TopCoder Statistics - Problem Statement

Problem Statement

Shopping before Christmas comes with many perils. One of them is the endless barrage of Christmas songs.

A few weeks ago Tommy needed to visit a shopping mall and spend visitTime consecutive seconds inside.

The shopping mall has a single tape with several distinct Christmas songs. They play the tape in an endless loop. The lengths of the songs on the tape (in seconds, in the order in which they are on the tape) are given in the int[] playTime.

Depending on when Tommy enters the mall, he may hear a different number of songs. Hearing only a part of a song still counts as hearing the song.

We are only interested in distinct songs Tommy hears, so if he hears the same song multiple times, we only count it once.

Assume Tommy will enter the mall at some integer offset from the beginning of the tape, and that all offsets are equally likely. (The set of possible offsets includes zero = the start of the tape but excludes the end of the tape, as that is already also the beginning of its next loop.)

Tommy is lucky if the number of distinct songs he has to hear is minimal. (I.e., there is no other entrance time for which he would hear fewer distinct songs.)

Return the probability that Tommy will be lucky.

Definition

Class:: ChristmasSongTrauma
Method:: fewest
Parameters:: int[], int
Returns:: double
Method signature:: double fewest(int[] playTime, int visitTime)
(be sure your method is public)

Notes

Return values with an absolute difference at most 1e-9 will be accepted as correct.
If a song stops playing at the same time Tommy enters the mall, or starts playing at the same time he leaves the mall, we do NOT count the song as heard by him.

Constraints

playTime will have between 1 and 1000 elements, inclusive.
Each element of playTime will be between 1 and 10^6, inclusive.
visitTime will be between 1 and 10^9, inclusive.

Examples

{50, 100, 100, 100, 70, 90, 90}

300

Returns: 0.0016666666666666668

The loop of Christmas songs has 600 seconds. Tommy will be in the mall for 300 seconds. If he enters exactly 50 seconds after the start of the tape, he will only hear the three longest songs while at the mall. If he enters at any other time, he will hear at least four distinct songs.
{100, 100, 100, 100, 100, 100, 100, 100}

301

Returns: 1.0

All songs on this tape have the exact same length. Regardless of when Tommy enters he will hear four of the songs.
{312}

15241235

Returns: 1.0

Tommy will be in the mall for a very long time. Regardless of when he enters, he will be treated to an endless loop of Last Christmas and he will slowly lose his remaining shreds of sanity and his will to live. Regardless of when he enters, he will hear exactly one distinct song. Per our definition in the problem statement this means that for all possible offsets he will be lucky.
{50, 100, 100, 100, 70, 90, 90}

310

Returns: 0.41

The extra 10 seconds (in comparison to Example 0) now mean that Tommy will hear at least four distinct songs. His chance of hearing exactly four is somewhat less than 50 percent.
{50, 100, 100, 100, 70, 90, 90}

550

Returns: 0.0016666666666666668
{50, 100, 100, 100, 70, 90, 90}

551

Returns: 1.0
{312}

47

Returns: 1.0
{2, 3}

1

Returns: 1.0
{2, 3}

2

Returns: 0.6
{2, 3}

3

Returns: 0.2
{2, 3}

4

Returns: 1.0
{2, 3}

5

Returns: 1.0
{2, 3}

6

Returns: 1.0
{2, 3}

7

Returns: 1.0
{2, 3}

8

Returns: 1.0
{2, 3}

111

Returns: 1.0
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459089299

Returns: 5.888728602465533E-5
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677837919

Returns: 1.0
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52299786

Returns: 1.1010601068776998E-4
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639474482

Returns: 0.00260365937413399
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58988259

Returns: 0.0028725952881848552
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405265501

Returns: 0.0031326054560611746
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531155025

Returns: 0.0026706222935259528
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6418271

Returns: 0.0051395890986667275
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802352634

Returns: 1.0
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298036304

Returns: 0.0014055445497238646
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303007229

Returns: 0.009271288237787013
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319147551

Returns: 5.162048568756263E-4
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1011, 269534, 85656, 13414, 315402, 741, 9410, 61, 382506, 247769, 12, 158, 21911, 465, 3870, 1677, 1, 3392, 160, 565225, 8, 44, 3323, 6096, 22, 15115, 85, 5838, 22, 26, 13, 23407, 23, 106171, 1, 4926, 394698, 56772, 805821, 6, 7}

696208

Returns: 0.053724611712489316
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343554, 10768, 537740, 305064, 882783, 448306, 752669, 57863, 112259, 828462, 841208, 577951, 522324, 44767, 299327, 411880, 4223, 690765, 782702, 938652, 698145, 173233, 283150, 194541}

444909798

Returns: 2.579764346187978E-4
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16351, 4078, 326785, 2, 79248, 19568, 1, 20, 118, 436, 42349, 2, 1, 637, 24, 42, 1442, 439398, 4, 1236, 320716, 5, 762839, 229614, 789134, 28238, 1, 1933, 125, 1, 7, 670, 1380, 49996, 10}

3

Returns: 0.9999775085466593
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27859683

Returns: 0.016224540419413864
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25843802

Returns: 0.00415102113437104
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674598027

Returns: 1.0
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170311613

Returns: 8.617098742321346E-4
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256408164

Returns: 0.006169638617978899
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345363764

Returns: 0.004168022581612513
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252781893

Returns: 5.763110782663884E-4
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37472460

Returns: 0.004513425645435734
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15118248

Returns: 0.0025132194519421895
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11172

Returns: 0.9852206171593704
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2920

Returns: 0.9961216656934858
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44010653

Returns: 4.97424502093016E-4
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27290952

Returns: 0.008201099285866686
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615063, 663000, 581714, 964677, 523451, 529910, 855218, 542372, 891714, 670500, 671849, 571295, 902604, 873312, 573295, 824147, 773721, 555053, 970153}

492801603

Returns: 0.0023074669478969677
{992553, 233, 62647, 201910, 1, 196654, 223, 1, 246, 2001, 47, 294806, 107, 179594, 2, 386598, 368786, 134, 70961, 1, 13, 63380, 10024, 134204, 78690, 6, 3, 395, 12345, 851389, 5, 6, 13829, 24937, 28735, 26800, 1801, 80, 13, 48861, 3047, 407, 84439, 5553, 15, 644316, 3, 4, 87, 10777, 468715, 212, 78213, 6283, 3, 57067, 54758, 9565, 560000, 789, 552, 466, 808522, 7, 9846, 44, 125373, 5002, 13993, 797170, 29, 124, 840, 4075, 2728, 105, 5, 7043, 973, 342004, 400167, 97761, 1, 11, 13, 1813, 18, 433, 64821, 5, 246, 783, 234, 4670, 424, 6, 1579, 95113, 6722, 1960, 144689, 7468, 3289, 2800, 1951, 2446, 211974, 90365, 10328, 21, 15, 356063, 8108, 15363, 252734, 2160, 48484, 5561, 13729, 6, 63361, 838254, 3630, 5, 99, 19, 34431, 36, 692568, 4, 1410, 293, 1, 16652, 11, 311, 1, 103394, 140836, 470, 121315, 31, 11, 1574, 8987, 98, 1, 28, 67377, 32823, 56285, 167751, 526506, 108, 285452, 57013, 13797, 273011, 151, 199, 7449, 5, 43833, 397, 60473, 403411, 1, 13930, 1, 41, 16710, 134, 32301, 7342, 24, 750818, 7, 485, 262262, 1982, 73825, 468174, 211, 1, 33900, 210, 5, 990570, 3193, 1053, 2, 1, 105826, 235, 1422, 1, 88, 36779, 680, 14310, 1, 4042, 74587, 83731, 247, 3986, 59, 470, 1761, 186, 1, 656916, 237, 1, 441, 3, 410, 15, 11, 474439, 808, 45406, 813625, 35, 4951, 2, 78, 51, 10, 275, 151446, 598812, 392190, 101185, 4387, 48, 2265, 175, 1010, 421515, 9, 508820, 1, 14, 33, 926, 9, 63, 6419, 71764, 1591, 180126, 37400, 321438, 8101, 990899, 63, 205327, 1878, 48686, 7, 434, 395, 2149, 59723, 24503, 3465, 1575, 2, 1434, 1, 219886, 3, 935441, 2, 3, 6, 128683, 424, 1, 98943, 17, 8, 22, 1, 834, 32, 1389, 294, 15823, 136, 66, 56, 475, 75341, 49246, 115, 24, 149619, 405, 1, 47236, 1, 1949, 919, 182587, 16791, 2689, 43, 38059, 26, 10399, 511, 111458, 7518, 8, 97668, 21571, 3, 192, 394644, 266, 11, 14068, 12576, 13, 11, 554300, 163677, 2227, 5934, 190, 8653, 247257, 214472, 383, 1256, 561964, 872, 994397, 2047, 1336, 300, 267, 2562, 917655, 51, 7, 7747, 28661, 225, 186, 225, 190550, 1, 729, 3394, 4, 7763, 89, 11833, 1846, 1, 198904, 6353, 41213, 28, 1, 2, 486221, 28, 1009, 2, 24981, 1, 2, 37, 17857, 33441, 1, 27, 13, 1541, 2542, 1928, 422470, 7120, 407, 24931, 45594, 406, 51, 1, 2, 9225, 2465, 94, 102, 3, 487, 2315, 847, 3, 65466, 130909, 2089, 5563, 5510, 329, 209595, 252, 544380, 138184, 9986, 237406, 2034, 297, 8, 155046, 6, 150, 593474, 485, 392401, 164, 13, 565, 636685, 103, 67996, 1981, 255, 945790, 1040, 129751, 30086, 3, 223, 61171, 120319, 15459, 5, 58, 941, 19, 1, 228, 5, 75500, 60, 200084, 3, 27362, 22, 90, 30854, 61, 1, 2, 844760, 753, 26, 88, 213266, 1, 573, 943, 65, 101644, 1039, 12, 478540, 24, 35082, 13332, 14, 888648, 107725, 413931, 64857, 18641, 6949, 1, 245, 236, 43324, 99343, 2907, 27, 195927, 199354, 14, 4366, 4, 318, 1, 58, 847, 691274, 219287, 4891, 65094, 17, 10, 6, 503606, 110000, 15, 71709, 7, 55, 1, 4, 536, 277446, 212, 473805, 1749, 165, 554070, 2, 103939, 11, 10, 59121, 3538, 1524, 13, 398, 3862, 43, 10591, 804148, 6, 5746, 591, 1, 2, 37341, 4, 43280, 3372, 47, 195, 15, 19419, 3, 5429, 5, 927757, 880397, 32313, 992, 73351, 125, 130424, 5, 77, 42, 235482, 6572, 37, 5, 15, 968142, 39, 206845, 94067, 1, 37827, 7520, 46, 7, 103918, 111, 1, 186434, 23437, 13067, 168173, 2, 5, 9, 12338, 3442, 57, 1273, 14268, 1, 3, 110649, 109, 96, 22847, 3668, 9, 9, 545785, 336, 2027, 7, 55, 1188, 18922, 335, 758, 106535, 1, 6, 2838, 43, 6576, 413549, 3804, 1301, 135484, 25, 399584, 1241, 3, 215, 2, 3499, 671077, 220, 20, 86, 133, 10720, 4511, 14351, 10, 3212, 19275, 29899, 1, 73, 28, 165, 49808, 975304, 6642, 686}

31626

Returns: 0.8672780365860135
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139490185

Returns: 8.211426845879876E-4
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436458683

Returns: 6.453421237345907E-4
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531321966

Returns: 1.0
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52708716

Returns: 0.0018111630331807019
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28582719

Returns: 1.541515504015728E-4
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178145689

Returns: 6.309301524078419E-4
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273053286

Returns: 0.0077065825416426465
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27458561

Returns: 0.0035622897977518097
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277177391

Returns: 7.702489737251271E-5
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457026460

Returns: 4.361754096559752E-4
{94, 920279, 92, 8, 169, 4, 5532, 26, 212630, 39294, 637825, 3035, 299005, 31, 60, 69, 2, 328299, 47231, 328, 586, 7, 23246, 14, 446222, 101896, 699, 31, 30, 12494, 33, 48724, 46, 1507, 4, 113, 1, 198, 134469, 92330, 45, 20, 43163, 90, 1, 20859, 605, 3, 56305, 56, 5395, 31, 30, 31, 14827, 27136, 118133, 34, 4326, 1457, 36644, 24633, 126, 43900, 215, 121, 136173, 1102, 150722, 1, 436017, 3408, 233394, 15, 47, 100, 1804, 89, 681064, 321946, 196071, 6, 25733, 268, 218225, 116903, 9404, 5845, 114, 98, 5957, 9255, 851, 4969, 586656, 14, 198, 2194, 93, 11, 76439, 2, 1, 18597, 1, 354, 11518, 636, 1, 401, 114905, 1116, 341087, 18, 465297, 876, 1, 363, 757265, 6, 413361, 25706, 53631, 29, 59, 3, 46, 12, 101923, 2659, 37, 664, 7, 364, 3104, 152, 1260, 1, 12, 2082, 127, 17, 10243, 174800, 691, 93, 39, 1309, 166137, 6389, 162, 8560, 2, 58818, 23, 431, 57, 1, 1851, 139938, 1, 25, 14999, 929, 353, 5, 26546, 890, 973, 13, 14, 10, 6203, 49409, 4175, 38970, 11574, 1, 57992, 83, 1108, 20, 6, 10, 1744, 1, 56, 11169, 456023, 291173, 379873, 1043, 533684, 2, 1, 150703, 6179, 22252, 1676, 1793, 17, 19907, 291346, 12386, 438201, 108, 108, 2027, 4, 1159, 64, 903, 15, 2, 3266, 30883, 819782, 236623, 3879, 10, 167, 110, 33, 57000, 37674, 743, 6413, 7, 79, 307, 85, 3, 5326, 85, 57, 1, 1, 156, 6, 456, 13, 63745, 103108, 30, 727808, 104, 1, 7, 3, 2119, 3813, 11, 245, 644339, 40, 10, 487, 7587, 89201, 88824, 28, 4397, 6670, 310770, 97, 1557, 11569, 359414, 7876, 393491, 297, 1, 1764, 117, 1778, 22, 4, 7645, 68988, 55, 4695, 41934, 153, 7, 3, 1468, 941377, 12877, 15616, 19, 12, 10, 57061, 412462, 202097, 648762, 1, 741, 3727, 27, 983391, 3, 333, 169, 109248, 441417, 6, 12437, 949244, 1020, 1, 107622, 3, 43, 119, 13444, 517084, 1, 141, 128122, 3, 39, 177588, 13471, 7549, 13883, 8, 685122, 134, 34, 22, 41240, 64478, 29506, 13, 907910, 6406, 8568, 913983, 285, 1019, 104094, 2, 236, 5513, 2035, 22, 4, 15301, 781613, 290, 2682, 1, 5, 61861, 192, 52751, 47487, 60, 3, 143, 1822, 1, 1, 6, 127, 510, 72590, 23953, 1799, 261671, 3040, 81, 1717, 29175, 2, 9236, 119, 14, 14039, 31919, 136, 95029, 335, 1, 638610, 9213, 161, 19275, 5771, 266624, 1, 17652, 1, 447443, 253, 37, 35181, 15, 42371, 1, 7823, 29, 63, 223639, 154, 28807, 52, 946605, 7, 104, 111, 27836, 6, 359671, 60080, 163, 167377, 1542, 1962, 3, 4, 43680, 55, 39570, 1, 641, 947634, 824624, 557479, 58956, 14, 49749, 2, 588, 40890, 610298, 1534, 24, 13, 713998, 25, 17, 159, 58358, 242, 35, 640, 2163, 37, 2028, 61, 174, 251948, 29373, 18, 38, 414, 11418, 10291, 636, 6, 21835, 1, 1, 272657, 30, 43474, 136, 59, 234545, 1187, 132943, 11121, 1011, 977, 7116, 378851, 3588, 1, 59306, 75287, 1, 1049, 2347, 1835, 1, 5, 15234, 14, 8612, 69, 1, 5, 63, 2370, 160684, 3, 5857, 2585, 20789, 29835, 36, 22115, 245, 5820, 505, 83, 60679, 19, 135, 23, 396695, 6, 305, 22510, 82510, 98, 82, 151, 67, 1, 1, 6, 589390, 28146, 13, 60, 557456, 77846, 294, 257693, 4557, 27, 17, 5, 212218, 39743, 6188, 60994, 3, 735, 2, 16803, 3763, 1, 11, 913779, 15, 61714, 1046, 99, 242, 13341, 10, 16, 32, 62431, 363463, 8733, 764, 7644, 2254, 23, 6, 694741, 7619, 3807, 12, 84938, 1550, 1, 5406, 3055, 6, 28, 296633, 1546, 210988, 10, 4296, 301524, 77029, 766, 509, 88935, 165405, 410, 362, 8, 30596, 13, 2976, 652182, 354453, 49, 5083, 7, 9584, 928, 13317, 924287, 138, 1652, 658, 1, 65360, 7, 303, 13, 54, 476760, 891621, 25658, 56, 928081, 505171, 13999, 502, 1, 1187, 194, 124, 27426, 1, 1275, 11820, 12213, 107, 10, 4, 959517, 270464, 2066, 101740, 213, 1681, 60029, 16009, 155, 53456, 51, 74, 23, 4, 56, 181, 10, 567228, 5, 57, 199497, 5, 34, 191, 30904, 22, 7, 7462, 1, 23, 61, 133, 850, 387845, 35, 1, 1, 3, 22038, 20487, 1, 178025, 967540, 1, 19757, 15448, 31910, 103, 4, 12511, 6361, 424395, 530, 105, 388, 156, 863525, 473, 89, 21, 335976, 7123, 84, 1426, 1, 511, 223700, 3027, 2840, 905739, 13649, 138, 262417, 887, 15614, 240836, 3104, 27499, 98, 4794, 43685, 106717, 5, 28191, 453, 37, 8, 493212, 242665, 273412, 5102, 969558, 918, 158484, 564, 65, 1, 678948}

57102778

Returns: 1.0
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29213210

Returns: 0.0024627367151699043
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293254324

Returns: 0.014437986467291285
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4

Returns: 0.9999939211088039
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64965592

Returns: 0.0015232932028422825
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4

Returns: 0.9999959954647595
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180256904

Returns: 1.0077653466452991E-6
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25358097

Returns: 0.008143024985319205
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49148039

Returns: 6.536122258626354E-4
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342536924

Returns: 0.0012770934255849251
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620035, 521236, 772769, 588412}

359980207

Returns: 1.9111481840446327E-4
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133390927

Returns: 1.0
{440, 26429, 249, 217, 4859, 6021, 2, 10, 1, 24731, 116977, 924, 218, 32, 11, 5, 56, 91137, 1, 228, 1, 244575, 27703, 52177, 58, 1, 263, 169046, 261, 364502, 6452, 14486, 86309, 12, 455, 753964, 395928, 757, 201, 14, 1, 9, 2368, 7, 94, 97, 53132, 134272, 542246, 113, 579888, 6460, 244409, 8200, 37, 298642, 1, 842, 6, 2, 859188, 73756, 8, 5938, 65323, 189271, 451, 3, 83, 278213, 14, 229, 58773, 23492, 13, 4072, 15393, 4102, 632, 889, 12317, 237, 2614, 28123, 80, 496, 871, 3, 6112, 424, 1887, 1, 140, 29, 960, 3, 132266, 1382, 15523, 12226, 156386, 46, 431, 93167, 11059, 3, 205, 1, 37148, 1633, 454, 24, 656, 106, 11, 3, 26, 925, 945, 42, 9860, 1, 865212, 81159, 9, 254, 8, 860159, 65, 10, 215, 638223, 18916, 2955, 307819, 13, 3, 209619, 1, 229, 205, 14, 12720, 421, 2, 639, 5, 17245, 1737, 10, 1, 2216, 52448, 3, 13205, 2, 6027, 484078, 240, 2, 955, 64002, 53897, 691074, 3, 7, 2302, 588, 11, 118, 8, 15083, 4, 54213, 31, 744157, 1, 254, 2650, 2249, 176301, 165353, 3080, 12703, 3049, 256865, 13, 1301, 18565, 8, 25971, 155, 342, 12202, 16274, 22, 18, 38, 1, 54217, 63, 184, 6, 158438, 5, 712, 12636, 8473, 8, 43896, 3893, 8091, 37980, 962, 3, 34, 81, 9416, 712, 12, 236032, 33, 6, 61, 2, 30, 2827, 104359, 24, 4862, 24, 124187, 55, 237352, 28, 1, 4167, 41, 779691, 4054, 2, 5068, 222880, 584, 97, 291, 1537, 124, 24548, 25039, 2615, 309, 4515, 218, 2342, 103467, 123270, 152287, 8, 946, 59428, 15, 61, 9851, 545117, 12709, 90869, 26494, 1898, 484, 83487, 24457, 2, 302, 893, 521401, 242, 7, 308963, 5598, 2, 56815, 26, 491401, 597726, 870478, 78, 13, 67454, 10207, 14719, 31970, 387390, 14, 374, 453565, 1, 4, 21, 212771, 24941, 383, 85939, 1836, 84600, 42078, 43, 50099, 1, 4020, 7, 134044, 63, 31484, 777913, 1, 37754, 210, 1771, 39527, 729, 43528, 10533, 146248, 47, 8908, 5747, 88, 11, 269, 101, 2209, 3, 94696, 6233, 118381, 12, 958020, 5186, 920242, 360, 341, 10, 151932, 151, 603, 391, 9, 71481, 87, 10, 7939, 2, 6, 24, 35, 38, 756132, 613810, 171472, 3340, 130468, 17, 94, 124033, 10977, 58731, 221202, 879, 749964, 27, 1, 9168, 8, 164, 1, 129897, 545936, 583789, 1, 26699, 227, 14, 1, 3433, 193455, 68548, 1271, 51, 33912, 117704, 127, 779, 572325, 74927, 26624, 27, 12, 1, 3, 777626, 1, 14, 11395, 98237, 75951, 122927, 88781, 89702, 106, 127, 65316, 15, 2179, 15, 14, 2, 14501, 1895, 4, 1754, 7073, 7, 8, 133, 96, 172, 2317, 107, 3, 514754, 343, 13, 640, 59840, 1, 89408, 6344, 3009, 2710, 294, 54, 57, 178410, 61458, 278, 50, 16, 1, 354937, 2820, 342, 8978, 6, 2358, 29, 25451, 819517, 226, 1952, 118126, 636363, 662, 30, 232, 3, 13, 36, 61935, 3, 1, 11, 132, 21083, 38745, 23, 2200, 130084, 668044, 837, 1, 3, 1, 5, 117496, 10510, 1, 455305, 1, 1, 646707, 112336, 2029, 19619, 1, 410815, 4, 21170, 217, 264, 91248, 894, 9, 42, 203, 355, 55, 9543, 6, 3, 1566, 109886, 188341, 17, 1, 5, 197, 336689, 224, 38, 2226, 3472, 11853, 1508, 510, 472209, 356, 12331, 69, 98032, 235, 15, 275749, 50, 12, 3, 251101, 32, 186826, 1215, 21, 7, 59137, 63727, 4, 25249, 3, 60, 10834, 801437, 519861, 184, 183893, 200091, 280563, 4391, 21275, 448410, 3869, 12, 31743, 700, 39662, 559, 31, 1396, 112, 27143, 5171, 621814, 155, 1912, 36633, 27, 43393, 872429, 27, 15465, 155776, 899, 94539, 108460, 3751, 19, 307, 225, 16191, 31774, 2943, 12, 551926, 1, 228, 14, 97, 221740, 11, 5, 647269, 1018, 50934, 73, 308090, 596023, 61, 219, 3, 320, 7376, 47469, 38, 4925, 1371, 633, 47115, 1405, 182, 11957, 227406, 495, 186, 4, 23548, 257, 15, 246155, 1, 1537, 910, 9765, 52594, 2, 443861, 16706, 3075, 27802, 16800, 24, 7, 224, 143024, 231512, 565, 79, 116, 55, 1796, 775, 231, 10, 24, 568, 20, 64, 11104, 19591, 498383, 162, 3545, 426, 3, 10, 929, 199374, 940, 2, 39, 872056, 172, 410992, 120304, 496, 1, 14962, 571180, 12306, 167, 19870, 3730, 29, 700, 1, 10721, 416505, 1, 15, 1, 37757, 1145, 41, 7494, 151438}

44152262

Returns: 2.4126463497819824E-4
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14

Returns: 0.9999731890254614
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427731, 525264, 370296}

166572796

Returns: 0.0024004253317635857
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814166, 733784, 253865, 523946, 599469, 869036, 367765, 576048, 472974, 304403, 388126, 430472, 379744, 701601, 170203, 308241, 716238, 825833, 905443, 43563, 453254, 961460, 58807, 813626, 269004, 380467, 69935, 542440, 801977, 164168, 59300, 764979, 407060, 348237, 575083, 669705, 976309, 267616, 414850, 735590, 420121, 392358, 86615, 179400, 146723, 112721, 951365, 973073, 161151, 403917}

64481969

Returns: 3.71153956287924E-4
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834137, 354461, 742690, 621975, 950793, 947200, 901043, 111064, 628545, 581306, 364679, 463840, 403494, 394282, 36097, 163721, 156373, 737275, 464517, 125455, 960988, 530516, 610748, 254970, 654814, 243722, 253387, 478083, 258884, 658620, 378773, 612147, 833055, 431737, 12018, 346024, 327081, 544946, 475232}

234319063

Returns: 0.0027634966140623577
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5250454

Returns: 0.0015605325480096378
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43249100

Returns: 0.0013004493578296827
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24565582

Returns: 0.007191331902160517
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223316224

Returns: 0.0030821145173624614
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3040171

Returns: 0.0026200932487000714
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172932051

Returns: 0.0023887958623061705
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91606757

Returns: 0.00238109440578106
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418426257

Returns: 0.006612605303406635
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20894262

Returns: 0.004698078034737038
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459558374

Returns: 0.016649740247622537
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937802, 262126, 136277, 154429, 264170, 193766, 916089}

69084467

Returns: 3.40806813608877E-4
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474837998

Returns: 2.0263623825673247E-4
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2697579

Returns: 0.0025657617912610018
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19921842

Returns: 0.001280083422960352
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420

Returns: 0.9991709299942583
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382591274

Returns: 0.0016344071055697325
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1

Returns: 1.0
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320119

Returns: 0.45931438637388666
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1442

Returns: 0.9980784436341942
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115945

Returns: 0.7837188137467046
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42572834

Returns: 0.0020082609949447047
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366579723

Returns: 0.006404402343436667
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17380

Returns: 0.9145324262808068
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936282, 963255, 919026, 581980, 531243, 625335, 913181, 828302, 800991, 760291, 619254, 800374, 680671, 583160, 746579, 527104, 567594, 964426, 674593, 824781, 643410, 914252, 799140, 817280}

101977817

Returns: 2.950208920420302E-4
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249038430

Returns: 0.004226496880743773
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17

Returns: 0.999815969158432
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318849993

Returns: 7.041300838922061E-4
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325762194

Returns: 6.43062608267053E-4
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667

Returns: 0.9991178353255128
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450744613

Returns: 0.013765181782341294
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2378718

Returns: 0.012737638297164185
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844413556

Returns: 1.0
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489049790

Returns: 0.01670150184388418
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948515}

62

Returns: 0.9998741994929314
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237802023

Returns: 0.002592870823204428
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1

Returns: 1.0
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5887

Returns: 0.988163488358745
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502373748

Returns: 0.006496274748019876
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651266787

Returns: 1.0
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333025124

Returns: 0.014121495052928878
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581033, 936940, 644257, 671847, 928355, 804662, 531426, 840684, 774843, 829631, 655295, 600729, 815718, 836033, 982797, 708806, 629817, 672919, 572303, 971876, 846566, 912581, 546319, 574448, 554771, 658435, 926894, 707408, 808396, 968496, 687127, 784591, 702347, 925442, 537202}

381

Returns: 0.9994944147833321
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44732068

Returns: 5.61063191508605E-5
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125713357

Returns: 0.0034478353538829934
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56806494

Returns: 0.11725039417107658
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18254

Returns: 0.10714285714285714
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500728837

Returns: 0.2750037012585056
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54332638

Returns: 0.23638347298550133
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5643

Returns: 0.9317738438761696
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91279057

Returns: 0.2007555839325495
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769285525

Returns: 1.0
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1

Returns: 1.0
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299722534

Returns: 0.001984187011321324
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359968

Returns: 0.019876457108697818
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531283

Returns: 0.004578134060339965
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129011

Returns: 0.7583604399506755
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346281324

Returns: 0.7004518478231397
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415538893

Returns: 0.16272905889234018
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716992

Returns: 0.2501136080449155
{27, 1777, 27, 2, 27, 1777, 27, 27, 1777, 2, 1777, 27, 2, 2, 1777, 2, 27, 2, 27, 27, 2, 27, 27, 2, 27, 27, 1777, 1777, 2, 2, 27, 2, 2, 2, 2, 1777, 1777, 2, 27, 2, 1777, 27, 27, 1777, 2, 2, 27, 1777, 2, 1777, 2, 27, 1777, 27, 2, 2, 1777, 1777, 2, 27, 1777, 27, 27, 2, 2, 1777, 2, 2, 2, 2, 2, 27, 1777, 2, 27, 1777, 1777, 1777, 27, 27, 1777, 1777, 1777, 27, 2, 27, 1777, 1777, 1777, 1777, 27, 1777, 1777, 2, 1777, 1777, 2, 27, 2, 2, 2, 1777, 27, 1777, 2, 27, 27, 1777, 27, 1777, 27, 1777, 1777, 2, 2, 2, 27, 1777, 2, 2, 27, 2, 27, 27, 2, 2, 1777, 1777, 27, 2, 1777, 2, 27, 2, 1777, 27, 27, 27, 2, 27, 2, 2, 2, 27, 1777, 1777, 27, 2, 2, 2, 27, 27, 27, 2, 27, 27, 1777, 27, 27, 2, 27, 27, 2, 27, 2, 27, 1777, 2, 1777, 27, 27, 1777, 1777, 2, 1777, 27, 27, 27, 2, 27, 27, 27, 2, 2, 27, 27, 27, 2, 27, 1777, 1777, 2, 27, 1777, 27, 1777, 2, 27, 1777, 27, 1777, 1777, 27, 1777, 1777, 1777, 2, 1777, 27, 27, 2, 27, 2, 27, 27, 1777, 27, 27, 27, 2, 1777, 27, 2, 2, 27, 1777, 1777, 2, 2, 27, 2, 1777, 1777, 2, 1777, 1777, 27, 1777, 1777, 27, 2, 2, 27, 2, 1777, 1777, 2, 2, 2, 1777, 2, 1777, 2, 1777, 2, 27, 27, 1777, 2, 1777, 2, 27, 1777, 2, 27, 1777, 1777, 1777, 2, 27, 1777, 1777, 27, 27, 1777, 27, 2, 27, 27, 27, 27, 1777, 2, 1777, 1777, 27, 27, 2, 27, 2, 1777, 1777, 27, 2, 2, 27, 27, 27, 2, 2, 1777, 2, 2, 27, 27, 1777, 1777, 2, 1777, 27, 27, 27, 1777, 1777, 2, 2, 27, 2, 1777, 27, 27, 27, 27, 1777, 27, 2, 2, 2, 1777, 2, 27, 2, 27, 27, 1777, 27, 27, 27, 1777, 27, 27, 1777, 1777, 27, 2, 1777, 27, 1777, 27, 1777, 2, 27, 2, 1777, 2, 2, 1777, 27, 2, 2, 2, 1777, 1777, 1777, 1777, 2, 1777, 27, 2, 27, 2, 1777, 1777, 27, 27, 1777, 27, 27, 2, 2, 27, 2, 1777, 1777, 27, 27, 2, 27, 1777, 2, 27, 2, 2, 27, 1777, 27, 27, 2, 1777, 1777, 27, 27, 2, 2, 1777, 1777, 1777, 27, 1777, 27, 2, 27, 2, 27, 27, 27, 1777, 27, 1777, 2, 2, 1777, 27, 27, 2, 2, 27, 2, 27, 1777, 2, 1777, 1777, 1777, 2, 1777, 2, 1777, 27, 27, 2, 2, 27, 27, 2, 27, 2, 1777, 2, 2, 1777, 27, 27, 2, 27, 1777, 2, 1777, 2, 1777, 27, 2, 27, 2, 1777, 1777, 27, 2, 2, 2, 2, 27, 27, 1777, 1777, 2, 2, 1777, 27, 27, 1777, 1777, 2, 2, 27, 1777, 27, 27, 27, 27, 1777, 2, 27, 2, 2, 27, 27, 27, 1777, 27, 27, 27, 1777, 1777, 1777, 1777, 2, 27, 1777, 1777, 2, 27, 2, 1777, 2, 1777, 27, 27, 2, 1777, 2, 2, 2, 2, 2, 27, 1777, 27, 1777, 27, 2, 27, 2, 27, 27, 1777, 27, 27, 1777, 1777, 1777, 2, 2, 27, 27, 1777, 2, 2, 1777, 2, 1777, 1777, 1777, 27, 27, 1777, 1777, 2, 2, 27, 27, 1777, 27, 1777, 27, 27, 1777, 2, 2, 2, 27, 2, 27, 1777, 1777, 27, 27, 1777, 27, 1777, 2, 27, 27, 27, 2, 27, 2, 2, 27, 2, 1777, 27, 27, 2, 1777, 2, 2, 2, 2, 27, 1777, 2, 1777, 1777, 27, 2, 1777, 27, 1777, 2, 1777, 2, 2, 1777, 1777, 1777, 2, 27, 27, 27, 27, 2, 1777, 27, 27, 1777, 1777, 1777, 27, 1777, 27, 1777, 2, 2, 1777, 1777, 27, 2, 27, 2, 27, 1777, 27, 27, 27, 2, 1777, 2, 27, 27, 2, 2, 27, 1777, 2, 27, 1777, 27, 27, 1777, 27, 2, 1777, 27, 27, 27, 27, 27, 2, 1777, 2, 1777, 2, 2, 27, 27, 27, 27, 1777, 27, 2, 27, 2, 27, 1777, 2, 1777, 27, 2, 1777, 1777, 2, 1777, 1777, 27, 1777, 1777, 1777, 27, 2, 1777, 2, 1777, 27, 2, 2, 2, 1777, 27, 1777, 1777, 1777, 27, 2, 2, 27, 27, 2, 2, 27, 2, 2, 27, 27, 2, 1777, 27, 2, 27, 1777, 2, 27, 2, 1777, 2, 1777, 27, 1777, 1777, 27, 1777, 1777, 2, 2, 1777, 27, 2, 27, 2, 2, 1777, 27, 2, 2, 1777, 1777, 27, 1777, 1777, 2, 2, 2, 27, 27, 27, 2, 1777, 27, 27, 2, 2, 1777, 27, 1777, 27, 1777, 27, 1777, 1777, 2, 1777, 1777, 2, 2, 1777, 27, 1777, 2, 2, 27, 27, 1777, 2, 2, 2, 1777, 1777, 27, 27, 27, 2, 2, 1777, 27, 27, 27, 27, 27, 27, 1777, 27, 2, 27, 2, 1777, 27, 1777, 1777, 2, 1777, 1777, 1777, 1777, 1777, 2, 27, 1777, 1777, 2, 27, 2, 27, 27, 2, 27, 2, 1777, 2, 27, 2}

395042

Returns: 0.004883815821136
{7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493, 7, 174493}

51917795

Returns: 0.4767851002865329
{555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312, 555428, 503552, 441312}

11627894

Returns: 0.7487402452322615
{395509, 395509, 432424, 432424, 696258, 205599, 613041, 205599, 613041, 432424, 205599, 696258, 205599, 613041, 696258, 432424, 696258, 613041, 613041, 395509, 205599, 205599, 613041, 613041, 613041, 613041, 696258, 613041, 613041, 696258, 395509, 613041, 613041, 613041, 613041, 432424, 696258, 613041, 395509, 205599, 432424, 205599, 395509, 395509, 432424, 432424, 205599, 613041, 205599, 205599, 205599, 432424, 205599, 432424, 205599, 205599, 613041, 432424, 696258, 432424, 613041, 205599, 395509, 696258, 395509, 696258, 613041, 395509, 432424, 432424, 395509, 395509, 432424, 432424, 205599, 395509, 432424, 696258, 432424, 613041, 696258, 696258, 696258, 395509, 432424, 613041, 696258, 205599, 205599, 613041, 696258, 432424, 205599, 205599, 613041, 613041, 205599, 395509, 395509, 432424, 613041, 395509, 205599, 205599, 696258, 205599, 395509, 696258, 395509, 395509, 205599, 395509, 696258, 205599, 205599, 395509, 613041, 696258, 395509, 432424, 395509, 205599, 613041, 696258, 395509, 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395509, 432424, 432424, 432424, 696258, 205599, 432424, 432424, 432424, 432424}

48955728

Returns: 1.2217235469455187E-4
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24309313

Returns: 0.004839015712233496
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1000000 }

999000100

Returns: 1.0
{3 }

2

Returns: 1.0
{50, 100, 100, 100, 70, 90, 90 }

1337

Returns: 1.0
{100, 100, 100 }

210

Returns: 1.0
{3, 3, 3 }

8

Returns: 1.0
{312 }

300

Returns: 1.0

Statistics

Problem Statement for "ChristmasSongTrauma"

Problem Statement

Definition

Notes

Constraints

Examples