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Power 2 The Cards (Posted on 2007-09-16) Difficulty: 3 of 5
88,889 cards are consecutively numbered from 11,111 to 99,999.

These cards are now arranged in a line in any arbitrary order.

Can the 444,445 digit number formed in this manner be a power of 2?

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

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Solution re: solution Comment 2 of 2 |
(In reply to solution by Charlie)

My previous comment focused on disproving the possibility of forming the three powers of two in the given range (444,445 digits long) from the set of cards by showing instances of cards that would need leading zeros.  The other initial probabilistic argument against such formation was the "birthday problem". That is, with what's actually 100,000 possible groups of 5, with 88,889 random examples taken, it's a virtual certainty that there will be duplicates needed.

UBASIC is limited to about 2000-digit accuracy, but in the processes of division and taking logarithms this was effectively reduced to about 1000, or 200 groups of 5. Within such a short run, it's less than 10% likely to find any match.

Then I remembered there was MIRACL CALC.  I don't remember on which discussion it appeared, but I redid the calculations using that software, getting the first 5,000 digits of each of the three powers of 2 in question.  They appear below, in the groups of 5 that would purport to be cards. Sets of 5 involved in duplicates are bolded:

2^1476412
19859 03996 52092 54123 81315 09567 19966 03134 14381 71202
61421 54620 00201 35035 08851 69240 30855 05479 23960 95466
67079 34872 13348 79475 35340 85635 54132 52094 20641 41922
50236 97043 63582 91797 32315 39873 79502 81533 21900 73636
60306 41897 76003 27543 45056 89885 59044 59615 93357 88158
35178 54033 50147 33723 41091 97260 86284 78402 37858 36645
84975 98975 36208 44666 60487 06586 46605 89779 54725 59054
07238 97718 53782 53813 92550 87754 67562 32230 07806 19810
30305 71820 61489 79922 66682 05367 17375 67795 37144 08695
41269 33371 59817 76897 44673 58640 07570 72372 87085 79734
06477 82744 96817 24637 96879 75240 55037 88629 62663 66245
21896 39878 95898 37413 14983 09312 70059 81686 68443 42189
40714 32000 65385 31086 75745 54642 32272 10873 63936 22272
80259 84057 05370 28731 08345 92369 31818 46832 87620 31939
24985 74138 21176 76079 06648 14668 06129 46612 04984 38174
55044 67983 08550 99839 11274 95094 04978 18726 89985 83683
97409 34233 61032 39598 59841 32977 42495 03672 29307 31250
42827 26626 36265 29256 81016 75831 30990 46755 19680 41971
30128 11697 36497 65534 68729 13801 28913 79926 76823 06958
91164 45475 63320 04786 09263 50581 01187 04690 10638 72612
47317 29030 04630 82913 83383 66027 59801 31285 77955 56114
90341 05850 68044 10420 58586 08564 54247 01950 53715 49432
26810 25278 56255 41901 32469 88235 64263 01094 57516 79524
75496 86216 31223 19394 53472 48696 00369 49960 27930 23520
85431 97591 69917 33107 68805 27879 09542 05960 00930 60709
94300 56896 13216 78576 49822 76488 14914 22736 24519 90767
87501 25888 83089 68507 80689 23236 35902 45259 96870 57410
77586 66716 35607 53859 20679 35177 83970 28830 97940 54768
23862 26481 29871 14362 44500 45647 04581 18343 44858 29273
12136 49014 63320 64479 32808 04084 48736 26592 58158 72068
68738 20828 18693 80488 99221 70318 96593 32752 63605 95277
03862 76665 66095 26227 52586 14708 67496 43440 74308 14181
14188 42650 77636 59175 54719 48430 68621 31143 81046 26035
61492 36434 80503 90282 06407 48286 37582 20246 48074 38454
77456 90853 52255 70153 39996 46945 67133 83116 17488 58582
71258 68681 04755 89675 03528 50182 48516 37787 50667 88702
88959 71306 42549 79758 75622 39243 25693 05702 60469 58848
76304 67409 35909 00148 51258 30037 25616 43072 98573 05128
62987 05299 05695 58232 91674 36743 97409 87587 23685 31742
48563 70287 18920 96261 86132 18163 26450 73491 92153 73326
31884 92475 86416 74240 82148 39106 88706 76563 69383 70024
63526 69415 59897 94273 69526 59731 55867 70255 68936 63900
59832 09301 86529 10344 61630 51335 18683 11927 03694 32300
78008 96334 27403 92300 78472 08832 79530 66642 60687 45822
89534 82026 69893 79160 90021 19825 67495 72247 85037 63569
90305 71458 97162 62424 59028 11842 86236 05722 40047 75115
55587 85639 05907 68993 03504 12412 84006 31150 19756 61981
60321 81911 61193 10240 32766 64880 07982 93142 45888 37511
27071 47053 59712 48756 32210 99249 74565 03656 52698 14298
85329 96415 59420 84112 11983 40350 58524 26158 75907 51198
75758 10674 83379 35599 78440 48897 18755 75833 55092 20270
49231 05021 11726 72727 81433 36304 60375 47465 91063 74162
92207 95306 51202 97799 70040 21610 30901 31216 93914 15102
15881 20224 40278 30693 25379 66000 25177 92056 42336 98944
04254 20521 63422 75453 37970 29259 08460 24190 01611 75491
15168 69458 26352 81196 45267 71661 00007 03533 99439 30152
97696 95433 77093 29715 27430 04333 19891 80933 95322 11037
47035 42933 41446 41754 63149 08814 23566 08491 72637 05084
08558 89815 49955 04038 01104 50115 71046 01940 52807 40786
01762 11512 27180 04855 56193 10022 54050 45620 24119 99284
91042 92421 22022 47997 10625 53112 44479 96308 18993 40191
20404 63834 97677 52424 75074 22585 79244 87356 61297 26405
18741 27189 93166 33044 36287 10281 71887 14867 93720 90431
81507 04872 99797 49425 32732 51919 33871 65948 01761 98884
35121 03079 49877 39995 53875 77930 49693 41446 09064 08744
16410 57281 99937 94601 31575 21575 65576 25239 91819 51966
96157 89338 41000 33260 74534 79898 04528 80459 72264 31156
36352 22616 53596 80475 94985 72335 46218 91870 16211 38705
67277 76247 09318 19524 70835 19505 39342 05402 64380 89184
20184 66996 06160 01380 40489 45372 88293 13873 53721 99546
20402 04780 15981 72736 79356 82711 93176 39976 39535 42344
86815 42366 28418 65641 11959 57024 88691 50049 37140 03449
01022 03585 15534 72293 61105 67027 31571 79695 84760 24401
57580 73534 59301 39239 45487 13237 36463 56115 63827 31365
75250 80414 90536 29297 12328 12475 14401 33464 33350 94203
70050 91550 46395 82144 74361 89969 91474 78829 72975 15084
35059 10345 86663 39258 54179 31859 71831 57194 49504 83962
05323 08164 81843 28764 29223 45631 23562 67115 78730 14617
26728 56687 39341 36579 63536 05385 98488 80391 40924 57247
03215 77594 70199 80741 03777 13372 59106 06983 26610 00132
81992 12128 58862 79658 06086 57306 26186 78083 21068 48495
42969 95939 23903 88425 77256 82036 31215 57366 84738 02528
92802 05240 38538 20185 62703 65079 97961 15718 35507 26145
36178 80003 45674 64103 27860 69567 05208 29370 10189 99959
03916 11998 65107 09030 57822 63187 09810 33440 47322 41861
69934 93035 24848 99444 78399 14878 93374 88139 76424 82072
26801 70492 30607 91699 05103 51231 37880 55379 12317 54858
12306 24706 63606 84474 88744 14645 73995 51598 11518 13092
21825 39021 24523 55296 08827 71534 99768 48441 40953 80319
31496 60266 80192 01644 18359 94822 72448 44561 85506 02617
53015 69824 97708 05403 73882 44265 06072 22964 98051 82066
93169 55930 08551 52475 85716 54273 46577 17179 17187 37853
10361 22250 07605 63353 96436 98058 69325 25923 31577 10367
50044 75662 86207 49295 44181 45388 08042 74696 45914 55827
80380 39977 83130 96598 16434 79487 65481 42887 87173 00330
10849 27271 94733 07545 82748 38319 89005 63963 47914 60155
83499 27645 62340 56255 58343 58644 72687 70297 40518 73921
91164 62449 79837 96722 73909 10854 35641 56283 78557 25029
24776 22323 03749 73343 69811 81053 64711 84093 60692 53068
91269 48770 72555 69793 89636 40518 85274 53838 30916 23007
2^1476413
39718 07993 04185 08247 62630 19134 39932 06268 28763 42405
22843 09240 00402 70070 17703 38480 61710 10958 47921 90933
34158 69744 26697 58950 70681 71271 08265 04188 41282 83845
00473 94087 27165 83594 64630 79747 59005 63066 43801 47273
20612 83795 52006 55086 90113 79771 18089 19231 86715 76316
70357 08067 00294 67446 82183 94521 72569 56804 75716 73291
69951 97950 72416 89333 20974 13172 93211 79559 09451 18108
14477 95437 07565 07627 85101 75509 35124 64460 15612 39620
60611 43641 22979 59845 33364 10734 34751 35590 74288 17390
82538 66743 19635 53794 89347 17280 15141 44745 74171 59468
12955 65489 93634 49275 93759 50481 10075 77259 25327 32490
43792 79757 91796 74826 29966 18625 40119 63373 36886 84378
81428 64001 30770 62173 51491 09284 64544 21747 27872 44545
60519 68114 10740 57462 16691 84738 63636 93665 75240 63878
49971 48276 42353 52158 13296 29336 12258 93224 09968 76349
10089 35966 17101 99678 22549 90188 09956 37453 79971 67367
94818 68467 22064 79197 19682 65954 84990 07344 58614 62500
85654 53252 72530 58513 62033 51662 61980 93510 39360 83942
60256 23394 72995 31069 37458 27602 57827 59853 53646 13917
82328 90951 26640 09572 18527 01162 02374 09380 21277 45224
94634 58060 09261 65827 66767 32055 19602 62571 55911 12229
80682 11701 36088 20841 17172 17129 08494 03901 07430 98864
53620 50557 12510 83802 64939 76471 28526 02189 15033 59049
50993 72432 62446 38789 06944 97392 00738 99920 55860 47041
70863 95183 39834 66215 37610 55758 19084 11920 01861 21419
88601 13792 26433 57152 99645 52976 29828 45472 49039 81535
75002 51777 66179 37015 61378 46472 71804 90519 93741 14821
55173 33432 71215 07718 41358 70355 67940 57661 95881 09536
47724 52962 59742 28724 89000 91294 09162 36686 89716 58546
24272 98029 26641 28958 65616 08168 97472 53185 16317 44137
37476 41656 37387 60977 98443 40637 93186 65505 27211 90554
07725 53331 32190 52455 05172 29417 34992 86881 48616 28362
28376 85301 55273 18351 09438 96861 37242 62287 62092 52071
22984 72869 61007 80564 12814 96572 75164 40492 96148 76909
54913 81707 04511 40306 79992 93891 34267 66232 34977 17165
42517 37362 09511 79350 07057 00364 97032 75575 01335 77405
77919 42612 85099 59517 51244 78486 51386 11405 20939 17697
52609 34818 71818 00297 02516 60074 51232 86145 97146 10257
25974 10598 11391 16465 83348 73487 94819 75174 47370 63484
97127 40574 37841 92523 72264 36326 52901 46983 84307 46652
63769 84951 72833 48481 64296 78213 77413 53127 38767 40049
27053 38831 19795 88547 39053 19463 11735 40511 37873 27801
19664 18603 73058 20689 23261 02670 37366 23854 07388 64601
56017 92668 54807 84601 56944 17665 59061 33285 21374 91645
79069 64053 39787 58321 80042 39651 34991 44495 70075 27139
80611 42917 94325 24849 18056 23685 72472 11444 80095 50231
11175 71278 11815 37986 07008 24825 68012 62300 39513 23963
20643 63823 22386 20480 65533 29760 15965 86284 91776 75022
54142 94107 19424 97512 64421 98499 49130 07313 05396 28597
70659 92831 18841 68224 23966 80701 17048 52317 51815 02397
51516 21349 66758 71199 56880 97794 37511 51667 10184 40540
98462 10042 23453 45455 62866 72609 20750 94931 82127 48325
84415 90613 02405 95599 40080 43220 61802 62433 87828 30204
31762 40448 80556 61386 50759 32000 50355 84112 84673 97888
08508 41043 26845 50906 75940 58518 16920 48380 03223 50982
30337 38916 52705 62392 90535 43322 00014 07067 98878 60305
95393 90867 54186 59430 54860 08666 39783 61867 90644 22074
94070 85866 82892 83509 26298 17628 47132 16983 45274 10168
17117 79630 99910 08076 02209 00231 42092 03881 05614 81572
03524 23024 54360 09711 12386 20045 08100 91240 48239 98569
82085 84842 44044 95994 21251 06224 88959 92616 37986 80382
40809 27669 95355 04849 50148 45171 58489 74713 22594 52810
37482 54379 86332 66088 72574 20563 43774 29735 87441 80863
63014 09745 99594 98850 65465 03838 67743 31896 03523 97768
70242 06158 99754 79991 07751 55860 99386 82892 18128 17488
32821 14563 99875 89202 63150 43151 31152 50479 83639 03933
92315 78676 82000 66521 49069 59796 09057 60919 44528 62312
72704 45233 07193 60951 89971 44670 92437 83740 32422 77411
34555 52494 18636 39049 41670 39010 78684 10805 28761 78368
40369 33992 12320 02760 80978 90745 76586 27747 07443 99092
40804 09560 31963 45473 58713 65423 86352 79952 79070 84689
73630 84732 56837 31282 23919 14049 77383 00098 74280 06898
02044 07170 31069 44587 22211 34054 63143 59391 69520 48803
15161 47069 18602 78478 90974 26474 72927 12231 27654 62731
50501 60829 81072 58594 24656 24950 28802 66928 66701 88407
40101 83100 92791 64289 48723 79939 82949 57659 45950 30168
70118 20691 73326 78517 08358 63719 43663 14388 99009 67924
10646 16329 63686 57528 58446 91262 47125 34231 57460 29234
53457 13374 78682 73159 27072 10771 96977 60782 81849 14494
06431 55189 40399 61482 07554 26745 18212 13966 53220 00265
63984 24257 17725 59316 12173 14612 52373 56166 42136 96990
85939 91878 47807 76851 54513 64072 62431 14733 69476 05057
85604 10480 77076 40371 25407 30159 95922 31436 71014 52290
72357 60006 91349 28206 55721 39134 10416 58740 20379 99918
07832 23997 30214 18061 15645 26374 19620 66880 94644 83723
39869 86070 49697 98889 56798 29757 86749 76279 52849 64144
53603 40984 61215 83398 10207 02462 75761 10758 24635 09716
24612 49413 27213 68949 77488 29291 47991 03196 23036 26184
43650 78042 49047 10592 17655 43069 99536 96882 81907 60638
62993 20533 60384 03288 36719 89645 44896 89123 71012 05235
06031 39649 95416 10807 47764 88530 12144 45929 96103 64133
86339 11860 17103 04951 71433 08546 93154 34358 34374 75706
20722 44500 15211 26707 92873 96117 38650 51846 63154 20735
00089 51325 72414 98590 88362 90776 16085 49392 91829 11655
60760 79955 66261 93196 32869 58975 30962 85775 74346 00660
21698 54543 89466 15091 65496 76639 78011 27926 95829 20311
66998 55291 24681 12511 16687 17289 45375 40594 81037 47843
82329 24899 59675 93445 47818 21708 71283 12567 57114 50058
49552 44646 07499 46687 39623 62107 29423 68187 21385 06137
82538 97541 45111 39587 79272 81037 70549 07676 61832 46015
2^1476414
79436 15986 08370 16495 25260 38268 79864 12536 57526 84810
45686 18480 00805 40140 35406 76961 23420 21916 95843 81866
68317 39488 53395 17901 41363 42542 16530 08376 82565 67690
00947 88174 54331 67189 29261 59495 18011 26132 87602 94546
41225 67591 04013 10173 80227 59542 36178 38463 73431 52633
40714 16134 00589 34893 64367 89043 45139 13609 51433 46583
39903 95901 44833 78666 41948 26345 86423 59118 18902 36216
28955 90874 15130 15255 70203 51018 70249 28920 31224 79241
21222 87282 45959 19690 66728 21468 69502 71181 48576 34781
65077 33486 39271 07589 78694 34560 30282 89491 48343 18936
25911 30979 87268 98551 87519 00962 20151 54518 50654 64980
87585 59515 83593 49652 59932 37250 80239 26746 73773 68757
62857 28002 61541 24347 02982 18569 29088 43494 55744 89091
21039 36228 21481 14924 33383 69477 27273 87331 50481 27756
99942 96552 84707 04316 26592 58672 24517 86448 19937 52698
20178 71932 34203 99356 45099 80376 19912 74907 59943 34735
89637 36934 44129 58394 39365 31909 69980 14689 17229 25001
71309 06505 45061 17027 24067 03325 23961 87020 78721 67885
20512 46789 45990 62138 74916 55205 15655 19707 07292 27835
64657 81902 53280 19144 37054 02324 04748 18760 42554 90449
89269 16120 18523 31655 33534 64110 39205 25143 11822 24459
61364 23402 72176 41682 34344 34258 16988 07802 14861 97729
07241 01114 25021 67605 29879 52942 57052 04378 30067 18099
01987 44865 24892 77578 13889 94784 01477 99841 11720 94083
41727 90366 79669 32430 75221 11516 38168 23840 03722 42839
77202 27584 52867 14305 99291 05952 59656 90944 98079 63071
50005 03555 32358 74031 22756 92945 43609 81039 87482 29643
10346 66865 42430 15436 82717 40711 35881 15323 91762 19072
95449 05925 19484 57449 78001 82588 18324 73373 79433 17092
48545 96058 53282 57917 31232 16337 94945 06370 32634 88274
74952 83312 74775 21955 96886 81275 86373 31010 54423 81108
15451 06662 64381 04910 10344 58834 69985 73762 97232 56724
56753 70603 10546 36702 18877 93722 74485 24575 24185 04142
45969 45739 22015 61128 25629 93145 50328 80985 92297 53819
09827 63414 09022 80613 59985 87782 68535 32464 69954 34330
85034 74724 19023 58700 14114 00729 94065 51150 02671 54811
55838 85225 70199 19035 02489 56973 02772 22810 41878 35395
05218 69637 43636 00594 05033 20149 02465 72291 94292 20514
51948 21196 22782 32931 66697 46975 89639 50348 94741 26969
94254 81148 75683 85047 44528 72653 05802 93967 68614 93305
27539 69903 45666 96963 28593 56427 54827 06254 77534 80098
54106 77662 39591 77094 78106 38926 23470 81022 75746 55602
39328 37207 46116 41378 46522 05340 74732 47708 14777 29203
12035 85337 09615 69203 13888 35331 18122 66570 42749 83291
58139 28106 79575 16643 60084 79302 69982 88991 40150 54279
61222 85835 88650 49698 36112 47371 44944 22889 60191 00462
22351 42556 23630 75972 14016 49651 36025 24600 79026 47926
41287 27646 44772 40961 31066 59520 31931 72569 83553 50045
08285 88214 38849 95025 28843 96998 98260 14626 10792 57195
41319 85662 37683 36448 47933 61402 34097 04635 03630 04795
03032 42699 33517 42399 13761 95588 75023 03334 20368 81081
96924 20084 46906 90911 25733 45218 41501 89863 64254 96651
68831 81226 04811 91198 80160 86441 23605 24867 75656 60408
63524 80897 61113 22773 01518 64001 00711 68225 69347 95776
17016 82086 53691 01813 51881 17036 33840 96760 06447 01964
60674 77833 05411 24785 81070 86644 00028 14135 97757 20611
90787 81735 08373 18861 09720 17332 79567 23735 81288 44149
88141 71733 65785 67018 52596 35256 94264 33966 90548 20336
34235 59261 99820 16152 04418 00462 84184 07762 11229 63144
07048 46049 08720 19422 24772 40090 16201 82480 96479 97139
64171 69684 88089 91988 42502 12449 77919 85232 75973 60764
81618 55339 90710 09699 00296 90343 16979 49426 45189 05620
74965 08759 72665 32177 45148 41126 87548 59471 74883 61727
26028 19491 99189 97701 30930 07677 35486 63792 07047 95537
40484 12317 99509 59982 15503 11721 98773 65784 36256 34976
65642 29127 99751 78405 26300 86302 62305 00959 67278 07867
84631 57353 64001 33042 98139 19592 18115 21838 89057 24625
45408 90466 14387 21903 79942 89341 84875 67480 64845 54822
69111 04988 37272 78098 83340 78021 57368 21610 57523 56736
80738 67984 24640 05521 61957 81491 53172 55494 14887 98184
81608 19120 63926 90947 17427 30847 72705 59905 58141 69379
47261 69465 13674 62564 47838 28099 54766 00197 48560 13796
04088 14340 62138 89174 44422 68109 26287 18783 39040 97606
30322 94138 37205 56957 81948 52949 45854 24462 55309 25463
01003 21659 62145 17188 49312 49900 57605 33857 33403 76814
80203 66201 85583 28578 97447 59879 65899 15318 91900 60337
40236 41383 46653 57034 16717 27438 87326 28777 98019 35848
21292 32659 27373 15057 16893 82524 94250 68463 14920 58469
06914 26749 57365 46318 54144 21543 93955 21565 63698 28988
12863 10378 80799 22964 15108 53490 36424 27933 06440 00531
27968 48514 35451 18632 24346 29225 04747 12332 84273 93981
71879 83756 95615 53703 09027 28145 24862 29467 38952 10115
71208 20961 54152 80742 50814 60319 91844 62873 42029 04581
44715 20013 82698 56413 11442 78268 20833 17480 40759 99836
15664 47994 60428 36122 31290 52748 39241 33761 89289 67446
79739 72140 99395 97779 13596 59515 73499 52559 05699 28289
07206 81969 22431 66796 20414 04925 51522 21516 49270 19432
49224 98826 54427 37899 54976 58582 95982 06392 46072 52368
87301 56084 98094 21184 35310 86139 99073 93765 63815 21277
25986 41067 20768 06576 73439 79290 89793 78247 42024 10470
12062 79299 90832 21614 95529 77060 24288 91859 92207 28267
72678 23720 34206 09903 42866 17093 86308 68716 68749 51412
41444 89000 30422 53415 85747 92234 77301 03693 26308 41470
00179 02651 44829 97181 76725 81552 32170 98785 83658 23311
21521 59911 32523 86392 65739 17950 61925 71551 48692 01320
43397 09087 78932 30183 30993 53279 56022 55853 91658 40623
33997 10582 49362 25022 33374 34578 90750 81189 62074 95687
64658 49799 19351 86890 95636 43417 42566 25135 14229 00116
99104 89292 14998 93374 79247 24214 58847 36374 42770 12275
65077 95082 90222 79175 58545 62075 41098 15353 23664 92031
The summary of duplicates is:
   "card" number            value
--- 2^1476412
 193           293          63320
 161           387          97409
 573           648          41446
 223           964          56255
 191           971          91164
 969           996          40518
--- 2^1476413
 464           609          37986
 239           646          55860
 573           648          82892
 184           723          31069
 91            991          82538
 969           996          81037
--- 2^1476414
 460           586          00462
 536           663          64001
 184           723          62138
 112           856          59515
 91            991          65077

  Posted by Charlie on 2007-09-21 10:35:13
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