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.
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 |
computer solution
