All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers > Sequences
Square String Settlement (Posted on 2010-01-28) Difficulty: 2 of 5
All the positive perfect squares 1, 4, 9, 16, 25.... are written in strictly ascending order of magnitude and without the commas and spaces, resulting in the following infinite string:

149162536496481100121144......

Reading left to right, what is the 2010th digit in the above pattern?

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 3 of 8 |

There are 3 1-digit squares, 6 2-digit squares, etc., per the following table:

                          digits     total digits
   digits     number   accounted     accounted for
                           for         thus far
      1           3           3              3
      2           6          12             15
      3          22          66             81
      4          68         272            353
      5         217        1085           1438
      6         683        4098           5536
      7        2163       15141          20677
      8        6837       54696          75373
      9       21623      194607         269980
     10       68377      683770         953750
    


So for example, the six 2-digit squares account for 12 digits in all, and added to the 3 digits from the single-digit squares, the result is 15, representing the total of the numbers of digits in all squares of two or fewer digits.

By the table, the squares with 5 or fewer digits account for 1438 digits, so we need to go the the (2010 - 1438)th or 572nd digit thereafter, among the 6-digit squares, making that the second digit (ten thousands position) in the 96th 6-digit square.

The last 5-digit square is the square of 3+6+22+68+217, or 316^2 = 99856, so the 96th 6-digit square is the square of 316 + 96, or 412^2 = 169744.

The digit in question, therefore, is 6.

For the above table:

DEFDBL A-Z
pwr = 10
FOR n = 1 TO 10
  num = INT(SQR(pwr - 1))
  numDiff = num - prevNum
  prevNum = num
  pwr = pwr * 10
  totDig = totDig + n * numDiff
  PRINT USING "############"; n; numDiff; n * numDiff; totDig
NEXT


Computer (brute force) verification:

list
    5   S1="":S2=""
   10   while len(S1)+len(S2)<2010
   20    I=I+1
   30    if len(S1)<1050 then S1=S1+cutspc(str(I*I)):else S2=S2+cutspc(str(I*I))

   40   wend
   60   print mid(S2,2010-len(S1),1)
OK
run
6
OK
?i*i
 169744
OK

 


  Posted by Charlie on 2010-01-28 14:18:33
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (10)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information