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888...888 (Posted on 2022-08-02) Difficulty: 3 of 5
Determine the smallest positive perfect cube whose base ten representation begins and ends with 888.

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution computer-aided solution | Comment 2 of 10 |
Just to make sure the front 888 do not overlap with the ending 888, the cube roots of all such numbers are:

    888      9.61179106741066
   8888      20.7141928538955
  88888      44.6287145754376
 888888      96.1499393038261
 
So all the answers must be larger than 888888.

Then, from these cube roots:

    8880000    207.079761089681
   88800000    446.139820895124
  888000000    961.179106741066
    8889999     207.15745680462
   88899999    446.307226299642
  888999999    961.539773996649

can be seen that the first three digits of the cube root will be 207, 446 or 961, depending on the length of the perfect cube.

The first part of the program below determines that there are four possible endings to the cube root: 192, 442, 692, and 942.

As a start it tries all 12 combinations of lead and trailing digits, not knowing the length of the resulting perfect cube, with no intervening digits:

endings=[];
for e=2:2:998
  p=e;
  for i=1:2
    p=mod(p*e,1000);
    if p==888
      disp(e)
      endings(end+1)=e;
    end
  end
end
beginnings=[207,446,961]
 
  for front=beginnings
    for bsub=1:length(endings)
      cr=sym(front)*1000+endings(bsub);
      v=cr^3;
      disp([cr v])
    end
  end

finding   
   
[207192, 8894446923621888]
[207442, 8926682181394888]
[207692, 8958995229917888]
[207942, 8991386162940888]
[446192, 88831160947109888]  This is it!
[446442, 88980560099382888]
[446692, 89130126667405888]
[446942, 89279860744928888]
[961192, 888035735381989888]
[961442, 888728833166762888]
[961692, 889422291492285888]
[961942, 890116110452308888]


88,831,160,947,109,888 is the smallest such perfect cube, its cube root being 446192,

But hold on; what about the overlapping cases: 44692 and 96192. Well, their cubes are 89266677421888 and 890055039909888 respectively and so don't quite meet one of the criteria.

  Posted by Charlie on 2022-08-02 09:23:26
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