In the following layout, each letter is to be replaced by a decimal digit. As there are more than ten letters, there's obviously repetition of digits.

  A B  
C D E F
G H I J
  K L  

The digits can be considered as forming eight numbers, read left to right and top to bottom. Four have four digits each and four have two digits each.

Seven of these eight numbers are perfect squares. Only AB is not.

What are the digits to replace the letters?

using Qbasic I searched all possible combinations and find the following single solution
 71
2304
5929
 64

code follows:
CLS 0
FOR a = 1 TO 9
 FOR b = 1 TO 9
  v = SQR(10 * a + b)
  IF INT(v) <> v THEN
   FOR s1 = 4 TO 9
    sq1 = s1 ^ 2
    g = sq1 MOD 10
    c = (sq1 - g) / 10
    FOR s2 = 4 TO 9
     sq2 = s2 ^ 2
     j = sq2 MOD 10
     f = (sq2 - j) / 10
     FOR s3 = 4 TO 9
      sq3 = s3 ^ 2
      l = sq3 MOD 10
      k = (sq3 - l) / 10
      FOR d = 0 TO 9
       FOR e = 0 TO 9
        v = SQR(f + 10 * e + 100 * d + 1000 * c)
        IF INT(v) = v THEN
         FOR h = 0 TO 9
          v = SQR(k + 10 * h + 100 * d + 1000 * a)
          IF INT(v) = v THEN
           FOR i = 0 TO 9
            v = SQR(l + 10 * i + 100 * e + 1000 * b)
            v2 = SQR(j + 10 * i + 100 * h + 1000 * g)
            IF INT(v) = v AND INT(v2) = v2 THEN
             PRINT "   " + STR$(a) + " " + STR$(b)
             PRINT STR$(c) + " " + STR$(d) + " "; STR$(e) + " " + STR$(f)
             PRINT STR$(g) + " " + STR$(h) + " " + STR$(i) + " "; STR$(j)
             PRINT "   " + STR$(k) + " " + STR$(l)
            END IF
           NEXT i
          END IF
         NEXT h
        END IF
       NEXT e
      NEXT d
     NEXT s3
    NEXT s2
   NEXT s1
  END IF
 NEXT b
NEXT a

Posted by Daniel on 2009-09-06 23:03:43