Substitute each of the letters by a different decimal digit from 0 to 9 to satisfy this alphametic square root extraction problem. None of O and P can be 0.

                P I E R
  --------------------
√ O P E R A T O R |
   C T
------------
      B E R
      B A C
------------------
         E  P A T
         T  I A  I
----------------------
             P I  E O R
             P I  E O R
----------------------

Note: Try to derive a non computer assisted method, although computer program/spreadsheet solutions are welcome.

Well, I know how to manually extract a square root.

1) Subtraction works just the way you expect.  With respect to this problem, that means that:

   BER
  -BEC
   ----
     AP

  From which we can conclude that A = 0 and R = C + P

  Also,

   EPAT
  -TIAI
   -----
     PIE
 
  Because A = 0, we can conclude that I = 9 and E = T + 1

2) Also, we know that P*P = CT

  Possible values are

  P  CT
  -  --
  4  16
  5  25 -- not possible, because P <> T
  6  36 -- not possible, because P <> T
  7  49 -- not possible, because I = 9
  8  64
  9  81 -- not possible, because I = 9

  So really, the only possible values are

  P  CT
  -  --
  4  16
  8  64

3)  Finally,

   OP
  -CT
   ---
     B
 
  P must be less than T, so we can rule out P = 8
 
  Therefore P = 4, C = 1, T = 6, B = 8, O = 2
 
  We already know that
     A              = 0
     I               = 9
     R = C + P  = 5
     E = T + 1   = 7
    
  So, PIER = 4975
      OPERATOR = 24750625
      and 4975*4975 does indeed = 24750625
     
  /******************************************/
 
  Extra credit:  Although we did not use it above, doing square root by hand means (among other things that)
 
  R*(20*PIE + R) = PIEOR.
 
  We could have used this to conclude (as a first step) that R = 5 and O = 2

Posted by Steve Herman on 2008-12-14 12:18:49