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4 sixes (Posted on 2013-03-05) Difficulty: 2 of 5
What is the smallest number whose cube contains four sixes?

No Solution Yet Submitted by Ady TZIDON    
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Solution 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
                      sixes
716     367061696       4
806     523606616       4
874     667627624       4
982     946966168       4
989     967361669       4
1185    1664006625      4
1188    1676676672      5
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
2536    16309766656     5
2553    16639966377     4
2556    16698695616     5
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
3863    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                      3
716       367061696                  4
1188      1676676672                 5
4055      66676466375                6
13832     2646396666368              7
18821     6666963601661              8
190806    6946660616126616           9
1542023   3666676156163166167       10
3971816   62656677666653866496      11
13881356  2674826866666660366016    12
55009989  166465666639716668628669  13
154057624 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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