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5 cubes (Posted on 2016-01-26) Difficulty: 4 of 5
Find the smallest cube for which exactly five permutations of its digits are cubes.
Rem: Runtime heavily dependent of solution's concept.

Source - disclosed later.

No Solution Yet Submitted by Ady TZIDON    
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Solution computer solution Comment 1 of 1
Thanks for the remark about runtimes: it made me think: going through all permutations of all cubes would be horrendous.  Much better to list all the cubes of a given number of digits and check pairs for being anagrams.

The first found is the 12-digit 127035954683, which has four other permutations that are cubes: 352045367981 = 7061^3, 373559126408 = 7202^3, 569310543872 = 8288^3 and 589323567104 = 8384^3.

The next set of 5 is 140283769536, 536178930624, 613258407936, 913237656408 and 936302451687.


If we want a cube with five other permutations that are cubes, it would be the 13-digit 1000600120008, which has permutations 1006012008000, 1061208000000, 8001200060001, 8012006001000 and 8120601000000.  Next would be the set 1426487591593, 1432197595648, 3496581419752, 4275981654391, 4813967954125 and 7591941538264.

The above were all found within a runtime of less than one minute. It took only a minute or two more to check the rest of the 13-digit cubes, which included only three other sets of five:

1961574655832 2981631556457 5657831164259 8631525145697 9625537115648

2097643558401 2501609484375 2595641083407 6140550947328 9083540714625

2546097986375 2567976059384 5429503678976 6587579024936 7736546098952



DefDbl A-Z
Dim crlf$, cube(20000) As String, hit(10) As String

Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For digs = 2 To 13
   DoEvents
   lower = -Int(-10 ^ ((digs - 1) / 3))
   upper = Int((10 ^ digs - 1) ^ (1 / 3))
   Text1.Text = Text1.Text & digs & Str(lower) & Str(upper) & Str(upper - lower + 1) & crlf
   
   cubect = 0
   For cr = lower To upper
     cubect = cubect + 1
     cube(cubect) = LTrim(Str(cr * cr * cr))
   Next
   
   For i = 1 To cubect
     DoEvents
     hitct = 0
     For j = i + 1 To cubect
       If isPerm(cube(i), cube(j)) Then hitct = hitct + 1: hit(hitct) = cube(j)
     Next
     If hitct >= 4 Then
       Text1.Text = Text1.Text & cube(i)
       For j = 1 To hitct
         Text1.Text = Text1.Text & " " & hit(j)
       Next
       Text1.Text = Text1.Text & crlf
     End If
   Next i
   Text1.Text = Text1.Text & crlf
 Next
 
 

 Text1.Text = Text1.Text & crlf & " done"
  
End Sub
Function isPerm(a$, b$)
  x$ = a$
  good = 1
  If Len(a$) <> Len(b$) Then isPerm = 0: Exit Function
  For i = 1 To Len(b$)
    ix = InStr(x, Mid(b, i, 1))
    If ix = 0 Then good = 0: Exit For
    x = Left(x, ix - 1) + Mid(x, ix + 1)
  Next
  isPerm = good
End Function

  Posted by Charlie on 2016-01-26 15:29:18
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