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3 features (Posted on 2014-07-08) Difficulty: 3 of 5
My integer number N can be found just by referring to its peculiar features:

1. Eight successive integer powers of two can be written using only
the digits of N.

2. There is only one digit of N not needed to express those powers.

3. N=K*(K+1)*(K+2)

Find K.

See The Solution Submitted by Ady TZIDON    
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Solution computer solution Comment 1 of 1
The digits used by eight successive powers of 2 are shown:

range
of powers   digits
of 2     used
0-7        124863
1-8     2481635
2-9     4816325
3-10     81632450
4-11     16324850
5-12     326418509
6-13     64128509
7-14     128560493
8-15     2561048937

Any set of eight successive powers of 2 from then on also uses all ten decimal digits.

Several values of k under 3000 work:

		 range	   digits used
of powers   in those 
  k     n of 2     powers
 182     6128304 0- 7      124863
 231    12486936 0- 7   124863
 306    28934136 0- 7   124863
 336    38272416 0- 7   124863
 344    41063280 0- 7   124863
 487   116213784 0- 7   124863
 567   183249864 0- 7   124863
 597   213846594 1- 8   2481635
 597   213846594 2- 9   4816325
 617   236028414 0- 7   124863
 753   428660310 0- 7   124863
 754   430368120 0- 7   124863
 857   631627854 1- 8   2481635
 857   631627854 2- 9   4816325
 932   812165304 1- 8   2481635
 932   812165304 2- 9   4816325
1064  1207948560 6-13   64128509
1067  1218185364 0- 7   124863
1102  1341918624 0- 7   124863
1113  1382468430 0- 7   124863
1155  1544803260 1- 8   2481635
1155  1544803260 2- 9   4816325
1179  1643030820 0- 7   124863
1220  1820315640 1- 8   2481635
1220  1820315640 2- 9   4816325
1224  1838264400 0- 7   124863
1225  1842769950 6-13   64128509
1241  1915863246 1- 8   2481635
1241  1915863246 2- 9   4816325
1267  2038719564 5-12   326418509
1267  2038719564 7-14   128560493
1287  2136718584 1- 8   2481635
1287  2136718584 2- 9   4816325
1322  2315683944 1- 8   2481635
1322  2315683944 2- 9   4816325
1332  2368591704 5-12   326418509
1332  2368591704 7-14   128560493
1341  2416892346 0- 7   124863
1351  2471324856 1- 8   2481635
1351  2471324856 2- 9   4816325
1464  3144218160 0- 7   124863
1480  3248366160 0- 7   124863
1483  3268146420 0- 7   124863
1490  3314612280 0- 7   124863
1504  3408861120 0- 7   124863
1511  3456648216 0- 7   124863
1538  3645152280 1- 8   2481635
1538  3645152280 2- 9   4816325
1567  3855120864 1- 8   2481635
1567  3855120864 2- 9   4816325
1568  3862501440 1- 8   2481635
1568  3862501440 2- 9   4816325
1576  3921885456 1- 8   2481635
1576  3921885456 2- 9   4816325
1600  4103683200 0- 7   124863
1619  4251526380 1- 8   2481635
1619  4251526380 2- 9   4816325
1800  5841723600 3-10   81632450
1800  5841723600 4-11   16324850
1895  6815769240 6-13   64128509
1902  6891539424 1- 8   2481635
1902  6891539424 2- 9   4816325
1905  6924183510 3-10   81632450
1905  6924183510 4-11   16324850
1905  6924183510 6-13   64128509
2045  8564791290 6-13   64128509
2231 11119428936 0- 7   124863
2236 11194323816 0- 7   124863
2238 11224375680 3-10   81632450
2238 11224375680 4-11   16324850
2295 12103628040 0- 7   124863
2306 12278426136 0- 7   124863
2311 12358433016 1- 8   2481635
2311 12358433016 2- 9   4816325
2332 12698257704 6-13   64128509
2336 12763684416 0- 7   124863
2341 12845783346 1- 8   2481635
2341 12845783346 2- 9   4816325
2367 13278377664 0- 7   124863
2378 13464283560 1- 8   2481635
2378 13464283560 2- 9   4816325
2381 13515284586 0- 7   124863
2387 13617672684 0- 7   124863
2416 14119843296 0- 7   124863
2425 14278162350 3-10   81632450
2425 14278162350 4-11   16324850
2455 14814432360 0- 7   124863
2456 14832535536 0- 7   124863
2489 15438246510 1- 8   2481635
2489 15438246510 2- 9   4816325
2502 15681315024 1- 8   2481635
2502 15681315024 2- 9   4816325
2521 16041138126 0- 7   124863
2540 16406423880 0- 7   124863
2547 16542388044 1- 8   2481635
2547 16542388044 2- 9   4816325
2549 16581372450 3-10   81632450
2549 16581372450 4-11   16324850
2552 16639963824 0- 7   124863
2562 16836264984 0- 7   124863
2581 17213478786 0- 7   124863
2620 18005326440 1- 8   2481635
2620 18005326440 2- 9   4816325
2621 18025943226 3-10   81632450
2621 18025943226 4-11   16324850
2621 18025943226 6-13   64128509
2632 18253767504 3-10   81632450
2632 18253767504 4-11   16324850
2642 18462539064 3-10   81632450
2642 18462539064 4-11   16324850
2642 18462539064 6-13   64128509
2680 19270384560 5-12   326418509
2680 19270384560 7-14   128560493
2683 19335146820 3-10   81632450
2683 19335146820 4-11   16324850
2683 19335146820 6-13   64128509
2697 19639289694 0- 7   124863
2707 19858476204 6-13   64128509
2719 20123645280 1- 8   2481635
2719 20123645280 2- 9   4816325
2726 20279411856 6-13   64128509
2741 20615899746 6-13   64128509
2773 21346138050 1- 8   2481635
2773 21346138050 2- 9   4816325
2777 21438612174 0- 7   124863
2784 21601083840 0- 7   124863
2787 21670965084 6-13   64128509
2808 22164358320 1- 8   2481635
2808 22164358320 2- 9   4816325
2811 22235448516 0- 7   124863
2826 22593180456 3-10   81632450
2826 22593180456 4-11   16324850
2826 22593180456 6-13   64128509
2836 22833787416 0- 7   124863
2863 23491945680 3-10   81632450
2863 23491945680 4-11   16324850
2863 23491945680 6-13   64128509
2868 23615198040 3-10   81632450
2868 23615198040 4-11   16324850
2868 23615198040 6-13   64128509
2890 24162631080 0- 7   124863
2892 24212813064 0- 7   124863
2896 24313385376 1- 8   2481635
2896 24313385376 2- 9   4816325
2936 25334468016 1- 8   2481635
2936 25334468016 2- 9   4816325
2940 25438120680 1- 8   2481635
2940 25438120680 2- 9   4816325
2984 26597018640 6-13   64128509
2992 26811437664 0- 7   124863

Note that in a few instances, nine successive powers of two are covered, such as k =  597; n = 213846594 for range 1- 9, where the powers of 2 use  2481635.

In this regard, k = 2863; n = 23491945680 has nine different digits, lacking only a 7. For powers of 2 from 3 to 11 (a range of nine), it has only the 9 that's not used in that range; for powers of 2 from 6 to 13 it has only the 3 that's not used in that range. Powers of 2 from 5 to 12, all its digits are used, including the 3 and the 9, so that range does not qualify.

Note that k = 182,  n = 6128304, is the only instance where there are no repeat digits in n. For example, k = 231, n = 12486936, has the digit 6 appear twice in n, even though the seven different digits that make it up satisfy the criteria of the puzzle: all the digits except the 9 are used in forming the powers of 2 from zero to seven.

DefDbl A-Z
Dim crlf$, pwr2dig(50) As String

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()
 Text1.Text = "": Text2.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 
 p2 = 1
 For i = 0 To 16
   s$ = LTrim(Str(p2))
   pwr2dig(i) = s$
   p2 = p2 * 2
 Next i
 For i = 0 To 8
   s$ = pwr2dig(i)
   For j = i + 1 To i + 7
     s$ = s$ + pwr2dig(j)
   Next
   k = 1
   Do
    c$ = Mid(s$, k, 1)
    If InStr(s$, c$) < k Then
     s$ = Left(s$, k - 1) + Mid(s$, k + 1)
    Else
     k = k + 1
    End If
   Loop Until k > Len(s$)
   pwr2dig(i) = s$
   Text1.Text = Text1.Text & i & "-" & i + 7 & "  " & pwr2dig(i) & crlf
 Next
 
 For k = 1 To 3000
  ReDim used(9)
  n = k * (k + 1) * (k + 2)
  ns$ = LTrim(Str(n))
  usedct = 0
  For i = 1 To Len(ns$)
   If used(Val(Mid(ns$, i, 1))) = 0 Then usedct = usedct + 1
   used(Val(Mid(ns$, i, 1))) = 1
  Next
  For i = 0 To 7
    If usedct = Len(pwr2dig(i)) + 1 Then
     good = 1
     For j = 1 To Len(pwr2dig(i))
      If InStr(ns$, Mid(pwr2dig(i), j, 1)) = 0 Then good = 0: Exit For
     Next
     If good Then
      Text1.Text = Text1.Text & mform(k, "###0") & mform(n, "###########0") & mform(i, "#0") & "-" & mform(i + 7, "##") & " " & pwr2dig(i) & crlf
     End If
    End If
  Next
 Next k
 Text1.Text = Text1.Text & "done"


End Sub


  Posted by Charlie on 2014-07-08 20:32:07
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