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More than a million, less than ten (Posted on 2015-01-06) Difficulty: 2 of 5
This number can be expressed: - by 4 distinct digits in bases 4,8,10; - by 5 distinct digits in bases 6,7,11,15 and - by 6 distinct digits in base 16.

There is no data about other bases.

Find the number.

No Solution Yet Submitted by Ady TZIDON    
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Solution More than 3000, less than 8, computer solutions | Comment 1 of 4
To have six distinct digits in base 16 means the number has at least six digits in base 15, so the search was limited to base-15 numbers with six digits, exactly one of which was a repeat of another.

DefDbl A-Z
Dim crlf$, used(15)

Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

Private Sub Form_Load()
 ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 
 For a = 1 To 14
  used(a) = 1
  For b = 0 To 14
   If used(b) < 2 And (used(b) = 0 Or dd = 0) Then
     used(b) = used(b) + 1: If used(b) = 2 Then dd = 1
  For c = 0 To 14
   If used(c) < 2 And (used(c) = 0 Or dd = 0) Then
     used(c) = used(c) + 1: If used(c) = 2 Then dd = 1
  For d = 0 To 14
   If used(d) < 2 And (used(d) = 0 Or dd = 0) Then
     used(d) = used(d) + 1: If used(d) = 2 Then dd = 1
  For e = 0 To 14
   If used(e) < 2 And (used(e) = 0 Or dd = 0) Then
     used(e) = used(e) + 1: If used(e) = 2 Then dd = 1
  For f = 0 To 14
   If used(f) < 2 And (used(f) = 0 Or dd = 0) Then
     used(f) = used(f) + 1: If used(f) = 2 Then dd = 1
     
     If dd Then
       vinner = 15 * (c + 15 * (b + 15 * a))
       v = f + 15 * (e + 15 * (d + vinner))
       v15$ = base$(v, 15)
       v6$ = base$(v, 6)
       If dnumct(v6$) = 5 Then
         v7$ = base$(v, 7)
         If dnumct(v7$) = 5 Then
         v11$ = base$(v, 11)
         If dnumct(v11$) = 5 Then
         v16$ = base$(v, 16)
         If dnumct(v16$) = 6 Then
         v10$ = base$(v, 10)
         If dnumct(v10$) = 4 Then
         v8$ = base$(v, 8)
         If dnumct(v8$) = 4 Then
         v4$ = base$(v, 4)
         If dnumct(v4$) = 4 Then
           good = 1
           For i = 1 To Len(v11$)
            If InStr("0123456789", Mid(v11$, i, 1)) = 0 Then good = 0: Exit For
           Next
           For i = 1 To Len(v15$)
            If InStr("0123456789", Mid(v15$, i, 1)) = 0 Then good = 0: Exit For
           Next
           For i = 1 To Len(v16$)
            If InStr("0123456789", Mid(v16$, i, 1)) = 0 Then good = 0: Exit For
           Next
           ct = ct + 1
           If ct Mod 300 = 1 Or good = 1 Then
                Text1.Text = Text1.Text & v4$
                Text1.Text = Text1.Text & "  " & v8$
                Text1.Text = Text1.Text & "  " & v10$
                Text1.Text = Text1.Text & "  " & v6$
                Text1.Text = Text1.Text & "  " & v7$
                Text1.Text = Text1.Text & "  " & v11$
                Text1.Text = Text1.Text & "  " & v15$
                Text1.Text = Text1.Text & "  " & v16$
                If good Then Text1.Text = Text1.Text & "  ****"
                Text1.Text = Text1.Text & crlf
           End If
         End If
         End If
         End If
         End If
         DoEvents
         End If
         End If
       End If
       
     End If
     
     used(f) = used(f) - 1: If used(f) = 1 Then dd = 0
   End If
  Next f
     used(e) = used(e) - 1: If used(e) = 1 Then dd = 0
   End If
  Next e
     used(d) = used(d) - 1: If used(d) = 1 Then dd = 0
   End If
  Next d
     used(c) = used(c) - 1: If used(c) = 1 Then dd = 0
   End If
  Next c
     used(b) = used(b) - 1: If used(b) = 1 Then dd = 0
   End If
  Next b
  used(a) = 0
 Next



 Text1.Text = Text1.Text & ct & " done"
End Sub

Function base$(n, b)
  v$ = ""
  n2 = n
  Do
    d = n2 Mod b
    n2 = n2 \ b
    v$ = Mid("0123456789abcdefghijklmnopqrstuvwxyz", d + 1, 1) + v$
  Loop Until n2 = 0
  base$ = v$
End Function

Function dnumct(s$)

   diffnums = 0
   For i = 1 To Len(s$)
     If InStr(s$, Mid(s$, i, 1)) = i Then diffnums = diffnums + 1
   Next
   dnumct = diffnums
End Function

The program found 3659 solutions. Every 300th one is listed below as a sampling, but also listed are seven special solutions: ones in which, even in bases 11, 15 and 16, only digits valid in base 10 are used.

      4         8        10        6          7       11       15     16
10010333123  4047733  1069019  34525055  12041450  670196  161b2e  104fdb
11230331332  5547576  1494910  52012514  15464224  93116a  1e7e0a  16cf7e
13233113100  7572720  2029008  111253320  23150322  1166473  2a12c3  1ef5d0
21012000320  11060070  2383928  123032412  26156141  1389098  321538  246038  ****
21012000321  11060071  2383929  123032413  26156142  1389099  321539  246039  ****
21020031121  11101531  2392921  123142201  26224306  1394923  324031  248359  ****
21300012010  11600604  2556292  130442404  30504514  1496642  357647  270184  ****
22232112301  12562661  2811313  140131201  32616151  16501aa  3a7ead  2ae5b1
32112302033  16266217  3763343  212354515  43662563  2140501  4e50e8  396c8f
100012132120  20063630  4220824  230244504  50606416  2423193  585934  406798  ****
100013020320  20071070  4223544  230305240  50620353  2425236  586649  407238  ****
102113012000  22270600  4813184  251055132  55624415  2798242  6511de  497180
111210333102  25447722  5656530  321123350  66036225  3213920  76b020  564fd2
121000130201  31003441  6555425  352301105  106502012  3778208  897535  640721  ****
121102323122  31227332  6631130  354043402  110235512  381a080  8aeba5  652eda
123222303311  33526365  7253237  415243445  115436315  4104512  984192  6eacf5
132332012000  36760600  8118656  450002212  126002360  4645737  aa57db  7be180
202111330102  42257422  9002770  520543254  136344100  5099a07  bcc74a  895f12
210313112312  44672666  9663926  543044222  145066466  5500718  cad5bb  9375b6
222123100102  52332022  11121682  1034213134  163350505  6306980  e9a4a7  a9b412
3659 done


  Posted by Charlie on 2015-01-06 09:30:05
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