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 4 sixes (Posted on 2013-03-05)
What is the smallest number whose cube contains four sixes?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution | Comment 1 of 2

716 would seem to be the desired answer.

10   for N=1 to 999999999
20     Cu=N*N*N
30     Ct6=0
40     S\$=str(Cu)
50     for I=1 to len(S\$)
60       if mid(S\$,I,1)="6" then inc Ct6
70     next
80     if Ct6>=4 then print N,Cu,Ct6:inc Pct
81     if Pct>40 then cancel for:end
90   next

finds the first 41 that have four or five sixes:

` n         n^3      number of                      sixes716     367061696       4806     523606616       4874     667627624       4982     946966168       4989     967361669       41185    1664006625      41188    1676676672      51332    2363266368      41378    2616662152      41387    2668267603      41456    3086626816      41466    3150662696      41786    5696975656      41841    6239666321      41863    6466042647      41882    6665900968      42054    8665653464      42201    10662526601     42286    11946169656     42306    12262468616     42331    12665630691     42385    13566416625     42521    16022066761     42523    16060229667     42536    16309766656     52553    16639966377     42556    16698695616     52786    21624363656     42907    24566036643     42954    25776946664     42966    26092364696     42986    26623761256     43306    36133376616     43323    36693659267     43438    40636623672     43454    41206620664     43556    44966103616     43606    46889669016     43821    55786756661     43832    56269946368     43863    57646656647     5`

A little tweaking of the program shows the lowest for the given number of sixes in the cube:

`4         64                         1         36        46656                      3716       367061696                  41188      1676676672                 54055      66676466375                613832     2646396666368              718821     6666963601661              8190806    6946660616126616           91542023   3666676156163166167       103971816   62656677666653866496      1113881356  2674826866666660366016    1255009989  166465666639716668628669  13154057624 3656365366626065666266624 14`

The 2 entry is missing as no number under 36 has a cube with two 6's. The first cube with exactly two sixes is 55^3 = 166375.

Of course we can also get smaller and smaller numbers with exactly four sixes, so there is no smallest:

`-716    -367061696       4-806    -523606616       4-874    -667627624       4-982    -946966168       4-989    -967361669       4-1185   -1664006625      4-1332   -2363266368      4-1378   -2616662152      4-1387   -2668267603      4-1456   -3086626816      4-1466   -3150662696      4-1786   -5696975656      4-1841   -6239666321      4-1863   -6466042647      4-1882   -6665900968      4-2054   -8665653464      4-2201   -10662526601     4-2286   -11946169656     4-2306   -12262468616     4-2331   -12665630691     4-2385   -13566416625     4-2521   -16022066761     4-2523   -16060229667     4-2553   -16639966377     4-2786   -21624363656     4-2907   -24566036643     4-2954   -25776946664     4-2966   -26092364696     4-2986   -26623761256     4-3306   -36133376616     4-3323   -36693659267     4-3438   -40636623672     4-3454   -41206620664     4-3556   -44966103616     4-3606   -46889669016     4-3821   -55786756661     4-3832   -56269946368     4-3883   -58546666387     4-3896   -59136667136     4-3941   -61209566621     4-3951   -61676694351     4`

 Posted by Charlie on 2013-03-05 13:56:51

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