by $$10$$.
04-09-2010, 03:13 PM
Post: #2
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
RE: A min problem of the year 2010
I think that we need the following result:
If $$p$$ is prime and $$m = \displaystyle{\sum_{k=1}^{\infty} \left[\frac{n}{p^k} \right]}$$, then $$n! = p^m G$$ for some $$G \in \mathbf{N}$$ with $$p \not | G$$
05-01-2010, 09:36 PM
Post: #3
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
RE: A min problem of the year 2010
There are 2712 solutions to make a!b!c! contain the smallest possible power of 10^m
where m = 493. Among them [624, 624, 762] makes a!b!c! the maximum.

There are 336675 arrangements of $$a,b,c \in \mathbb{N}^+$$ with $$a \le b \le c, \quad a+b+c=2010$$
And so the probability to hit a solution here is about $$0.008$$

[137, 624, 1249], [138, 623, 1249], [138, 624, 1248], [139, 622, 1249], [139, 623, 1248],
[139, 624, 1247], [142, 619, 1249], [142, 624, 1244], [143, 618, 1249], [143, 619, 1248],
[143, 623, 1244], [143, 624, 1243], [144, 617, 1249], [144, 618, 1248], [144, 619, 1247],
[144, 622, 1244], [144, 623, 1243], [144, 624, 1242], [147, 614, 1249], [147, 619, 1244],
[147, 624, 1239], [148, 613, 1249], [148, 614, 1248], [148, 618, 1244], [148, 619, 1243],
[148, 623, 1239], [148, 624, 1238], [149, 612, 1249], [149, 613, 1248], [149, 614, 1247],
[149, 617, 1244], [149, 618, 1243], [149, 619, 1242], [149, 622, 1239], [149, 623, 1238],
[149, 624, 1237], [162, 599, 1249], [162, 624, 1224], [163, 598, 1249], [163, 599, 1248],
[163, 623, 1224], [163, 624, 1223], [164, 597, 1249], [164, 598, 1248], [164, 599, 1247],
[164, 622, 1224], [164, 623, 1223], [164, 624, 1222], [167, 594, 1249], [167, 599, 1244],
[167, 619, 1224], [167, 624, 1219], [168, 593, 1249], [168, 594, 1248], [168, 598, 1244],
[168, 599, 1243], [168, 618, 1224], [168, 619, 1223], [168, 623, 1219], [168, 624, 1218],
[169, 592, 1249], [169, 593, 1248], [169, 594, 1247], [169, 597, 1244], [169, 598, 1243],
[169, 599, 1242], [169, 617, 1224], [169, 618, 1223], [169, 619, 1222], [169, 622, 1219],
[169, 623, 1218], [169, 624, 1217], [172, 589, 1249], [172, 594, 1244], [172, 599, 1239],
[172, 614, 1224], [172, 619, 1219], [172, 624, 1214], [173, 588, 1249], [173, 589, 1248],
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[173, 614, 1223], [173, 618, 1219], [173, 619, 1218], [173, 623, 1214], [173, 624, 1213],
[174, 587, 1249], [174, 588, 1248], [174, 589, 1247], [174, 592, 1244], [174, 593, 1243],
[174, 594, 1242], [174, 597, 1239], [174, 598, 1238], [174, 599, 1237], [174, 612, 1224],
[174, 613, 1223], [174, 614, 1222], [174, 617, 1219], [174, 618, 1218], [174, 619, 1217],
[174, 622, 1214], [174, 623, 1213], [174, 624, 1212], [187, 574, 1249], [187, 599, 1224],
[187, 624, 1199], [188, 573, 1249], [188, 574, 1248], [188, 598, 1224], [188, 599, 1223],
[188, 623, 1199], [188, 624, 1198], [189, 572, 1249], [189, 573, 1248], [189, 574, 1247],
[189, 597, 1224], [189, 598, 1223], [189, 599, 1222], [189, 622, 1199], [189, 623, 1198],
[189, 624, 1197], [192, 569, 1249], [192, 574, 1244], [192, 594, 1224], [192, 599, 1219],
[192, 619, 1199], [192, 624, 1194], [193, 568, 1249], [193, 569, 1248], [193, 573, 1244],
[193, 574, 1243], [193, 593, 1224], [193, 594, 1223], [193, 598, 1219], [193, 599, 1218],
[193, 618, 1199], [193, 619, 1198], [193, 623, 1194], [193, 624, 1193], [194, 567, 1249],
[194, 568, 1248], [194, 569, 1247], [194, 572, 1244], [194, 573, 1243], [194, 574, 1242],
[194, 592, 1224], [194, 593, 1223], [194, 594, 1222], [194, 597, 1219], [194, 598, 1218],
[194, 599, 1217], [194, 617, 1199], [194, 618, 1198], [194, 619, 1197], [194, 622, 1194],
[194, 623, 1193], [194, 624, 1192], [197, 564, 1249], [197, 569, 1244], [197, 574, 1239],
[197, 589, 1224], [197, 594, 1219], [197, 599, 1214], [197, 614, 1199], [197, 619, 1194],
[197, 624, 1189], [198, 563, 1249], [198, 564, 1248], [198, 568, 1244], [198, 569, 1243],
[198, 573, 1239], [198, 574, 1238], [198, 588, 1224], [198, 589, 1223], [198, 593, 1219],
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[198, 618, 1194], [198, 619, 1193], [198, 623, 1189], [198, 624, 1188], [199, 562, 1249],
[199, 563, 1248], [199, 564, 1247], [199, 567, 1244], [199, 568, 1243], [199, 569, 1242],
[199, 572, 1239], [199, 573, 1238], [199, 574, 1237], [199, 587, 1224], [199, 588, 1223],
[199, 589, 1222], [199, 592, 1219], [199, 593, 1218], [199, 594, 1217], [199, 597, 1214],
[199, 598, 1213], [199, 599, 1212], [199, 612, 1199], [199, 613, 1198], [199, 614, 1197],
[199, 617, 1194], [199, 618, 1193], [199, 619, 1192], [199, 622, 1189], [199, 623, 1188],
[199, 624, 1187], [212, 549, 1249], [212, 574, 1224], [212, 599, 1199], [212, 624, 1174],
[213, 548, 1249], [213, 549, 1248], [213, 573, 1224], [213, 574, 1223], [213, 598, 1199],
[213, 599, 1198], [213, 623, 1174], [213, 624, 1173], [214, 547, 1249], [214, 548, 1248],
[214, 549, 1247], [214, 572, 1224], [214, 573, 1223], [214, 574, 1222], [214, 597, 1199],
[214, 598, 1198], [214, 599, 1197], [214, 622, 1174], [214, 623, 1173], [214, 624, 1172],
[217, 544, 1249], [217, 549, 1244], [217, 569, 1224], [217, 574, 1219], [217, 594, 1199],
[217, 599, 1194], [217, 619, 1174], [217, 624, 1169], [218, 543, 1249], [218, 544, 1248],
[218, 548, 1244], [218, 549, 1243], [218, 568, 1224], [218, 569, 1223], [218, 573, 1219],
[218, 574, 1218], [218, 593, 1199], [218, 594, 1198], [218, 598, 1194], [218, 599, 1193],
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[219, 567, 1224], [219, 568, 1223], [219, 569, 1222], [219, 572, 1219], [219, 573, 1218],
[219, 574, 1217], [219, 592, 1199], [219, 593, 1198], [219, 594, 1197], [219, 597, 1194],
[219, 598, 1193], [219, 599, 1192], [219, 617, 1174], [219, 618, 1173], [219, 619, 1172],
[219, 622, 1169], [219, 623, 1168], [219, 624, 1167], [222, 539, 1249], [222, 544, 1244],
[222, 549, 1239], [222, 564, 1224], [222, 569, 1219], [222, 574, 1214], [222, 589, 1199],
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[223, 573, 1214], [223, 574, 1213], [223, 588, 1199], [223, 589, 1198], [223, 593, 1194],
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[223, 618, 1169], [223, 619, 1168], [223, 623, 1164], [223, 624, 1163], [224, 537, 1249],
[224, 538, 1248], [224, 539, 1247], [224, 542, 1244], [224, 543, 1243], [224, 544, 1242],
[224, 547, 1239], [224, 548, 1238], [224, 549, 1237], [224, 562, 1224], [224, 563, 1223],
[224, 564, 1222], [224, 567, 1219], [224, 568, 1218], [224, 569, 1217], [224, 572, 1214],
[224, 573, 1213], [224, 574, 1212], [224, 587, 1199], [224, 588, 1198], [224, 589, 1197],
[224, 592, 1194], [224, 593, 1193], [224, 594, 1192], [224, 597, 1189], [224, 598, 1188],
[224, 599, 1187], [224, 612, 1174], [224, 613, 1173], [224, 614, 1172], [224, 617, 1169],
[224, 618, 1168], [224, 619, 1167], [224, 622, 1164], [224, 623, 1163], [224, 624, 1162],
[237, 524, 1249], [237, 549, 1224], [237, 574, 1199], [237, 599, 1174], [237, 624, 1149],
[238, 523, 1249], [238, 524, 1248], [238, 548, 1224], [238, 549, 1223], [238, 573, 1199],
[238, 574, 1198], [238, 598, 1174], [238, 599, 1173], [238, 623, 1149], [238, 624, 1148],
[239, 522, 1249], [239, 523, 1248], [239, 524, 1247], [239, 547, 1224], [239, 548, 1223],
[239, 549, 1222], [239, 572, 1199], [239, 573, 1198], [239, 574, 1197], [239, 597, 1174],
[239, 598, 1173], [239, 599, 1172], [239, 622, 1149], [239, 623, 1148], [239, 624, 1147],
[242, 519, 1249], [242, 524, 1244], [242, 544, 1224], [242, 549, 1219], [242, 569, 1199],
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[243, 518, 1249], [243, 519, 1248], [243, 523, 1244], [243, 524, 1243], [243, 543, 1224],
[243, 544, 1223], [243, 548, 1219], [243, 549, 1218], [243, 568, 1199], [243, 569, 1198],
[243, 573, 1194], [243, 574, 1193], [243, 593, 1174], [243, 594, 1173], [243, 598, 1169],
[243, 599, 1168], [243, 618, 1149], [243, 619, 1148], [243, 623, 1144], [243, 624, 1143],
[244, 517, 1249], [244, 518, 1248], [244, 519, 1247], [244, 522, 1244], [244, 523, 1243],
[244, 524, 1242], [244, 542, 1224], [244, 543, 1223], [244, 544, 1222], [244, 547, 1219],
[244, 548, 1218], [244, 549, 1217], [244, 567, 1199], [244, 568, 1198], [244, 569, 1197],
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[244, 618, 1148], [244, 619, 1147], [244, 622, 1144], [244, 623, 1143], [244, 624, 1142],
[247, 514, 1249], [247, 519, 1244], [247, 524, 1239], [247, 539, 1224], [247, 544, 1219],
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[247, 594, 1169], [247, 599, 1164], [247, 614, 1149], [247, 619, 1144], [247, 624, 1139],
[248, 513, 1249], [248, 514, 1248], [248, 518, 1244], [248, 519, 1243], [248, 523, 1239],
[248, 524, 1238], [248, 538, 1224], [248, 539, 1223], [248, 543, 1219], [248, 544, 1218],
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[264, 623, 1123], [264, 624, 1122], [267, 494, 1249], [267, 499, 1244], [267, 619, 1124],
[267, 624, 1119], [268, 493, 1249], [268, 494, 1248], [268, 498, 1244], [268, 499, 1243],
[268, 618, 1124], [268, 619, 1123], [268, 623, 1119], [268, 624, 1118], [269, 492, 1249],
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[317, 594, 1099], [317, 599, 1094], [317, 619, 1074], [317, 624, 1069], [318, 443, 1249],
[318, 444, 1248], [318, 448, 1244], [318, 449, 1243], [318, 468, 1224], [318, 469, 1223],
[318, 473, 1219], [318, 474, 1218], [318, 493, 1199], [318, 494, 1198], [318, 498, 1194],
[318, 499, 1193], [318, 568, 1124], [318, 569, 1123], [318, 573, 1119], [318, 574, 1118],
[318, 593, 1099], [318, 594, 1098], [318, 598, 1094], [318, 599, 1093], [318, 618, 1074],
[318, 619, 1073], [318, 623, 1069], [318, 624, 1068], [319, 442, 1249], [319, 443, 1248],
[319, 444, 1247], [319, 447, 1244], [319, 448, 1243], [319, 449, 1242], [319, 467, 1224],
[319, 468, 1223], [319, 469, 1222], [319, 472, 1219], [319, 473, 1218], [319, 474, 1217],
[319, 492, 1199], [319, 493, 1198], [319, 494, 1197], [319, 497, 1194], [319, 498, 1193],
[319, 499, 1192], [319, 567, 1124], [319, 568, 1123], [319, 569, 1122], [319, 572, 1119],
[319, 573, 1118], [319, 574, 1117], [319, 592, 1099], [319, 593, 1098], [319, 594, 1097],
[319, 597, 1094], [319, 598, 1093], [319, 599, 1092], [319, 617, 1074], [319, 618, 1073],
[319, 619, 1072], [319, 622, 1069], [319, 623, 1068], [319, 624, 1067], [322, 439, 1249],
[322, 444, 1244], [322, 449, 1239], [322, 464, 1224], [322, 469, 1219], [322, 474, 1214],
[322, 489, 1199], [322, 494, 1194], [322, 499, 1189], [322, 564, 1124], [322, 569, 1119],
[322, 574, 1114], [322, 589, 1099], [322, 594, 1094], [322, 599, 1089], [322, 614, 1074],
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[469, 473, 1068], [469, 474, 1067], [469, 492, 1049], [469, 493, 1048], [469, 494, 1047],
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[474, 617, 919], [474, 618, 918], [474, 619, 917], [474, 622, 914], [474, 623, 913],
[474, 624, 912], [487, 499, 1024], [487, 524, 999], [487, 549, 974], [487, 574, 949],
[487, 599, 924], [487, 624, 899], [488, 498, 1024], [488, 499, 1023], [488, 523, 999],
[488, 524, 998], [488, 548, 974], [488, 549, 973], [488, 573, 949], [488, 574, 948],
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[489, 547, 974], [489, 548, 973], [489, 549, 972], [489, 572, 949], [489, 573, 948],
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[492, 524, 994], [492, 544, 974], [492, 549, 969], [492, 569, 949], [492, 574, 944],
[492, 594, 924], [492, 599, 919], [492, 619, 899], [492, 624, 894], [493, 493, 1024],
[493, 494, 1023], [493, 498, 1019], [493, 499, 1018], [493, 518, 999], [493, 519, 998],
[493, 523, 994], [493, 524, 993], [493, 543, 974], [493, 544, 973], [493, 548, 969],
[493, 549, 968], [493, 568, 949], [493, 569, 948], [493, 573, 944], [493, 574, 943],
[493, 593, 924], [493, 594, 923], [493, 598, 919], [493, 599, 918], [493, 618, 899],
[493, 619, 898], [493, 623, 894], [493, 624, 893], [494, 494, 1022], [494, 497, 1019],
[494, 498, 1018], [494, 499, 1017], [494, 517, 999], [494, 518, 998], [494, 519, 997],
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[494, 544, 972], [494, 547, 969], [494, 548, 968], [494, 549, 967], [494, 567, 949],
[494, 568, 948], [494, 569, 947], [494, 572, 944], [494, 573, 943], [494, 574, 942],
[494, 592, 924], [494, 593, 923], [494, 594, 922], [494, 597, 919], [494, 598, 918],
[494, 599, 917], [494, 617, 899], [494, 618, 898], [494, 619, 897], [494, 622, 894],
[494, 623, 893], [494, 624, 892], [497, 499, 1014], [497, 514, 999], [497, 519, 994],
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[497, 569, 944], [497, 574, 939], [497, 589, 924], [497, 594, 919], [497, 599, 914],
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[498, 524, 988], [498, 538, 974], [498, 539, 973], [498, 543, 969], [498, 544, 968],
[498, 548, 964], [498, 549, 963], [498, 563, 949], [498, 564, 948], [498, 568, 944],
[498, 569, 943], [498, 573, 939], [498, 574, 938], [498, 588, 924], [498, 589, 923],
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[498, 614, 898], [498, 618, 894], [498, 619, 893], [498, 623, 889], [498, 624, 888],
[499, 499, 1012], [499, 512, 999], [499, 513, 998], [499, 514, 997], [499, 517, 994],
[499, 518, 993], [499, 519, 992], [499, 522, 989], [499, 523, 988], [499, 524, 987],
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[499, 598, 913], [499, 599, 912], [499, 612, 899], [499, 613, 898], [499, 614, 897],
[499, 617, 894], [499, 618, 893], [499, 619, 892], [499, 622, 889], [499, 623, 888],
[499, 624, 887], [512, 624, 874], [513, 623, 874], [513, 624, 873], [514, 622, 874],
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[522, 619, 869], [522, 624, 864], [523, 613, 874], [523, 614, 873], [523, 618, 869],
[523, 619, 868], [523, 623, 864], [523, 624, 863], [524, 612, 874], [524, 613, 873],
[524, 614, 872], [524, 617, 869], [524, 618, 868], [524, 619, 867], [524, 622, 864],
[524, 623, 863], [524, 624, 862], [537, 599, 874], [537, 624, 849], [538, 598, 874],
[538, 599, 873], [538, 623, 849], [538, 624, 848], [539, 597, 874], [539, 598, 873],
[539, 599, 872], [539, 622, 849], [539, 623, 848], [539, 624, 847], [542, 594, 874],
[542, 599, 869], [542, 619, 849], [542, 624, 844], [543, 593, 874], [543, 594, 873],
[543, 598, 869], [543, 599, 868], [543, 618, 849], [543, 619, 848], [543, 623, 844],
[543, 624, 843], [544, 592, 874], [544, 593, 873], [544, 594, 872], [544, 597, 869],
[544, 598, 868], [544, 599, 867], [544, 617, 849], [544, 618, 848], [544, 619, 847],
[544, 622, 844], [544, 623, 843], [544, 624, 842], [547, 589, 874], [547, 594, 869],
[547, 599, 864], [547, 614, 849], [547, 619, 844], [547, 624, 839], [548, 588, 874],
[548, 589, 873], [548, 593, 869], [548, 594, 868], [548, 598, 864], [548, 599, 863],
[548, 613, 849], [548, 614, 848], [548, 618, 844], [548, 619, 843], [548, 623, 839],
[548, 624, 838], [549, 587, 874], [549, 588, 873], [549, 589, 872], [549, 592, 869],
[549, 593, 868], [549, 594, 867], [549, 597, 864], [549, 598, 863], [549, 599, 862],
[549, 612, 849], [549, 613, 848], [549, 614, 847], [549, 617, 844], [549, 618, 843],
[549, 619, 842], [549, 622, 839], [549, 623, 838], [549, 624, 837], [562, 574, 874],
[562, 599, 849], [562, 624, 824], [563, 573, 874], [563, 574, 873], [563, 598, 849],
[563, 599, 848], [563, 623, 824], [563, 624, 823], [564, 572, 874], [564, 573, 873],
[564, 574, 872], [564, 597, 849], [564, 598, 848], [564, 599, 847], [564, 622, 824],
[564, 623, 823], [564, 624, 822], [567, 569, 874], [567, 574, 869], [567, 594, 849],
[567, 599, 844], [567, 619, 824], [567, 624, 819], [568, 568, 874], [568, 569, 873],
[568, 573, 869], [568, 574, 868], [568, 593, 849], [568, 594, 848], [568, 598, 844],
[568, 599, 843], [568, 618, 824], [568, 619, 823], [568, 623, 819], [568, 624, 818],
[569, 569, 872], [569, 572, 869], [569, 573, 868], [569, 574, 867], [569, 592, 849],
[569, 593, 848], [569, 594, 847], [569, 597, 844], [569, 598, 843], [569, 599, 842],
[569, 617, 824], [569, 618, 823], [569, 619, 822], [569, 622, 819], [569, 623, 818],
[569, 624, 817], [572, 574, 864], [572, 589, 849], [572, 594, 844], [572, 599, 839],
[572, 614, 824], [572, 619, 819], [572, 624, 814], [573, 573, 864], [573, 574, 863],
[573, 588, 849], [573, 589, 848], [573, 593, 844], [573, 594, 843], [573, 598, 839],
[573, 599, 838], [573, 613, 824], [573, 614, 823], [573, 618, 819], [573, 619, 818],
[573, 623, 814], [573, 624, 813], [574, 574, 862], [574, 587, 849], [574, 588, 848],
[574, 589, 847], [574, 592, 844], [574, 593, 843], [574, 594, 842], [574, 597, 839],
[574, 598, 838], [574, 599, 837], [574, 612, 824], [574, 613, 823], [574, 614, 822],
[574, 617, 819], [574, 618, 818], [574, 619, 817], [574, 622, 814], [574, 623, 813],
[574, 624, 812], [587, 599, 824], [587, 624, 799], [588, 598, 824], [588, 599, 823],
[588, 623, 799], [588, 624, 798], [589, 597, 824], [589, 598, 823], [589, 599, 822],
[589, 622, 799], [589, 623, 798], [589, 624, 797], [592, 594, 824], [592, 599, 819],
[592, 619, 799], [592, 624, 794], [593, 593, 824], [593, 594, 823], [593, 598, 819],
[593, 599, 818], [593, 618, 799], [593, 619, 798], [593, 623, 794], [593, 624, 793],
[594, 594, 822], [594, 597, 819], [594, 598, 818], [594, 599, 817], [594, 617, 799],
[594, 618, 798], [594, 619, 797], [594, 622, 794], [594, 623, 793], [594, 624, 792],
[597, 599, 814], [597, 614, 799], [597, 619, 794], [597, 624, 789], [598, 598, 814],
[598, 599, 813], [598, 613, 799], [598, 614, 798], [598, 618, 794], [598, 619, 793],
[598, 623, 789], [598, 624, 788], [599, 599, 812], [599, 612, 799], [599, 613, 798],
[599, 614, 797], [599, 617, 794], [599, 618, 793], [599, 619, 792], [599, 622, 789],
[599, 623, 788], [599, 624, 787], [612, 624, 774], [613, 623, 774], [613, 624, 773],
[614, 622, 774], [614, 623, 773], [614, 624, 772], [617, 619, 774], [617, 624, 769],
[618, 618, 774], [618, 619, 773], [618, 623, 769], [618, 624, 768], [619, 619, 772],
[619, 622, 769], [619, 623, 768], [619, 624, 767], [622, 624, 764], [623, 623, 764],
[623, 624, 763], [624, 624, 762]
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