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 3 features (Posted on 2014-07-08)
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 No Rating

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 computer solution Comment 1 of 1
The digits used by eight successive powers of 2 are shown:
`rangeof powers   digitsof 2	     used0-7        1248631-8   	   24816352-9   	   48163253-10  	   816324504-11  	   163248505-12  	   3264185096-13  	   641285097-14  	   1285604938-15  	   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 	   48163251064  1207948560 6-13 	   641285091067  1218185364 0- 7 	   1248631102  1341918624 0- 7 	   1248631113  1382468430 0- 7 	   1248631155  1544803260 1- 8 	   24816351155  1544803260 2- 9 	   48163251179  1643030820 0- 7 	   1248631220  1820315640 1- 8 	   24816351220  1820315640 2- 9 	   48163251224  1838264400 0- 7 	   1248631225  1842769950 6-13 	   641285091241  1915863246 1- 8 	   24816351241  1915863246 2- 9 	   48163251267  2038719564 5-12 	   3264185091267  2038719564 7-14 	   1285604931287  2136718584 1- 8 	   24816351287  2136718584 2- 9 	   48163251322  2315683944 1- 8 	   24816351322  2315683944 2- 9 	   48163251332  2368591704 5-12 	   3264185091332  2368591704 7-14 	   1285604931341  2416892346 0- 7 	   1248631351  2471324856 1- 8 	   24816351351  2471324856 2- 9 	   48163251464  3144218160 0- 7 	   1248631480  3248366160 0- 7 	   1248631483  3268146420 0- 7 	   1248631490  3314612280 0- 7 	   1248631504  3408861120 0- 7 	   1248631511  3456648216 0- 7 	   1248631538  3645152280 1- 8 	   24816351538  3645152280 2- 9 	   48163251567  3855120864 1- 8 	   24816351567  3855120864 2- 9 	   48163251568  3862501440 1- 8 	   24816351568  3862501440 2- 9 	   48163251576  3921885456 1- 8 	   24816351576  3921885456 2- 9 	   48163251600  4103683200 0- 7 	   1248631619  4251526380 1- 8 	   24816351619  4251526380 2- 9 	   48163251800  5841723600 3-10 	   816324501800  5841723600 4-11 	   163248501895  6815769240 6-13 	   641285091902  6891539424 1- 8 	   24816351902  6891539424 2- 9 	   48163251905  6924183510 3-10 	   816324501905  6924183510 4-11 	   163248501905  6924183510 6-13 	   641285092045  8564791290 6-13 	   641285092231 11119428936 0- 7 	   1248632236 11194323816 0- 7 	   1248632238 11224375680 3-10 	   816324502238 11224375680 4-11 	   163248502295 12103628040 0- 7 	   1248632306 12278426136 0- 7 	   1248632311 12358433016 1- 8 	   24816352311 12358433016 2- 9 	   48163252332 12698257704 6-13 	   641285092336 12763684416 0- 7 	   1248632341 12845783346 1- 8 	   24816352341 12845783346 2- 9 	   48163252367 13278377664 0- 7 	   1248632378 13464283560 1- 8 	   24816352378 13464283560 2- 9 	   48163252381 13515284586 0- 7 	   1248632387 13617672684 0- 7 	   1248632416 14119843296 0- 7 	   1248632425 14278162350 3-10 	   816324502425 14278162350 4-11 	   163248502455 14814432360 0- 7 	   1248632456 14832535536 0- 7 	   1248632489 15438246510 1- 8 	   24816352489 15438246510 2- 9 	   48163252502 15681315024 1- 8 	   24816352502 15681315024 2- 9 	   48163252521 16041138126 0- 7 	   1248632540 16406423880 0- 7 	   1248632547 16542388044 1- 8 	   24816352547 16542388044 2- 9 	   48163252549 16581372450 3-10 	   816324502549 16581372450 4-11 	   163248502552 16639963824 0- 7 	   1248632562 16836264984 0- 7 	   1248632581 17213478786 0- 7 	   1248632620 18005326440 1- 8 	   24816352620 18005326440 2- 9 	   48163252621 18025943226 3-10 	   816324502621 18025943226 4-11 	   163248502621 18025943226 6-13 	   641285092632 18253767504 3-10 	   816324502632 18253767504 4-11 	   163248502642 18462539064 3-10 	   816324502642 18462539064 4-11 	   163248502642 18462539064 6-13 	   641285092680 19270384560 5-12 	   3264185092680 19270384560 7-14 	   1285604932683 19335146820 3-10 	   816324502683 19335146820 4-11 	   163248502683 19335146820 6-13 	   641285092697 19639289694 0- 7 	   1248632707 19858476204 6-13 	   641285092719 20123645280 1- 8 	   24816352719 20123645280 2- 9 	   48163252726 20279411856 6-13 	   641285092741 20615899746 6-13 	   641285092773 21346138050 1- 8 	   24816352773 21346138050 2- 9 	   48163252777 21438612174 0- 7 	   1248632784 21601083840 0- 7 	   1248632787 21670965084 6-13 	   641285092808 22164358320 1- 8 	   24816352808 22164358320 2- 9 	   48163252811 22235448516 0- 7 	   1248632826 22593180456 3-10 	   816324502826 22593180456 4-11 	   163248502826 22593180456 6-13 	   641285092836 22833787416 0- 7 	   1248632863 23491945680 3-10 	   816324502863 23491945680 4-11 	   163248502863 23491945680 6-13 	   641285092868 23615198040 3-10 	   816324502868 23615198040 4-11 	   163248502868 23615198040 6-13 	   641285092890 24162631080 0- 7 	   1248632892 24212813064 0- 7 	   1248632896 24313385376 1- 8 	   24816352896 24313385376 2- 9 	   48163252936 25334468016 1- 8 	   24816352936 25334468016 2- 9 	   48163252940 25438120680 1- 8 	   24816352940 25438120680 2- 9 	   48163252984 26597018640 6-13 	   641285092992 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

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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