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 2011 Square End (Posted on 2011-10-06)
Determine the smallest octal (base 8) perfect square which ends with 2011 (reading left to right). What are the next two smallest octal perfect squares with this property?

***For an extra challenge, solve this puzzle without using a computer program.

 No Solution Yet Submitted by K Sengupta No Rating

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 computer solution | Comment 1 of 3

10   Md=8^4
20   Vl=2*8^3+8+1
30   for N=1 to 999999
40    if (N*N) @ Md=Vl then
50       :print N,N*N,
60       :Dc=N*N:Rp=""
70       :while Dc>0
80         :Rp=cutspc(str(Dc @ 8))+Rp
90         :Dc=Dc\8
100       :wend
110       :print Rp
120       :Ct=Ct+1:if Ct>40 then end
130   next

Decimal 509^2 = 259,081, which, represented in octal, is 772011, and is the lowest such. The next is  1539^2 = 2368521, which in octal is 11022011. The third is the 30742011 octal seen in the table below.

The decimal representations of such numbers are:

`   n      n^2        octal representation of square 509     259081         772011 1539    2368521        11022011 2557    6538249        30742011 3587    12866569       61052011 4605    21206025       120712011 5635    31753225       171102011 6653    44262409       250662011 7683    59028489       341132011 8701    75707401       440632011 9731    94692361       551162011 10749   115541001      670602011 11779   138744841      1021212011 12797   163763209      1160552011 13827   191185929      1331242011 14845   220374025      1510522011 15875   252015625      1701272011 16893   285373449      2100472011 17923   321233929      2311322011 18941   358761481      2530442011 19971   398840841      2761352011 20989   440538121      3220412011 22019   484836361      3471402011 23037   530703369      3750362011 24067   579220489      4241432011 25085   629257225      4540332011 26115   681993225      5051462011 27133   736199689      5370302011 28163   793154569      5721512011 29181   851530761      6260252011 30211   912704521      6631542011 31229   975250441      7210222011 32259   1040643081     7601572011 33277   1107358729     10200172011 34307   1176970249     10611622011 35325   1247855625     11230142011 36355   1321686025     11661652011 37373   1396741129     12320112011 38403   1474790409     12771702011 39421   1554015241     13450062011 40451   1636283401     14141732011 41469   1719677961     14640032011 `

The differences between successive values of n (the square roots of the intended numbers) alternate between 1030 and 1018, decimal.

 Posted by Charlie on 2011-10-06 14:16:03

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