Statistics

Problem Statement for "CubeOfDice"

Problem Statement

    +---+
    | D |
+---+---+---+---+
| E | A | B | F |
+---+---+---+---+
    | C |
    +---+

The ASCII art above shows the net of a general 6-sided die. There is a number written on each of the six sides. In the picture, these numbers are denoted A to F. (The net is folded so that the numbers are on the outside, but this information is not necessary to solve the following problem.)

The numbers A to F will be given to you as a int[] values, where A will be values[0], B will be values[1], and so on.

You have N^3 identical dice, each one matching the net shown above. You want to take all the dice, rotate some of them, and assemble a N×N×N cube. The cube will be standing on a table, hence only 5 of its sides will be visible.

You are given the int N, and the int[] values. Write a method that will return the smallest possible sum of the 5N^2 visible faces of the dice.

Definition

Class:
CubeOfDice
Method:
minimumSum
Parameters:
int, int[]
Returns:
long
Method signature:
long minimumSum(int N, int[] values)
(be sure your method is public)

Constraints

  • N will be between 1 and 1,000,000, inclusive.
  • values will contain exactly 6 elements.
  • Each element of values will be between 1 and 50, inclusive.

Examples

  1. 2

    {1,2,3,4,5,6}

    Returns: 36

    This input corresponds to 8 classical dice.

  2. 3

    {1,2,3,4,5,6}

    Returns: 69

    Now we have 27 classical dice.

  3. 1000000

    {50,50,50,50,50,50}

    Returns: 250000000000000

    The largest possible output. Note that all numbers are equal, hence in this case there is only one possible way to arrange the dice.

  4. 10

    {1,1,1,1,50,1}

    Returns: 500

    It is possible to rotate and arrange the dice so that no side with the 50 will be visible.

  5. 10

    {50,1,1,1,1,50}

    Returns: 696

  6. 10

    {50,1,1,1,50,1}

    Returns: 500

  7. 10

    {1,1,1,50,50,50}

    Returns: 500

  8. 1

    {1,2,3,4,5,6}

    Returns: 15

  9. 1

    {2,3,4,5,6,1}

    Returns: 15

  10. 1

    {3,4,5,6,1,2}

    Returns: 15

  11. 1

    {4,5,6,1,2,3}

    Returns: 15

  12. 1

    {5,6,1,2,3,4}

    Returns: 15

  13. 1

    {6,1,2,3,4,5}

    Returns: 15

  14. 565161

    { 1 , 1 , 1 , 1 , 1 , 1 }

    Returns: 1597034779605

  15. 921413

    { 1 , 1 , 1 , 1 , 1 , 2 }

    Returns: 4245009582845

  16. 798269

    { 1 , 1 , 1 , 1 , 2 , 1 }

    Returns: 3186166981805

  17. 220876

    { 1 , 1 , 1 , 1 , 2 , 2 }

    Returns: 243931036880

  18. 755646

    { 1 , 1 , 1 , 2 , 1 , 1 }

    Returns: 2855004386580

  19. 726415

    { 1 , 1 , 1 , 2 , 1 , 2 }

    Returns: 2638393761125

  20. 337414

    { 1 , 1 , 1 , 2 , 2 , 1 }

    Returns: 569241036980

  21. 622867

    { 1 , 1 , 1 , 2 , 2 , 2 }

    Returns: 1939816498445

  22. 196907

    { 1 , 1 , 2 , 1 , 1 , 1 }

    Returns: 193861833245

  23. 722074

    { 1 , 1 , 2 , 1 , 1 , 2 }

    Returns: 2606954307380

  24. 826498

    { 1 , 1 , 2 , 1 , 2 , 1 }

    Returns: 3415494720020

  25. 844774

    { 1 , 1 , 2 , 1 , 2 , 2 }

    Returns: 3568215555380

  26. 905125

    { 1 , 1 , 2 , 2 , 1 , 1 }

    Returns: 4096256328129

  27. 171279

    { 1 , 1 , 2 , 2 , 1 , 2 }

    Returns: 146682479209

  28. 751181

    { 1 , 1 , 2 , 2 , 2 , 1 }

    Returns: 2821364473809

  29. 223697

    { 1 , 1 , 2 , 2 , 2 , 2 }

    Returns: 250201739049

  30. 138408

    { 1 , 2 , 1 , 1 , 1 , 1 }

    Returns: 95783872320

  31. 913563

    { 1 , 2 , 1 , 1 , 1 , 2 }

    Returns: 4172986774845

  32. 913770

    { 1 , 2 , 1 , 1 , 2 , 1 }

    Returns: 4174878064504

  33. 500957

    { 1 , 2 , 1 , 1 , 2 , 2 }

    Returns: 1254789579249

  34. 884322

    { 1 , 2 , 1 , 2 , 1 , 1 }

    Returns: 3910126998420

  35. 772551

    { 1 , 2 , 1 , 2 , 1 , 2 }

    Returns: 2984175238005

  36. 657383

    { 1 , 2 , 1 , 2 , 2 , 1 }

    Returns: 2160762043449

  37. 72319

    { 1 , 2 , 1 , 2 , 2 , 2 }

    Returns: 26150188809

  38. 859083

    { 1 , 2 , 2 , 1 , 1 , 1 }

    Returns: 3690118004445

  39. 679948

    { 1 , 2 , 2 , 1 , 1 , 2 }

    Returns: 2311646413520

  40. 975054

    { 1 , 2 , 2 , 1 , 2 , 1 }

    Returns: 4753651514584

  41. 826128

    { 1 , 2 , 2 , 1 , 2 , 2 }

    Returns: 3412437361924

  42. 920225

    { 1 , 2 , 2 , 2 , 1 , 1 }

    Returns: 4234070253129

  43. 572673

    { 1 , 2 , 2 , 2 , 1 , 2 }

    Returns: 1639771824649

  44. 241984

    { 1 , 2 , 2 , 2 , 2 , 1 }

    Returns: 292783217148

  45. 256631

    { 1 , 2 , 2 , 2 , 2 , 2 }

    Returns: 329299403849

  46. 936606

    { 2 , 1 , 1 , 1 , 1 , 1 }

    Returns: 4386153996180

  47. 173328

    { 2 , 1 , 1 , 1 , 1 , 2 }

    Returns: 150212977924

  48. 280984

    { 2 , 1 , 1 , 1 , 2 , 1 }

    Returns: 394760041280

  49. 974241

    { 2 , 1 , 1 , 1 , 2 , 2 }

    Returns: 4745727630409

  50. 299591

    { 2 , 1 , 1 , 2 , 1 , 1 }

    Returns: 448773836405

  51. 666354

    { 2 , 1 , 1 , 2 , 1 , 2 }

    Returns: 2220138266584

  52. 87753

    { 2 , 1 , 1 , 2 , 2 , 1 }

    Returns: 38502945045

  53. 528655

    { 2 , 1 , 1 , 2 , 2 , 2 }

    Returns: 1397380545129

  54. 132483

    { 2 , 1 , 2 , 1 , 1 , 1 }

    Returns: 87758726445

  55. 976841

    { 2 , 1 , 2 , 1 , 1 , 2 }

    Returns: 4771091696409

  56. 728468

    { 2 , 1 , 2 , 1 , 2 , 1 }

    Returns: 2653328135120

  57. 806813

    { 2 , 1 , 2 , 1 , 2 , 2 }

    Returns: 3254736084849

  58. 478230

    { 2 , 1 , 2 , 2 , 1 , 1 }

    Returns: 1143519664504

  59. 671263

    { 2 , 1 , 2 , 2 , 1 , 2 }

    Returns: 2252975445945

  60. 125080

    { 2 , 1 , 2 , 2 , 2 , 1 }

    Returns: 78225032004

  61. 299192

    { 2 , 1 , 2 , 2 , 2 , 2 }

    Returns: 447581657852

  62. 381561

    { 2 , 2 , 1 , 1 , 1 , 1 }

    Returns: 727943983605

  63. 148513

    { 2 , 2 , 1 , 1 , 1 , 2 }

    Returns: 110280555849

  64. 199833

    { 2 , 2 , 1 , 1 , 2 , 1 }

    Returns: 199666139449

  65. 78446

    { 2 , 2 , 1 , 1 , 2 , 2 }

    Returns: 30769502144

  66. 709788

    { 2 , 2 , 1 , 2 , 1 , 1 }

    Returns: 2518995024720

  67. 565885

    { 2 , 2 , 1 , 2 , 1 , 2 }

    Returns: 1601129166129

  68. 125104

    { 2 , 2 , 1 , 2 , 2 , 1 }

    Returns: 78255054084

  69. 747516

    { 2 , 2 , 1 , 2 , 2 , 2 }

    Returns: 2793906831404

  70. 217562

    { 2 , 2 , 2 , 1 , 1 , 1 }

    Returns: 236666119220

  71. 495474

    { 2 , 2 , 2 , 1 , 1 , 2 }

    Returns: 1227472423384

  72. 407822

    { 2 , 2 , 2 , 1 , 2 , 1 }

    Returns: 831593918424

  73. 672621

    { 2 , 2 , 2 , 1 , 2 , 2 }

    Returns: 2262100429169

  74. 229636

    { 2 , 2 , 2 , 2 , 1 , 1 }

    Returns: 263663462484

  75. 836304

    { 2 , 2 , 2 , 2 , 1 , 2 }

    Returns: 3497028592508

  76. 715783

    { 2 , 2 , 2 , 2 , 2 , 1 }

    Returns: 2561732241705

  77. 140535

    { 2 , 2 , 2 , 2 , 2 , 2 }

    Returns: 197500862250

  78. 2

    { 15 , 7 , 16 , 12 , 8 , 35 }

    Returns: 212

  79. 3

    { 20 , 10 , 11 , 39 , 11 , 4 }

    Returns: 304

  80. 3

    { 37 , 49 , 9 , 34 , 47 , 41 }

    Returns: 1005

  81. 3

    { 23 , 3 , 27 , 46 , 6 , 26 }

    Returns: 551

  82. 2

    { 33 , 22 , 8 , 13 , 34 , 8 }

    Returns: 216

  83. 2

    { 48 , 23 , 15 , 31 , 46 , 22 }

    Returns: 388

  84. 2

    { 3 , 8 , 38 , 31 , 17 , 23 }

    Returns: 212

  85. 3

    { 17 , 37 , 10 , 20 , 2 , 24 }

    Returns: 278

  86. 3

    { 27 , 21 , 4 , 29 , 42 , 8 }

    Returns: 312

  87. 2

    { 24 , 1 , 14 , 13 , 21 , 11 }

    Returns: 148

  88. 3

    { 32 , 49 , 25 , 10 , 27 , 19 }

    Returns: 662

  89. 2

    { 22 , 2 , 9 , 10 , 39 , 23 }

    Returns: 176

  90. 2

    { 38 , 39 , 20 , 6 , 16 , 13 }

    Returns: 216

  91. 3

    { 3 , 24 , 46 , 28 , 41 , 27 }

    Returns: 571

  92. 2

    { 44 , 42 , 43 , 38 , 49 , 27 }

    Returns: 688

  93. 3

    { 11 , 14 , 47 , 20 , 16 , 24 }

    Returns: 579

  94. 3

    { 7 , 26 , 22 , 42 , 31 , 14 }

    Returns: 631

  95. 2

    { 44 , 32 , 2 , 6 , 48 , 2 }

    Returns: 160

  96. 3

    { 6 , 43 , 42 , 38 , 35 , 48 }

    Returns: 862

  97. 2

    { 43 , 42 , 42 , 10 , 12 , 34 }

    Returns: 312

  98. 3

    { 41 , 43 , 26 , 30 , 43 , 44 }

    Returns: 1478

  99. 3

    { 15 , 46 , 44 , 46 , 39 , 17 }

    Returns: 1175

  100. 2

    { 41 , 25 , 6 , 48 , 18 , 41 }

    Returns: 356

  101. 2

    { 16 , 11 , 44 , 38 , 24 , 7 }

    Returns: 296

  102. 2

    { 15 , 32 , 47 , 10 , 7 , 19 }

    Returns: 196

  103. 311153

    { 31 , 12 , 23 , 49 , 3 , 24 }

    Returns: 1452292625539

  104. 956244

    { 34 , 22 , 21 , 50 , 31 , 22 }

    Returns: 96012279341228

  105. 159959

    { 6 , 2 , 33 , 4 , 11 , 34 }

    Returns: 255871376154

  106. 382268

    { 6 , 4 , 37 , 18 , 7 , 7 }

    Returns: 2922582592808

  107. 862549

    { 17 , 46 , 5 , 43 , 11 , 13 }

    Returns: 18599810837361

  108. 125688

    { 39 , 40 , 17 , 39 , 48 , 32 }

    Returns: 1342800316772

  109. 90405

    { 7 , 30 , 2 , 42 , 48 , 7 }

    Returns: 81734256522

  110. 373761

    { 21 , 20 , 46 , 42 , 9 , 16 }

    Returns: 6286398761137

  111. 313941

    { 16 , 18 , 23 , 12 , 49 , 4 }

    Returns: 1971199121836

  112. 63089

    { 29 , 12 , 44 , 20 , 25 , 28 }

    Returns: 238817352956

  113. 331884

    { 48 , 43 , 11 , 23 , 36 , 43 }

    Returns: 6058150796808

  114. 36639

    { 30 , 1 , 36 , 30 , 17 , 10 }

    Returns: 6714719657

  115. 386220

    { 36 , 17 , 4 , 35 , 3 , 13 }

    Returns: 2237491415792

  116. 474619

    { 23 , 42 , 40 , 30 , 2 , 25 }

    Returns: 2252711687546

  117. 942051

    { 7 , 15 , 7 , 46 , 25 , 21 }

    Returns: 31061103031067

  118. 799803

    { 4 , 26 , 5 , 7 , 4 , 6 }

    Returns: 12793696776184

  119. 431465

    { 24 , 13 , 16 , 49 , 11 , 48 }

    Returns: 10238929800987

  120. 166775

    { 29 , 3 , 19 , 18 , 21 , 44 }

    Returns: 417228522359

  121. 969686

    { 22 , 32 , 41 , 2 , 36 , 33 }

    Returns: 9403064535680

  122. 516478

    { 23 , 1 , 21 , 32 , 43 , 50 }

    Returns: 1333830258828

  123. 538277

    { 16 , 49 , 29 , 15 , 8 , 47 }

    Returns: 11589715292648

  124. 189599

    { 26 , 6 , 13 , 8 , 50 , 30 }

    Returns: 1078436457678

  125. 14348

    { 44 , 9 , 47 , 33 , 16 , 37 }

    Returns: 9266684416

  126. 426998

    { 1 , 47 , 44 , 29 , 31 , 38 }

    Returns: 911732107468

  127. 326176

    { 14 , 2 , 6 , 46 , 44 , 24 }

    Returns: 1063918267408

  128. 979717

    { 41 , 27 , 14 , 47 , 19 , 22 }

    Returns: 67189217194902

  129. 572056

    { 48 , 44 , 7 , 25 , 9 , 36 }

    Returns: 11453691502756

  130. 278283

    { 26 , 25 , 10 , 38 , 50 , 5 }

    Returns: 1936046833585

  131. 603532

    { 6 , 35 , 3 , 15 , 4 , 13 }

    Returns: 5463767953620

  132. 144404

    { 16 , 48 , 19 , 29 , 5 , 39 }

    Returns: 521325587920

  133. 746736

    { 31 , 19 , 7 , 23 , 37 , 4 }

    Returns: 11152310995620

  134. 553406

    { 28 , 6 , 44 , 7 , 32 , 7 }

    Returns: 9187750452324

  135. 104330

    { 1 , 47 , 2 , 18 , 45 , 37 }

    Returns: 54424579308

  136. 656915

    { 48 , 40 , 27 , 13 , 9 , 24 }

    Returns: 19419200296433

  137. 17093

    { 28 , 40 , 2 , 45 , 26 , 50 }

    Returns: 2924988258

  138. 47284

    { 49 , 14 , 28 , 49 , 25 , 48 }

    Returns: 156509661752

  139. 141301

    { 36 , 39 , 15 , 32 , 16 , 21 }

    Returns: 1497449075499

  140. 285744

    { 50 , 26 , 31 , 13 , 17 , 20 }

    Returns: 5307235323644

  141. 392822

    { 31 , 8 , 31 , 28 , 31 , 38 }

    Returns: 6172427798812

  142. 686603

    { 42 , 14 , 44 , 15 , 16 , 14 }

    Returns: 32999657572634

  143. 1

    {1, 2, 3, 4, 5, 6 }

    Returns: 15

  144. 1

    {5, 15, 3, 2, 1, 6 }

    Returns: 17

  145. 10

    {1, 10, 10, 10, 10, 1 }

    Returns: 1184

  146. 1

    {2, 3, 4, 5, 6, 7 }

    Returns: 20

  147. 1000000

    {1, 2, 48, 50, 49, 1 }

    Returns: 5000008000180

  148. 999999

    {50, 49, 47, 46, 48, 49 }

    Returns: 229999556000210

  149. 2

    {1, 2, 1, 1, 2, 2 }

    Returns: 24

  150. 1000000

    {5, 4, 3, 1, 2, 6 }

    Returns: 5000008000008

  151. 2

    {1, 2, 3, 4, 5, 1 }

    Returns: 36

  152. 123

    {1, 1, 50, 50, 50, 1 }

    Returns: 75841

  153. 1

    {1, 2, 4, 8, 16, 32 }

    Returns: 31

  154. 50

    {35, 2, 1, 10, 4, 3 }

    Returns: 12900

  155. 10

    {2, 2, 1, 1, 2, 2 }

    Returns: 576

  156. 5

    {1, 50, 50, 50, 50, 1 }

    Returns: 1889

  157. 10

    {1, 5, 7, 6, 8, 1 }

    Returns: 808

  158. 2

    {1, 49, 48, 47, 1, 1 }

    Returns: 204

  159. 1000000

    {48, 1, 50, 10, 1, 10 }

    Returns: 5000071999964

  160. 123

    {6, 6, 6, 1, 1, 1 }

    Returns: 75645

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