# Investigation of the statistical probability of a bit in non-bitcoin

The sha256 (sha256 (BlockHeader)) function, which is used in the bitcoin hashing algorithm, is considered to be very reliable. It is so reliable that there is only one way to find a golden non for the next block in the blockchain - by going through the mining process.

Currently, the Bitcoin blockchain already has more than 530 thousand blocks. This is quite enough to conduct a statistical analysis and answer the question: "Are the values ​​of bits in previously found nones equally likely?".

The simplest statistical analysis can be done not on the whole blockchain, but only on the last few thousand blocks. To do this, you need to take this information, for example, from the site btc.com. This blockchain explorer has a well-developed api, which we will use. Let's write a simple script that, using the requests library, downloads information about the last 2000 blocks. I am interested in only one value - nonce, which I will write to a text file along with the block number.

``````import requests
target_blk_num=538800
num_block_for_study=2000
txt=open("bitcoin/nonces.txt","w")
blk_num=target_blk_num-num_block_for_study
print("Fetching blocks from "+str(blk_num) +" to "+str(target_blk_num) )
while blk_num<target_blk_num:
blk_num_=format(blk_num,'d')
url_hash="https://chain.api.btc.com/v3/block/"+blk_num_
block=requests.get(url_hash)
j=block.json()
data=j['data']
nonce=data['nonce']
print(blk_num,format(nonce,'x'))
txt.write("%d " % blk_num)
txt.write("0x%08X " % nonce)
txt.write("%d" % nonce)
txt.write("\n")
blk_num=blk_num+1
txt.close()
``````

You need to run the script and wait for it to finish. Let's get this file with 2 thousand lines:

Values ​​of nonse last blocks:
```536800 0x0A75F108 175501576 536801 0x64046EE3 1678012131 536802 0x43842181 1132732801 536803 0x69D1FE65 1775369829 536804 0x5F3A5706 1597658886 536805 0x77E7FC5A 2011692122 536806 0x29FC3572 704394610 536807 0x6FA83F41 1873297217 536808 0xB498D106 3029913862 536809 0xCFCADF12 3486179090 536810 0x3217C81D 840419357 536811 0x0A4DD1A5 172872101 536812 0xC422E8B4 3290622132 536813 0x9512DA88 2501040776 536814 0x5606B5C9 1443280329 536815 0x4EAD3CC6 1319976134 536816 0x421482FF 1108640511 536817 0x15311B34 355539764 536818 0xE2879E05 3800538629 536819 0x41EB3E73 1105935987 536820 0x29ED08CD 703400141 536821 0x8B4FB3F7 2337256439 536822 0x0FF2B634 267564596 536823 0x32E51D55 853876053 536824 0xE09DBAE1 3768433377 536825 0xD97F44E5 3648996581 536826 0x90A4EC5E 2426727518 536827 0xCBF74735 3421980469 536828 0x6D2F10F0 1831801072 536829 0xE595A073 3851788403 536830 0xC8765E94 3363200660 536831 0x09DC9851 165451857 536832 0xC2EFEA16 3270502934 536833 0x7897622F 2023186991 536834 0x706949F8 1885948408 536835 0x1B619051 459378769 536836 0xE707A346 3876037446 536837 0xB3B0D216 3014709782 536838 0x51B8F9E8 1371077096 536839 0xE2207F9D 3793780637 536840 0xD0D15245 3503379013 536841 0x904FE5C5 2421155269 536842 0xFF326B4B 4281494347 536843 0x312BAA45 824945221 536844 0x88702768 2289051496 536845 0xD8924803 3633465347 536846 0xEFAB5E6B 4020985451 536847 0x76779ACB 1987549899 536848 0x81436738 2168678200 536849 0x02896E6A 42561130 536850 0x76D3ED09 1993600265 536851 0x8F2052C7 2401260231 536852 0x6A9984CA 1788445898 536853 0xD541C515 3577857301 536854 0x6AC2384F 1791113295 536855 0x214031F1 557855217 536856 0xF028BB76 4029201270 536857 0xF821479C 4162930588 536858 0xEF9C2E42 4019990082 536859 0x9E02734E 2650960718 536860 0x20C786E9 549947113 536861 0xEB84DD18 3951353112 536862 0x422720B2 1109860530 536863 0xA59EC71A 2778646298 536864 0x2FFF3B42 805256002 536865 0x5D606431 1566598193 536866 0xF4FB4B01 4110109441 536867 0x68AC0CD5 1756105941 536868 0x6C35C2A6 1815462566 536869 0xF38F0409 4086236169 536870 0xABE37AC2 2883812034 536871 0xF60096CA 4127233738 536872 0x8E3B0BA3 2386234275 536873 0x2904FF1A 688193306 536874 0xCE6A1E32 3463061042 536875 0xB110BD24 2970664228 536876 0xF28289CC 4068641228 536877 0x2719E083 656007299 536878 0xD14BF4E7 3511416039 536879 0x046C418A 74203530 536880 0xF239BEC6 4063870662 536881 0x619CDB26 1637669670 536882 0x0FAB9F09 262905609 536883 0xC8A14FC1 3366014913 536884 0x2BDC9BF0 735878128 536885 0x551D0A18 1427966488 536886 0x43BDC8E5 1136511205 536887 0x93236DB3 2468572595 536888 0xA6DF3DD6 2799648214 536889 0xD956E81A 3646351386 536890 0xA292405D 2727493725 536891 0x4ECF97EA 1322227690 536892 0x6C289307 1814598407 536893 0x0FE7B408 266843144 536894 0xD7A54779 3617933177 536895 0x49A575B6 1235580342 536896 0x660B7018 1712025624 536897 0x5A90DE66 1519443558 536898 0x24602343 610280259 536899 0xEA91EE92 3935432338 536900 0x2A7DEBC9 712895433 536901 0x7F8919BF 2139691455 536902 0x85DD793B 2245884219 536903 0x508B343F 1351300159 536904 0xA7541C57 2807307351 536905 0x902EAF9E 2418978718 536906 0x2932BE12 691191314 536907 0x304C8808 810321928 536908 0xB0DC682A 2967234602 536909 0x4F53EFD7 1330900951 536910 0xF042D46A 4030911594 536911 0x84726E0D 2222091789 536912 0xC8667200 3362157056 536913 0x4CCEAD50 1288613200 536914 0x60D8E12E 1624826158 536915 0x40BC2101 1086071041 536916 0x5758F201 1465446913 536917 0x45BD335F 1170027359 536918 0x56E151A9 1457607081 536919 0x11DE7F25 299794213 536920 0x361D50CE 907890894 536921 0xFBC73CE0 4224138464 536922 0x57232881 1461921921 536923 0x83AB7ACF 2209053391 536924 0x8F7E47AF 2407417775 536925 0x6425150A 1680151818 536926 0x596C4C78 1500269688 536927 0xA2B5D934 2729826612 536928 0xBC673E6A 3160882794 536929 0xE8116AB6 3893455542 536930 0x92661734 2456164148 536931 0x65904416 1703953430 536932 0xD2DC8C64 3537669220 536933 0xAC14A926 2887035174 536934 0x98B9F2D6 2562323158 536935 0xB85796F5 3092748021 536936 0x08D3AC55 148089941 536937 0x08532D08 139668744 536938 0x58022FF2 1476538354 536939 0x2C1D0FA9 740102057 536940 0x83668A2B 2204535339 536941 0x938CBC67 2475474023 536942 0x49A14B37 1235307319 536943 0x8FA2492A 2409777450 536944 0x822227E8 2183276520 536945 0xB6AE2123 3064865059 536946 0xE83D8756 3896346454 536947 0xD6437316 3594744598 536948 0x731E7A99 1931377305 536949 0x9F63EDE8 2674126312 536950 0xA9B67DCD 2847309261 536951 0x2B77E7F6 729278454 536952 0x6F3E3B5F 1866349407 536953 0xF1FF973E 4060059454 536954 0x28AC9397 682398615 536955 0x5920CE0A 1495322122 536956 0x916C0AA2 2439776930 536957 0x5CDF71F2 1558147570 536958 0x9A41B702 2587997954 536959 0xB16D4B56 2976729942 536960 0x0C88EE3E 210300478 536961 0x6A6D7474 1785558132 536962 0x7544CFAE 1967443886 536963 0xD14824C2 3511166146 536964 0xA1FF9FFA 2717884410 536965 0xA1CABEE0 2714418912 536966 0xED5BD1E0 3982217696 536967 0x3095185B 815077467 536968 0x87A71339 2275873593 536969 0x877DB48E 2273162382 536970 0x4ABD04F2 1253901554 536971 0xC29C3B16 3265018646 536972 0xF0C21FF7 4039254007 536973 0x4B60F69E 1264645790 536974 0x1297C086 311935110 536975 0xB2DF972E 3000997678 536976 0xA3746A7F 2742315647 536977 0x5690FA49 1452341833 536978 0xDD69D9F1 3714701809 536979 0x16C9F432 382334002 536980 0x776F7186 2003792262 536981 0xE5DC34D6 3856413910 536982 0x07B9F683 129627779 536983 0x37ADFF70 934150000 536984 0x020655F6 33969654 536985 0x828F3054 2190422100 536986 0x6830DB63 1748032355 536987 0xC8BDAD1D 3367873821 536988 0x456172DD 1164014301 536989 0x4EEF786C 1324316780 536990 0x047313AB 74650539 536991 0x4A9C6F2E 1251766062 536992 0x16AE3E21 380517921 536993 0x75E6BBE9 1978055657 536994 0x4DB86F32 1303932722 536995 0x537A470F 1400522511 536996 0x09D57784 164984708 536997 0x225542F8 576013048 536998 0x9285CB22 2458241826 536999 0x8A5D251C 2321360156 537000 0x4EF17C51 1324448849 537001 0xF2EE1FD0 4075691984 537002 0x21F69ED9 569810649 537003 0xA3D38623 2748548643 537004 0x437747C6 1131890630 537005 0x2DAE22B4 766386868 537006 0xA564238C 2774803340 537007 0x9B0DA8F5 2601363701 537008 0x9BF14D85 2616282501 537009 0xEF7EFDCD 4018077133 537010 0x1D3654BE 490099902 537011 0x284DA6AE 676177582 537012 0xCCACDD6D 3433880941 537013 0x85DC2096 2245795990 537014 0x23720223 594674211 537015 0xBD5A6697 3176818327 537016 0x02ABD9F2 44816882 537017 0xCD810703 3447785219 537018 0x2594C210 630506000 537019 0x04D87CA6 81296550 537020 0x21BF189D 566171805 537021 0x0B3E23E2 188621794 537022 0x7852C9A1 2018691489 537023 0xE55F03D2 3848209362 537024 0xE0DE42DC 3772662492 537025 0xDDE9DF00 3723091712 537026 0x9A857C48 2592439368 537027 0x6506143F 1694897215 537028 0x8A1E2D28 2317233448 537029 0x9D56D3A8 2639713192 537030 0xE65D60AE 3864879278 537031 0xCC66C1EE 3429286382 537032 0x21A20619 564266521 537033 0x007C5EA1 8150689 537034 0x726CE93C 1919740220 537035 0x707D5E32 1887264306 537036 0x7AE9C992 2062141842 537037 0x161EA8C2 371108034 537038 0x5DD540CB 1574256843 537039 0x42DA383D 1121597501 537040 0x6A852D47 1787112775 537041 0xE003B830 3758340144 537042 0xA9CE3085 2848862341 537043 0xEFBD3268 4022153832 537044 0xF54E55A9 4115551657 537045 0x7149A940 1900652864 537046 0x701E3E47 1881030215 537047 0x758F5F80 1972330368 537048 0xFA3F88E0 4198467808 537049 0xA0C77A3A 2697427514 537050 0x97D20967 2547124583 537051 0x44CDB6A6 1154332326 537052 0x53281BD2 1395137490 537053 0x49305D48 1227906376 537054 0x843D2581 2218599809 537055 0x9440B117 2487267607 537056 0xC38E9622 3280901666 537057 0x82869D41 2189860161 537058 0x2DD924D9 769205465 537059 0xCE8C480A 3465299978 537060 0xECD53611 3973395985 537061 0xAD78CFBC 2910375868 537062 0xFFE73547 4293342535 537063 0x833FDB8A 2202000266 537064 0x60BB4400 1622885376 537065 0xDE2D780E 3727521806 537066 0x7373797F 1936947583 537067 0xAE525433 2924631091 537068 0x8A91E5AA 2324817322 537069 0x8CC62047 2361794631 537070 0x6092E498 1620239512 537071 0x83370EB6 2201423542 537072 0x19C2F477 432206967 537073 0x3536E714 892790548 537074 0x68FC8978 1761380728 537075 0xB146C1B2 2974204338 537076 0x86A5FF39 2259025721 537077 0xC7006B6A 3338693482 537078 0xA9B14363 2846966627 537079 0x43DA311B 1138372891 537080 0xEEC44C12 4005841938 537081 0x3CB1A0AC 1018273964 537082 0x9E875D82 2659671426 537083 0x5734A4B3 1463067827 537084 0x50449436 1346671670 537085 0x74A2CF44 1956826948 537086 0x161DDBE8 371055592 537087 0x85D01F7A 2245009274 537088 0x6DA2C81C 1839384604 537089 0x6A4E9C84 1783536772 537090 0x898D87F9 2307753977 537091 0xBE49168C 3192460940 537092 0x77CCF6DF 2009921247 537093 0x8BAB207C 2343247996 537094 0xD1C8CDCB 3519598027 537095 0xF85BEAEE 4166773486 537096 0xD3B6A308 3551961864 537097 0x1F830312 528679698 537098 0x6230DB94 1647369108 537099 0x5F9BC800 1604044800 537100 0x77AF542B 2007979051 537101 0x599B39D0 1503345104 537102 0x97121813 2534545427 537103 0x2FE9C1B5 803848629 537104 0xA58C9DC2 2777456066 537105 0xA650952E 2790298926 537106 0xA6DE4F97 2799587223 537107 0xDFF1F9B9 3757177273 537108 0x5980E9F1 1501620721 537109 0x94B2003D 2494693437 537110 0xB6FA8716 3069871894 537111 0x7532A699 1966253721 537112 0x717D6C2E 1904045102 537113 0x10B523B8 280306616 537114 0x25E93A02 636041730 537115 0x78600DE3 2019560931 537116 0xD0A4DFB9 3500466105 537117 0x1965022E 426050094 537118 0x8C2B3EFA 2351644410 537119 0xAF7FD1E9 2944389609 537120 0x420A3B31 1107966769 537121 0xABE37683 2883810947 537122 0x7832BBEB 2016590827 537123 0xC83E2CEB 3359517931 537124 0x8EA76ED7 2393337559 537125 0xE4655C3C 3831847996 537126 0x91DA6CBB 2447011003 537127 0x69A5ABC4 1772465092 537128 0x9879C300 2558116608 537129 0x02A7C845 44550213 537130 0x4622E285 1176691333 537131 0x352397C3 891525059 537132 0x9E2D2803 2653759491 537133 0xACB24867 2897365095 537134 0xCC4D68A9 3427625129 537135 0x005CD031 6082609 537136 0x228665C3 579233219 537137 0xB8C5D596 3099973014 537138 0x1B1599A4 454400420 537139 0xC33197EC 3274807276 537140 0x028C60A4 42754212 537141 0x0ADF69A2 182413730 537142 0xEE31513C 3996209468 537143 0x711A3D10 1897544976 537144 0x190F447D 420430973 537145 0x4A34F6CE 1244985038 537146 0xB8CF8EC1 3100610241 537147 0x096E8D7B 158240123 537148 0x4A41FA12 1245837842 537149 0x6F94338A 1871983498 537150 0x49582F6E 1230516078 537151 0x181CCB1C 404540188 537152 0x69991C6B 1771641963 537153 0x44C5E9AA 1153821098 537154 0x264DB929 642627881 537155 0xDF328C43 3744631875 537156 0xE62FF280 3861901952 537157 0x90FFD450 2432685136 537158 0xDF8FAC57 3750734935 537159 0xA1D156DF 2714851039 537160 0x71894209 1904820745 537161 0x9E8C4A0E 2659994126 537162 0x4EEAE41E 1324016670 537163 0xB39E40D2 3013492946 537164 0x83A1CB40 2208418624 537165 0xC5990BB2 3315141554 537166 0x62C16D29 1656843561 537167 0xDFBA4DD4 3753528788 537168 0xF27FF0AC 4068470956 537169 0x4D297BEF 1294564335 537170 0xD3CDC811 3553478673 537171 0xEB5BB554 3948655956 537172 0x1F035838 520312888 537173 0x0F8008F0 260049136 537174 0x1D0D54EB 487412971 537175 0x0AD7DAA8 181918376 537176 0xDB4243B5 3678553013 537177 0xEB87623B 3951518267 537178 0xA6E29A9D 2799868573 537179 0x657DEC41 1702751297 537180 0xDEBF78A9 3737090217 537181 0x9A0E3E42 2584624706 537182 0x4D81B479 1300345977 537183 0xEE237F29 3995303721 537184 0x64483418 1682453528 537185 0x70CE3A54 1892563540 537186 0x0407E621 67626529 537187 0xE87F58A8 3900659880 537188 0x8A572922 2320967970 537189 0xA1DA472C 2715436844 537190 0x1DB85112 498618642 537191 0xA5B332E0 2779984608 537192 0x8C6EB381 2356065153 537193 0x011D856C 18711916 537194 0x4FFB15C5 1341855173 537195 0x8BB55113 2343915795 537196 0x1F88FDBC 529071548 537197 0x71E8295D 1911040349 537198 0xDD326D1C 3711069468 537199 0x4997E7A1 1234692001 537200 0x0C25B238 203797048 537201 0xDC02F821 3691182113 537202 0x7E0272FA 2114089722 537203 0x7428DA05 1948834309 537204 0x5375694E 1400203598 537205 0xE4895B26 3834207014 537206 0xF2E27798 4074928024 537207 0x2750C3E9 659604457 537208 0xC491CE90 3297889936 537209 0xB27D5F42 2994560834 537210 0x48DEEAE7 1222568679 537211 0xD52B16F8 3576370936 537212 0x109162BA 277963450 537213 0x06F78C0D 116886541 537214 0xC3364C98 3275115672 537215 0x6BC8BF8E 1808318350 537216 0xDFB8B924 3753425188 537217 0x74FCDF85 1962729349 537218 0x89B3E526 2310268198 537219 0xB2B2756B 2998039915 537220 0x0BF45620 200562208 537221 0xF73D1ACF 4147976911 537222 0xC0C860CC 3234357452 537223 0xDE29DC19 3727285273 537224 0xB20C3EDC 2987146972 537225 0xE676C194 3866542484 537226 0x845B30F8 2220568824 537227 0x8B71E0B3 2339496115 537228 0xE8DB2A11 3906677265 537229 0x229AB7E1 580564961 537230 0x51D93418 1373189144 537231 0x74E81286 1961366150 537232 0x683E8338 1748927288 537233 0x94CAB9B2 2496313778 537234 0xA8A33C31 2829270065 537235 0x43B4A14E 1135911246 537236 0xA93CD929 2839337257 537237 0x5350FBDC 1397816284 537238 0xD261E675 3529631349 537239 0xC9E11D00 3386973440 537240 0x62D26A99 1657957017 537241 0x92A069C1 2459986369 537242 0x2517A475 622306421 537243 0xCE77FA34 3463969332 537244 0x0CCCB54F 214742351 537245 0x4CDF0B6B 1289685867 537246 0x23481E45 591928901 537247 0x9F02003A 2667708474 537248 0xD3BDA440 3552420928 537249 0x2DED0928 770509096 537250 0x2EA05107 782258439 537251 0x9A2B9830 2586548272 537252 0x098813F0 159912944 537253 0x245C53F8 610030584 537254 0xD29B8F24 3533410084 537255 0x890D4C28 2299350056 537256 0x685F61BA 1751081402 537257 0x12B43DA1 313802145 537258 0xD6882535 3599246645 537259 0x251D0329 622658345 537260 0x0E98327A 244855418 537261 0x5A2DA66F 1512941167 537262 0x00E45870 14964848 537263 0x91E53ADF 2447719135 537264 0xE501A114 3842089236 537265 0xB3119D1A 3004275994 537266 0x1FF020F0 535830768 537267 0xC748BF35 3343433525 537268 0xC876BF7D 3363225469 537269 0x450606F9 1158022905 537270 0x013786E5 20416229 537271 0xA964BC95 2841951381 537272 0x2CECAD8B 753708427 537273 0xB39E4070 3013492848 537274 0x421E0126 1109262630 537275 0x1261BDBD 308395453 537276 0xE17D1F46 3783073606 537277 0x8CAD3133 2360160563 537278 0xD8AE2C86 3635293318 537279 0x22E2EB8E 585296782 537280 0x7018748D 1880650893 537281 0x02DDDD7A 48094586 537282 0x2AFFB5E8 721401320 537283 0xD57FA399 3581911961 537284 0x6EBFE3CC 1858069452 537285 0xE904210B 3909361931 537286 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0x4F226E98 1327656600 538709 0x495E5CA5 1230920869 538710 0xE3ECCE49 3823947337 538711 0x15700867 359663719 538712 0x29D12C8A 701574282 538713 0xC7134F2A 3339931434 538714 0xEEB66B9F 4004932511 538715 0x4455B1D6 1146466774 538716 0xF27C58AA 4068235434 538717 0x80C2A8B2 2160240818 538718 0xAD76D91A 2910247194 538719 0x292ECBBC 690932668 538720 0xEB0CEB7A 3943492474 538721 0x0B39C986 188336518 538722 0xCBEAC47A 3421160570 538723 0xF520CF94 4112568212 538724 0x0E22DE71 237166193 538725 0xAACA82AD 2865398445 538726 0xC08E3613 3230545427 538727 0xE1AEB71C 3786323740 538728 0xF6A86355 4138230613 538729 0xE6409A01 3862993409 538730 0xF1470A28 4047964712 538731 0xDF9E739C 3751703452 538732 0x9F242F10 2669948688 538733 0xCA7A9E3E 3397033534 538734 0x9A762790 2591434640 538735 0xD3091FC1 3540590529 538736 0xF5D96F23 4124667683 538737 0x5B5F8BF8 1532988408 538738 0xCE69BC19 3463035929 538739 0xFA1B9BA8 4196113320 538740 0x9C8CB970 2626468208 538741 0xDC71095C 3698395484 538742 0x071F7BEC 119503852 538743 0x7400ED3A 1946217786 538744 0x1F0D3CAE 520961198 538745 0x6BC607AF 1808140207 538746 0x6B8D31AF 1804415407 538747 0x8629F8AE 2250897582 538748 0x14FF7400 352285696 538749 0xC3B5BC9D 3283467421 538750 0x29CB2575 701179253 538751 0x861AED3D 2249911613 538752 0x42AAA3CD 1118479309 538753 0x44C0B0B6 1153478838 538754 0x87AD78C1 2276292801 538755 0x5CE6831E 1558610718 538756 0xAD4B0CF7 2907376887 538757 0x958065C0 2508219840 538758 0x4179B5A0 1098495392 538759 0xA6766FBE 2792779710 538760 0xA945A40D 2839913485 538761 0x5F8801E3 1602748899 538762 0x5C18DED4 1545133780 538763 0x8D3B4284 2369471108 538764 0x60BA61CA 1622827466 538765 0x5C5DDEBC 1549655740 538766 0x4A307A20 1244690976 538767 0x4399D940 1134156096 538768 0x97E2F94F 2548234575 538769 0x1F48A827 524855335 538770 0x88C31D10 2294488336 538771 0xF44D4700 4098705152 538772 0xE464DA81 3831814785 538773 0x6E7A6187 1853514119 538774 0x448F6E9A 1150250650 538775 0xED003AA2 3976215202 538776 0x1AF65C81 452353153 538777 0x36DA69E0 920283616 538778 0x417C436A 1098662762 538779 0x8B5DEA00 2338187776 538780 0x97ECE049 2548883529 538781 0x52F85F9B 1392009115 538782 0x80370F89 2151092105 538783 0x2796E165 664199525 538784 0x8D019EE9 2365693673 538785 0xE1E18CCD 3789655245 538786 0x86B198AB 2259785899 538787 0x6BEBF3F0 1810625520 538788 0x3D4B8620 1028359712 538789 0x6F8F030C 1871643404 538790 0xC6798969 3329853801 538791 0xEB9FFD16 3953130774 538792 0xDD760436 3715499062 538793 0x2030335F 540029791 538794 0xC7E80160 3353870688 538795 0x0894B200 143962624 538796 0x326EE03C 846127164 538797 0x43C4DD73 1136975219 538798 0xE58D6273 3851248243 538799 0x1B71803F 460423231 ```

Now we divide this sample into groups of one hundred values ​​and produce a statistical analysis for each group. Nonse is a 32 bit number. According to common sense, every bit nonce is equally likely to take the value one or zero. That's what I'm going to check. I will write here such a script that counts the number of units in each bit position for a group of 100 nonsense.

``````import sys
import os
folder=""
stat_N=0
stat=[]
definit_stat():global stat
global stat_N
stat_N=0
i=0while i<64:
stat.append(0.0)
i=i+1defclear_stat():global stat
global stat_N
stat_N=0
i=0while i<64:
stat[i]=0.0
global stat_N
stat_N=stat_N+1
i=0while i<64:
if nonce & (1<<i):
stat[i]=stat[i]+1
i=i+1defnonce_num_bits():
i=32while i<64:
if stat[i]:
return64
i=i+1return32defprint_stat():global stat
global stat_N
i=0
sz = nonce_num_bits()
while i<sz:
print(i," ",stat[i]/stat_N)
i=i+1definit_stat_file():global folder
txt=open(folder+"/stat.txt","w")
txt.close()
defwrite_stat():global folder
global stat
global stat_N
txt=open(folder+"/stat.txt","a")
i=0
sz = nonce_num_bits()
while i<sz:
s=stat[i]/stat_N
txt.write("%.2f " % s )
i=i+1
txt.write("\n")
txt.close()
#--------------------------------
args = sys.argv
if len(args)<2:
print("Error: need argument name of folder with list of nonces")
exit()
folder=args[1]
print( "Working on \"" + folder +"\"" )
if os.path.isdir(folder)==False:
print("Error: folder \""+folder+"\" does not exist")
exit()
init_stat()
init_stat_file()
f = open(folder+"/nonces.txt","r")
for line in content:
word = line.split(" ")
if len(word)<2:
break
nonce=int(word[1],16)
if stat_N==100:
write_stat()
clear_stat()
f.close
``````

This script reads the bitcoin / nonces.txt file and creates a new bitcoin / stst.txt file of approximately the following form: Here, each column is the probability that a unit will appear in the corresponding bit position. For example, look at the column representing bit1, bit6, and bit7 of nonse: Expectancy 0.459, 0.446, and 0.458. Now it remains to somehow display the resulting statistics in the form of graphs. For this I have another plot.py script

```0.49 0.50 0.45 0.39 0.44 0.40 0.54 0.43 0.47 0.55 0.50 0.54 0.53 0.50 0.54 0.51 0.56 0.47 0.48 0.44 0.44 0.51 0.41 0.52 0.50 0.54 0.42 0.47 0.39 0.52 0.61 0.48 0.40 0.61 0.52 0.48 0.49 0.55 0.45 0.43 0.55 0.56 0.54 0.55 0.56 0.60 0.52 0.42 0.52 0.46 0.61 0.54 0.52 0.52 0.55 0.49 0.46 0.48 0.41 0.42 0.45 0.42 0.47 0.47 0.39 0.50 0.43 0.45 0.46 0.47 0.40 0.51 0.48 0.44 0.52 0.51 0.46 0.45 0.51 0.45 0.45 0.53 0.59 0.51 0.53 0.44 0.52 0.55 0.53 0.47 0.49 0.47 0.41 0.51 0.52 0.52 0.51 0.43 0.39 0.49 0.46 0.53 0.40 0.50 0.49 0.42 0.47 0.56 0.49 0.51 0.57 0.44 0.58 0.52 0.50 0.54 0.48 0.50 0.43 0.53 0.52 0.49 0.48 0.56 0.43 0.42 0.49 0.49 0.44 0.33 0.43 0.57 0.52 0.52 0.44 0.45 0.53 0.45 0.43 0.50 0.52 0.59 0.44 0.47 0.49 0.40 0.50 0.58 0.55 0.55 0.47 0.51 0.45 0.48 0.44 0.42 0.41 0.47 0.50 0.48 0.49 0.46 0.51 0.42 0.57 0.45 0.45 0.35 0.53 0.42 0.51 0.52 0.47 0.51 0.47 0.45 0.48 0.49 0.54 0.48 0.50 0.49 0.46 0.56 0.44 0.48 0.47 0.55 0.46 0.45 0.51 0.50 0.51 0.45 0.52 0.50 0.59 0.41 0.50 0.34 0.47 0.54 0.56 0.48 0.54 0.49 0.54 0.44 0.49 0.48 0.50 0.58 0.44 0.48 0.51 0.47 0.37 0.45 0.47 0.47 0.50 0.52 0.46 0.43 0.51 0.43 0.52 0.57 0.58 0.51 0.47 0.36 0.51 0.44 0.47 0.47 0.49 0.47 0.43 0.39 0.47 0.50 0.56 0.46 0.47 0.47 0.46 0.63 0.48 0.43 0.44 0.46 0.43 0.51 0.45 0.49 0.49 0.49 0.40 0.49 0.53 0.53 0.49 0.50 0.47 0.50 0.41 0.55 0.48 0.48 0.49 0.46 0.54 0.51 0.49 0.55 0.53 0.48 0.50 0.53 0.50 0.41 0.45 0.42 0.41 0.47 0.44 0.51 0.47 0.47 0.47 0.58 0.52 0.42 0.53 0.53 0.59 0.55 0.43 0.48 0.50 0.42 0.46 0.50 0.52 0.53 0.43 0.48 0.58 0.44 0.43 0.42 0.54 0.42 0.58 0.42 0.39 0.45 0.46 0.56 0.51 0.47 0.45 0.38 0.48 0.51 0.39 0.48 0.50 0.57 0.52 0.54 0.54 0.46 0.67 0.47 0.53 0.54 0.48 0.53 0.57 0.55 0.46 0.54 0.43 0.53 0.49 0.37 0.60 0.46 0.50 0.49 0.45 0.46 0.47 0.46 0.47 0.51 0.41 0.53 0.59 0.45 0.51 0.43 0.49 0.46 0.46 0.55 0.52 0.51 0.56 0.51 0.49 0.46 0.45 0.49 0.50 0.55 0.48 0.54 0.44 0.49 0.59 0.49 0.39 0.46 0.53 0.43 0.42 0.54 0.49 0.46 0.52 0.56 0.38 0.46 0.47 0.56 0.60 0.46 0.50 0.59 0.39 0.57 0.50 0.55 0.44 0.54 0.45 0.47 0.52 0.53 0.45 0.41 0.47 0.42 0.48 0.46 0.49 0.45 0.52 0.43 0.43 0.46 0.43 0.52 0.53 0.50 0.52 0.61 0.43 0.51 0.56 0.51 0.57 0.48 0.51 0.48 0.61 0.52 0.52 0.54 0.42 0.45 0.48 0.49 0.46 0.48 0.55 0.46 0.50 0.47 0.42 0.40 0.34 0.58 0.52 0.50 0.45 0.45 0.55 0.48 0.41 0.47 0.47 0.46 0.49 0.53 0.50 0.57 0.41 0.58 0.38 0.55 0.47 0.43 0.43 0.43 0.51 0.51 0.39 0.49 0.56 0.45 0.45 0.51 0.42 0.49 0.45 0.51 0.48 0.44 0.48 0.45 0.53 0.47 0.53 0.45 0.48 0.42 0.54 0.51 0.47 0.59 0.49 0.37 0.48 0.47 0.43 0.44 0.47 0.40 0.42 0.38 0.56 0.45 0.61 0.42 0.49 0.45 0.52 0.49 0.55 0.44 0.46 0.43 0.50 0.47 0.51 0.47 0.58 0.51 0.55 0.45 0.56 0.47 0.52 0.53 0.42 0.47 0.49 0.47 0.50 0.51 0.49 0.49 0.49 0.57 0.52 0.48 0.44 0.38 0.50 0.48 0.50 0.52 0.44 0.48 0.46 0.43 0.51 0.52 0.53 0.50 0.44 0.46 0.46 0.46 0.39 0.49 0.52 0.38 0.42 0.53 0.52 0.49 0.48 0.42 0.44 0.51 0.54 0.49 0.48 0.43 0.50 0.44 0.41 0.45 0.54 0.54 0.49 0.42 0.52 0.45 0.44 0.55 0.51 0.54 0.46 0.59 0.51 0.49 0.51 0.41 0.44 0.42 0.49 0.52 0.42 0.41 0.45 0.48 0.45 0.50 0.36 0.50 0.51 0.48 0.45 0.61 0.45 0.52 0.49 0.52 0.45 0.47 0.47 0.57 0.48 0.52 0.54 0.46 0.54 0.54 0.56 0.51 0.39 0.41 0.59 0.55 ```

``````import sys
import os
import numpy as np
import matplotlib.mlab as mlab
import matplotlib.pyplot as plt
folder=""#--------------------------------
args = sys.argv
if len(args)<2:
print("Error: need argument name of folder with list of nonces")
exit()
folder=args[1]
print( "Working on \"" + folder +"\"" )
if os.path.isdir(folder)==False:
print("Error: folder \""+folder+"\" does not exist")
exit()
stat_N=0
stat=[]
definit_stat():global stat
global stat_N
stat_N=0
i=0while i<64:
stat.append([])
i=i+1defdraw_stat(start_bit,ax=None):
print("draw_stat ",start_bit)
save_single=0if ax isNone:
fig = plt.figure( figsize=(16,9))
ax = fig.gca()
save_single=1
ax.set_yticks( np.arange( 0.0, 1.0, 0.1 ) )
plt.title("Probability of specific bit in nonce")
plt.ylabel("Probability")
plt.xlabel("Every X-point presents group nonces averaged")
plt.ylim(0.0,1.0)
plt.grid()
i=0while i<8:
color="#"if i&1:
color=color+"FF"else:
color=color+"00"if i&2:
color=color+"FF"else:
color=color+"00"if i&4:
color=color+"FF"else:
color=color+"00"if i==7:
color="#A0A0A0"
ax.plot(stat[start_bit+i],color,label="bit"+str(start_bit+i) )
i=i+1
ax.legend(loc="upper left",ncol=2)
#plt.show()if save_single :
plt.savefig(folder+"/bit"+str(start_bit)+"-"+str(start_bit+7)+".png",dpi=300)
defdraw_hist(start_bit,ax=None):
print("draw_hist ",start_bit)
save_single=0if ax isNone:
fig = plt.figure( figsize=(16,9))
ax = fig.gca()
save_single=1
ax.set_xticks( np.arange( 0.0, 1.0, 0.1 ) )
x=np.arange( 0.0, 1.0, 0.05 )
plt.title("Probability Density of specific bit in nonce")
plt.ylabel("Density")
plt.xlabel("Probability")
plt.grid()
i=0
max_val=0while i<8:
histogram=[0]*20
k=0
stat_column=stat[start_bit+i]
n=len(stat_column)
while k<n:
val=stat_column[k]
idx=int(round(val*20))
histogram[idx]=histogram[idx]+1if max_val<histogram[idx]:
max_val=histogram[idx];
k=k+1
color="#"if i&1:
color=color+"FF"else:
color=color+"00"if i&2:
color=color+"FF"else:
color=color+"00"if i&4:
color=color+"FF"else:
color=color+"00"if i==7:
color="#A0A0A0"
ax.plot(x,histogram,color,label="bit"+str(start_bit+i) )
i=i+1
plt.ylim(0.0,max_val*1.2)
ax.set_yticks( np.arange( 0.0, max_val*1.2, 1.0 ) )
ax.legend(loc="upper right")
#plt.show()if save_single :
plt.savefig(folder+"/density"+str(start_bit)+"-"+str(start_bit+7)+".png",dpi=300)
num_bits=32
init_stat()
f = open(folder+"/stat.txt","r")
for line in content:
word = line.split(" ")
nb=len(word)-1if nb==32or nb==64:
num_bits=nb
else:
break
i=0while i<num_bits:
stat[i].append( float(word[i]) )
i=i+1
f.close
print("Num Bits in Nonce:",num_bits)
draw_hist(0)
draw_hist(8)
draw_hist(16)
draw_hist(24)
fig = plt.figure( figsize=(16,9*num_bits/32))
draw_hist(0,ax1)
draw_hist(8,ax2)
draw_hist(16,ax3)
draw_hist(24,ax4)
plt.savefig(folder+"/density_all.png",dpi=300)
draw_stat(0)
draw_stat(8)
draw_stat(16)
draw_stat(24)
fig = plt.figure( figsize=(16,9*num_bits/32))
draw_stat(0,ax1)
draw_stat(8,ax2)
draw_stat(16,ax3)
draw_stat(24,ax4)
plt.savefig(folder+"/bits_all.png",dpi=300)
``````

Now it's time to review these charts. If you display them all 32 on one page, you will get incomprehensible porridge. Therefore, the script builds 4 graphs and each displays 8-bit statistics. So it will be more convenient to consider.

Here is everything:

It can be seen that some bits in non-Bitcoin are not equally probable in their meaning and the graphics are shifted down. So the value of the bit is still slightly more likely zero than one. Below are the probability distribution graphs.

Here you can see everything the same - on average, the probability of the value of some bits in Bitcoin's found golden nonce is not equal to 0.5, as I expected, but rather shifted towards zero. Bits 1, 15, 22 stand out in particular. What is the reason for this, unfortunately I cannot say. Nevertheless, this fact allows us to think about the redistribution of power miners across the ranges of values ​​more likely. For example, like this: there are a hundred hardware miners (video cards or asikov), 65 miners are sent to search nonce with values ​​of 22 bit equal to zero and 35 miners of minenits in ranges with value of bit 22 equal to one. Further, each selected range can be divided into smaller subranges according to the probability of bit 15 ... and so on. The idea is to look for golden nonses in more likely ranges according to the latest accumulated statistics.

Then the idea arose to look at the statistics more deeply, not only for the last 2000 blocks, but for the entire history of the blockchain bitcoin. The whole blockchain is already almost 200 GB, so all the sharp-witted terms “Bitcoin, blockchain, bigdata” are already present here.

I installed a bitcoin wallet and deflated the entire blockchain.

Further I used such a project .
This project allows the python script to go around the blockchain and extract the data of interest to us.

Slightly changed the script from the example so that it writes statistics to a text file. Now the entire range of blocks of about 530 thousand is divided into groups of 1000 blocks. In each block, I look at the average probability of a bit nonsense value.

``````import sys
sys.path.append('..')
from blockchain_parser.blockchain import Blockchain
stat_N=0
stat=[]
definit_stat():global stat
global stat_N
stat_N=0
i=0while i<32:
stat.append(0)
i=i+1defclear_stat():global stat
global stat_N
stat_N=0
i=0while i<32:
stat[i]=0
global stat_N
stat_N=stat_N+1
i=0while i<32:
if nonce & (1<<i):
stat[i]=stat[i]+1
i=i+1defprint_stat():global stat
global stat_N
i=0while i<32:
print(i," ",stat[i]/stat_N)
i=i+1definit_stat_file():
txt=open("stat.txt","w")
txt.close()
defwrite_stat():global stat
global stat_N
txt=open("stat.txt","a")
i=0while i<32:
s=stat[i]/stat_N
txt.write("%.4f " % s )
i=i+1
txt.write("\n")
txt.close()
init_stat()
init_stat_file()
# Instantiate the Blockchain by giving the path to the directory # containing the .blk files created by bitcoind
blockchain = Blockchain(sys.argv[1])
height=0# To get the blocks ordered by height, you need to provide the path of the# `index` directory (LevelDB index) being maintained by bitcoind. It contains# .ldb files and is present inside the `blocks` directoryfor block in blockchain.get_ordered_blocks(sys.argv[1] + '/index', end=538000):
print("")
height=block.height
print("height=%d" % (block.height) )
arr=block.hex[:80]
for a in arr:
print(format(a,'02X'),end='')
print("")
print("hash  =%s" %  (block.hash) )
nonce=hdr.nonce
print("nonce =", format(nonce,'08X'))
if stat_N==1000:
write_stat()
clear_stat()
print("")
``````

This script also generates a statistics file of probability non-bit throughout the entire history of the bitcoin blockchain. It turns out quite an interesting picture.

Here's how the probability of bits nonce from zero to fifteenth changed over time:

But from the sixteenth to thirty-first bits of nonse:

Here, each point horizontally is one thousand blocks in the blockchain, and for this thousand, the average probability of occurrence of one in the bit position nonce is considered.

Honestly, I don’t even know how to comment on this. At times, bits in non-oh how not equally probable. And if so, then why the miners are still just going over the nonsense? Is it possible to distribute the power of miners according to the pattern found?

If you are interested in such studies, you can still read my other articles:

UPDATE:
I did additional research on Ethereum. The picture is even more interesting.
If you look at the previously found nons, then with the naked eye you can see the uneven distribution: The hash algorithm of the ether is more complicated than that of Bitcoin. Non-air 64x bit. The middle of nonse (bits 28-36) is almost empty. At the same time, in the miner's source code that for Nvidia Cuda, that for AMD OpenCL, there seems to be no indication that the nonsense should be matched in the middle - there is a uniform build-up from zero to 0xFFFFFFFFFFFFFFFFFF. Here is the probability distribution of bits of non-ether:

```5844149 0xb431dfa8061eedc5 5844150 0xac0badf80082e518 5844151 0x64404b000042af4f 5844152 0x96d84c7bd66ae434 5844153 0xa6dd55a000364353 5844154 0x516a29f005a56197 5844155 0x4c0a1000006d50aa 5844156 0x9e61733806e46e10 5844157 0x8191b4d80925a334 5844158 0x53a42df800b24a72 5844159 0x9eb85a3c0031e03b 5844160 0xb707255405787b53 5844161 0xbf6bd50011069dc9 5844162 0x83171430153262b1 5844163 0x2e1c8a302ad94fc2 5844164 0x84c971180db7252e 5844165 0x47d606ф00110d3bb 5844166 0x7b1f78bc127bb7cb```