23
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NASA has spent hundreds of millions of dollars gathering data about cave systems in one region of the planet Mars. The project just finished up, and analysts will soon be ready to start analyzing the complete list of all the lengths of cave passageways.

However, the manager of the NASA data center was unhappy about his recent pay cut, and had decided to quit, but not before causing problems. First, he used his manager-level privileges to log on to the backup servers and delete all of the backup data for the cave system project, knowing that he could later blame that on another department and he would not be held responsible. However, he knew that if he altered or deleted the data on the main server, then he would be caught and spend many years in prison. Instead, he created fake data to sabotage the study, and put the text file in the same folder as the actual data, with a similar name.

But he didn't stop there! Knowing that he was not very skilled in creating data, he didn't know how believable his fake data would be, so he asked two friends for help. He explained the task to them and requested 500 fake data points, ranging between 10 and 99,999 (since any passageway at least 10 feet long was classified as its own passageway). One of the friends was a mathematically-inclined (but not brilliant) intern from the data center who had recently quit, and the other one wasn't very math-minded, but did work for the state lottery. They did as they were asked, and he put their fake data files into the same folder as the other 2 files. Then he submitted his 2-week notice that he was leaving his job.

[2 weeks later]

Today, when the analysts started analysis, they saw the 4 different data files in the folder, all with nearly the same time stamp. They figured out what had happened, but they are not sure which data file is the right one.

Here are the comma-separated data files, containing the lengths of cave passageways (rounded to the nearest foot). The order of numbers in each file appears random, but is actually the order in which the data was recorded (which is not important).


File A

41050,98026,41938,39381,1045,97472,51703,30377,75372,75309,78343,99036,75408,70667,89774,70002,5097,61742,94616,23463,84846,88702,18597,74446,93228,33341,28485,98423,61427,84341,9799,70743,48021,21144,22010,64840,7475,19554,10217,27524,76597,9070,12286,93984,60312,28973,94055,63258,76096,7887,12612,99797,58620,98829,22238,92713,55891,25876,82304,25952,98076,59944,48189,92035,68632,64751,61087,87282,7781,15843,92947,92493,99250,96661,16096,17061,94168,42362,46406,55357,8555,20205,57032,66867,84021,96631,4734,63426,48889,45497,58226,52729,52658,82993,56513,65993,55056,66841,58809,85054,32010,75380,463,61616,78582,83395,79596,30370,56880,53140,61191,55123,91865,31791,88886,61104,77691,47831,48680,43460,8464,7055,25197,58750,90168,42552,56945,28278,71074,32556,45801,71878,52522,9398,48117,21029,87662,27554,61111,49741,94721,82090,43066,35250,86231,70691,85977,92585,80862,85569,47080,9258,60444,8530,74430,80369,58381,94000,33628,82054,39398,71689,98519,56625,71930,93326,92301,30603,78417,71845,5715,74574,21810,45677,78871,80477,96839,55668,40428,90622,79301,90130,71499,95016,35910,93,86361,35809,53697,93082,8228,50428,86155,37872,51383,95949,9311,3171,34503,90655,74333,79247,7153,86971,97154,93701,65094,37023,94144,63786,4318,21926,28855,46348,58618,74274,35808,32079,66531,23969,5522,898,41166,41472,24671,9782,76597,5400,75648,91068,40815,57376,9554,96045,9562,26241,36632,61529,33499,31301,2005,61342,5694,4395,66230,61262,89484,96819,7613,4199,63792,22623,90801,90762,63008,63633,49323,37831,23594,50201,45283,69978,77300,34212,984,27867,82006,66391,51280,22889,31130,73150,20555,80150,43159,6137,75266,60612,28163,45344,69124,8354,51287,47606,84299,43180,64470,9139,19235,92058,73896,56535,26198,27038,61373,77262,9992,49708,3796,5158,69022,87934,76263,80053,20509,77691,71432,61867,39256,45213,98428,86327,34782,4799,31437,90828,80365,35442,1653,34143,84737,85459,76016,80901,35774,8868,87978,61669,11906,1698,11713,86946,29694,95622,99909,54504,99841,23658,32968,56220,47136,40976,96521,96319,41605,99829,45741,89666,69240,81010,85316,86587,10428,34052,64339,51147,51463,8608,12962,30626,58716,3685,18802,14130,67344,67377,82035,78767,70738,37714,40841,70226,99876,64362,61647,56810,36430,14136,86488,40407,65471,61720,22341,27308,59065,76543,14382,10678,58363,48516,48894,26914,13476,22203,48235,59081,96912,84838,80694,63956,60770,74846,70739,77694,58521,19777,8751,29012,86265,27631,43357,22594,7302,28441,62299,60896,24142,61573,38647,24448,70070,85524,89316,58467,54355,60345,98618,49783,48003,87225,65663,87880,84674,98442,55193,54593,3156,32392,45233,31554,41171,96207,14652,16297,31240,84013,54925,48578,44692,97775,59327,2925,32952,76215,61846,85000,81905,13932,59659,60745,94063,7202,11593,47043,18661,11788,35255,9622,50155,32101,47339,73778,5614,92081,569,63034,57246,87807,32915,1912,69437,4537,15466,95584,18446,68437,35389,60802,70909,95128,60537,69851,31124,71714,16042,40076,58640,31628,38410,14356

File B

91691,88462,65945,6184,34325,44339,69698,4286,22141,20186,44975,43597,96829,77866,7252,54008,21738,19743,8101,60443,82617,6851,35716,63086,36076,47458,98331,72521,35251,38997,48146,38296,22915,35492,76652,94287,58518,8231,56425,14023,98610,81957,64723,60408,63160,29617,28192,50648,85324,65164,79515,94465,45487,16505,57011,50404,61971,34423,33873,60873,43760,84050,52344,64020,61635,37047,83692,26705,34174,68583,31705,57819,99367,53781,93042,2584,73720,67228,30392,13506,58101,38355,44658,56364,67926,58956,21839,21690,11318,8899,28636,95459,97586,72035,87291,20492,65393,50878,54978,64335,38670,95348,92502,57695,7060,55568,18068,24166,43499,18730,34039,46185,75185,77576,73050,31232,29463,36601,317,14914,85335,98240,80874,82321,53747,96363,31671,80999,5246,24337,24367,82528,91680,18931,59898,46036,35777,95686,62359,65077,84826,53667,68313,25407,17211,19482,5642,80960,15082,87462,94179,27129,80813,70555,20023,94550,97800,63378,37137,19655,78046,92770,53552,44759,28259,58320,11076,10635,79768,44202,56989,78594,93293,28129,3229,57561,176,21060,41536,92535,71585,59230,89258,72566,91356,28949,57950,64062,53612,50107,5289,37000,10650,64444,42826,61001,39581,52143,53813,88926,42770,66730,61860,73477,29846,95376,24869,90696,40818,86561,98154,17851,32828,46486,13913,67501,92912,48088,25454,19737,91336,2561,87618,5332,38007,18249,1292,75057,48859,63883,22607,97957,75335,30694,62332,63525,52708,79295,69202,23584,88030,13634,12907,71531,15959,19372,61677,13021,59064,60147,23883,48380,23336,58929,15531,39795,71411,93948,64617,45541,69894,7336,34395,79809,12797,93800,39257,14708,45522,85826,3532,64933,6568,32807,35960,33565,60315,93438,2629,32594,36507,82399,27730,83577,43704,73512,38106,28205,18796,75407,21575,8155,69672,59273,52280,53999,4071,26300,15114,26193,95288,59232,31951,54460,85169,75012,1057,84003,70446,33623,16002,49505,56582,36807,92824,3890,39204,68168,91911,87683,74667,16026,84036,38859,47884,48955,47293,51690,12548,19201,6583,32916,69812,75262,51360,21814,3542,11904,5549,80243,59004,86368,13076,87519,92905,16420,30597,85887,36612,9278,37300,74286,17600,59874,20179,51240,43653,35821,94017,75815,93242,88656,15629,74356,52223,23899,20467,27719,51548,16602,62403,95977,8708,28605,18709,97767,44601,51425,45192,76236,77358,13087,69218,97381,48966,49755,39053,61344,94556,73018,8014,1261,49952,39087,66684,42116,45594,42306,92432,47997,51843,52026,50225,70509,90687,58022,90903,41685,92828,50644,11434,14357,38632,49443,63988,68154,43980,76108,57144,34881,79318,17168,91165,61167,86338,43114,3008,69265,63087,44647,24765,68231,93843,41002,12308,29837,73680,53948,7802,66507,98169,86749,52515,36734,88973,35273,43551,20971,11902,25951,46933,86801,24532,14094,85484,12119,649,86074,57796,40663,4373,12522,35569,45552,82769,88487,87849,41418,52744,88319,49029,36650,66149,51882,39211,43936,6292,87335,99410,41702,51750,37708,98231,31065,50180,24532,73126,62976,18342,62243,30702,62598,95874,53871,79319,60885,52998,18545,59121,64350

File C

96954,54615,28496,90692,46070,90096,35182,34835,30444,39432,1746,41055,12120,21303,37632,41930,970,39960,3926,15703,18380,2538,85552,869,10,88240,31341,4192,59198,21165,1781,63350,39670,1932,13610,11994,29728,11921,10086,43625,633,31758,54131,181,484,1110,24021,19811,16561,12483,664,123,219,3029,49997,26601,29318,3787,42574,48659,15382,1031,16117,25652,16918,28469,176,45567,25463,53151,1338,67662,658,3194,64703,22706,13129,47515,744,23043,12713,138,14663,7179,1440,13959,20788,7429,15271,4572,10473,14,24864,1782,89982,24256,60550,65475,43065,28938,26099,12043,15889,862,49217,55160,13080,33780,51251,48870,24073,32038,18,20129,16589,30301,26395,29836,11,16382,26514,595,374,82549,135,30149,48380,26171,25942,55089,53658,689,364,55221,17284,33114,57967,35500,574,50755,1252,39432,17937,13848,16369,22757,20726,19814,1216,49543,196,4143,54882,56734,11157,17685,10461,28370,17205,32508,31700,40495,17199,582,1886,44196,29891,1289,23334,32919,35218,43559,1903,225,14659,12230,28536,39024,24261,16456,16686,30170,152,7973,71284,813,83671,22212,10227,24243,748,172,24998,51693,181,46951,297,21645,17273,11278,1949,16351,23584,505,29505,11432,724,44522,45811,1586,60504,18062,670,18146,52856,61636,21983,976,80611,14191,84781,99125,35759,71589,11643,90843,3516,177,48,26312,27208,18124,12451,58201,72,31227,54201,13747,723,59987,15516,14795,9711,30930,3365,49528,88826,77243,95170,80774,32145,33390,31622,73089,5816,10310,89394,1004,24638,4923,25126,91059,15961,1826,40161,48490,51643,18550,22305,37355,1634,78750,6947,58777,15246,13178,13316,78603,28552,18625,15853,36304,25904,10702,79349,51567,78165,12072,17732,10618,160,25225,4788,46976,1667,75940,95315,49286,12689,73443,46367,77414,52191,60115,78232,94845,56429,77,7465,73710,60092,49601,49409,83002,86608,71563,28246,59807,13653,1210,3703,83780,57681,27322,37867,1000,4757,10652,19718,64762,63461,3891,9975,1580,38368,11342,73451,10996,72366,61801,6011,26471,26542,66239,18355,9413,27143,13028,22342,15816,33859,32837,40541,69729,14702,68239,22583,39337,15778,33081,49219,63797,12971,27552,37537,140,2448,13390,15744,28067,61129,1851,46811,50059,40624,105,1194,20040,39627,14447,20,33,2473,39144,19819,22229,44792,85437,172,2661,67807,34,35869,88637,3054,62108,85445,91064,62141,50342,93530,12752,31831,1464,279,37514,80745,18020,1221,93376,338,23846,350,16399,22489,8178,2485,66521,21909,56424,19558,46144,55297,52489,31303,27341,14972,14118,20930,11197,43125,65617,77243,29448,26393,80375,14699,47237,12987,5792,248,16264,25320,11228,60540,19430,41909,87067,11267,16434,27299,19195,6497,95398,39801,28648,3991,28,5385,38402,97163,61444,15500,99896,66699,47976,20559,10910,3696,24333,18087,82598,3917,81305,22458,56347,25021,19376,79014,27470,19400,1745,19671,91712,59696,26494,65125,13665,13740,11031,91653,72433,9313,50376,1056,21492,24154,25706,44787,46269

File D

96356,54607,27997,89863,45152,89915,35049,34534,29457,38995,1404,40187,11757,21238,37485,41537,241,39452,3513,15385,17899,1295,15410,804,297,87992,30658,4049,58916,20701,1193,12597,39667,1580,13524,11280,29162,11163,9223,1591,352,30928,54004,342,51,1025,23300,19455,16128,11652,15284,393,741,2393,49269,26043,28877,3209,41649,47779,15330,792,15713,24673,16430,27823,159,45024,24912,53016,474,66977,315,2823,64020,22082,12130,47227,74099,22912,12166,258,14289,6702,923,13068,20624,6774,14757,3973,10146,380,24176,824,89293,24050,59718,65169,42410,28474,26053,11563,15384,26,48517,54163,12349,33388,50791,47910,23960,31354,902,19172,16181,29891,26064,29616,953,15506,26127,330,320,82262,22,29991,47658,25514,25814,54956,53464,75,503,54465,16454,32583,57601,34889,164,49933,657,39149,17447,13026,16164,22441,20283,19467,925,49541,309,3437,54079,56025,10436,17682,10442,28325,16711,31979,30986,40401,16906,498,1281,43498,29242,354,22950,32525,34462,43417,1518,113,14278,11243,27624,38299,23860,16422,16429,29519,69,7262,71244,81584,83233,21381,10223,23940,245,468,24640,50995,81,46430,402,21065,17114,10925,1920,15692,23022,4284,28976,10839,119,43896,45224,935,59745,17296,11,17961,52536,61613,21908,97363,79659,13470,14027,19,35248,70805,11279,89899,3083,62,248,25375,27192,18038,11608,57458,21,31096,53430,13556,182,59382,15432,14680,8951,30894,2570,49521,87922,76923,1594,80359,31177,33270,30786,72274,5198,9831,88395,591,23751,4583,24308,90266,15880,1031,39696,48037,51380,18503,22126,37219,1406,78194,6821,58412,15084,12434,13026,77776,168,17842,15471,35945,25699,10489,78583,51515,77641,11680,17021,10289,351,24407,4039,46631,1223,75566,94609,48347,11843,72933,45400,76587,51754,59134,77398,94068,55577,469,6588,72748,59532,49408,49295,82093,86582,70569,28170,59458,13358,549,2836,83721,56993,26384,37399,530,4561,10617,19527,64562,63084,3474,9900,1368,37695,11234,72778,10516,71921,61259,5920,26015,26326,65805,18005,8564,26146,12256,21585,15153,33537,32335,40260,68820,14298,68018,22572,38959,15038,32394,49193,62804,12864,27251,37172,535,2410,12948,15564,27929,60539,1228,45843,49084,39820,119,639,19192,38931,14206,879,478,2261,38685,19011,21706,44,84558,705,2018,66901,326,34920,87856,2302,61514,84,90552,61745,49484,93362,12065,31430,936,76,36753,80355,17509,811,92853,578,23386,403,15845,21757,7908,2337,66385,21055,56304,19282,46090,54762,51919,30628,26465,14846,14029,20459,10562,42373,64,77103,28952,26152,79668,14067,46849,12441,5484,420,15571,24881,11112,60359,19334,41577,86574,10462,15930,26920,18405,6265,94873,38812,27834,3274,916,4865,37937,96333,60844,14525,99097,66118,47729,19664,10104,3050,23489,17592,82379,3857,80827,21751,55564,24973,18913,78279,26649,18781,1457,19296,90899,59215,25594,64951,13162,13504,10265,91326,71539,9247,49695,900,21420,23276,25458,199,45594

Can you figure out which data file is most likely the real one, and why?

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  • 2
    $\begingroup$ Why isn't NASA using the metric system? $\endgroup$ – Engineer Toast Jun 18 '15 at 12:40
  • $\begingroup$ @EngineerToast Good point, but it's because the setting for this question is pre-2007 :) $\endgroup$ – JLee Jun 18 '15 at 14:25
  • $\begingroup$ We have data on Mars cave systems and we're not in the future? You have discovered a major cover-up in the scientific community. What are they hiding? Are you in on it? Have we found sentient Martian life? Are you sentient Martian life? Quick, what am I thinking? Hrnnnnghh. Wrong! I was thinking if Humpty Dumpty is an egg , what species is the thing inside of him? Another egg? $\endgroup$ – Engineer Toast Jun 18 '15 at 15:27
  • $\begingroup$ @EngineerToast Another excellent point! However, cover-ups aren't so rare, and it seems that the larger they are, the less anyone will be able to see the truth: See the great points made in Zeitgeist: The Movie 2007 and Moon Landing Hoax for perhaps an overload to the world as you know it (or maybe not, since you posted Down the Rabbit Hole, which mentions that jet fuel can't melt steel beams). $\endgroup$ – JLee Jun 18 '15 at 17:30
34
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The real data set is:

File D

First clue:

If you look at the frequency of values that start with the number 1, the frequencies are (6.8%, 11.2%, 30.0%, 27.0%). For any real world measurements, the frequency of the first digit follows Benford's Law, where the frequency of 1s is much more than the frequency of 9s. Benford's Law predicts that the frequency of numbers starting with 1 should be around 30.1%. So files A and B are ruled out. Files C and D are within the approximate range that would be predicted.

Second clue:

Now compare File C and D and the frequency of each starting digit:

. Digit File C File D Benford Expected
. ----- ------ ------ ----------------
. 1 30.0 27.0 30.1
. 2 17.6 18.6 17.6
. 3 12.6 12.6 12.5
. 4 9.6 10.8 9.7
. 5 8.0 8.6 7.9
. 6 6.8 6.0 6.7
. 7 5.8 5.8 5.8
. 8 5.0 5.6 5.1
. 9 4.6 5.0 4.6

It looks like File C was manufactured to conform exactly to Benford's Law. Real data should be expected to have some variance (especially in a data set of only 500 numbers). It looks like the mathematically inclined coworker knew what the numbers should look like and faked File C to perfectly fit the expected pattern.

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  • $\begingroup$ I'm getting 6.8%, 11.2%, 30.0% and 27.0%. Which would make File C the more likely candidate. $\endgroup$ – CactusCake May 14 '15 at 20:14
  • 2
    $\begingroup$ Still a good answer, I wouldn't have thought of that. Though with file's C and D being so close I would want something to back up my hunch. Maybe the lottery guy knew about this law and overplayed his hand, it could still be File D... Need another method to double check, what could it be... $\endgroup$ – CactusCake May 14 '15 at 20:19
  • $\begingroup$ To add some information that may be helpful, File C and File D have similar exponential distributions - File C starts at a slightly lower peak than File D. File C has 496 unique values while File D has 498 unique values. $\endgroup$ – Duncan May 14 '15 at 21:07
  • 2
    $\begingroup$ @JS1 Nicely done and nicely explained $\endgroup$ – JLee May 14 '15 at 21:36
  • 1
    $\begingroup$ Informative puzzle and answer! +1s all round! $\endgroup$ – A E May 15 '15 at 8:30
2
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Just looking at the numbers, we got the answer:

File D

Explanation:

C and D's start are very similar. A and B is different and we did not see any similarities. The Mathematician and the Lottery guy did not see the real data set so we concluded that A and B (witch is very different) is the data sets of those two.
The manager did see the real data, so the data he created is very similar to the real data.If you take a look at D, you will see a lot more three digit integers. A person will most likely not include that because it will stand out.

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