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 Same digits - 3 cubes (Posted on 2015-10-14)
The digits of 5^3=125, can be permuted to form 8^3=512.

Find the smallest cube whose digits can be permuted to produce
two other cubes.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution Comment 1 of 1
41063625 is the smallest such cube, from

41063625   56623104 66430125
56623104   66430125 41063625
66430125   41063625 56623104

The three numbers are the cubes of 345, 384 and 405.

These are the first three rows of output from

DefDbl A-Z
Dim crlf\$, hist(90)

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For n = 3 To 9999
DoEvents
cubect = 0
cube = n * n * n
s\$ = LTrim(Str(cube)): h\$ = s
Do
permute s
If s = h Then Exit Do
v = Val(s)
cr = Int(v ^ (1 / 3) + 0.5)
If cr * cr * cr = v And Left(s, 1) <> "0" Then cubect = cubect + 1: hist(cubect) = v
Loop
If cubect > 1 Then
Text1.Text = Text1.Text & h & "  "
For i = 1 To cubect
Text1.Text = Text1.Text & Str(hist(i))
Next
Text1.Text = Text1.Text & crlf
End If
Next

Text1.Text = Text1.Text & crlf & " done"

End Sub

The program finds further values:

Filtering out duplicates (as each set is listed for each of its members):

1003003001   1030301000 1331000000
1006012008   1061208000 8012006001 8120601000
1086373952   1375036928 5097328361
1287913472   1897413272 3877292411
2062933417   3029741623 9274236301
3595640768   6740558369 7066834559
5204699837   6804992375 8402569937

The program was stopped before terminating. The list includes 1006012008 = 1002^3, which has 3 permutations that are cubes and are in fact cubes of permutations of 1002.

 Posted by Charlie on 2015-10-14 10:45:14

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