Substitute each of the capital letters by a different base six digit from 0 to 5 to satisfy the following set of alphametic relationships. None of the numbers can contain any leading zero.

• The base six number ICUS is a perfect square, and:
• A is a perfect square, and:
• The base six number MI is a perfect fifth power, and:
• The base six number US is a perfect square.

Determine the value assumed by the base six number AMICUS.

FOR i = 0 TO 5
  used(i) = 1
FOR c = 0 TO 5
 IF used(c) = 0 THEN
  used(c) = 1
FOR u = 0 TO 5
 IF used(u) = 0 THEN
  used(u) = 1
FOR s = 0 TO 5
 IF used(s) = 0 THEN
  used(s) = 1
FOR a = 0 TO 5
 IF used(a) = 0 THEN
  used(a) = 1
FOR m = 0 TO 5
 IF used(m) = 0 THEN
  used(m) = 1
 
  sr = INT(SQR(a) + .5)
  IF sr * sr = a THEN
    icus = i * 216 + c * 36 + u * 6 + s
    sr = INT(SQR(icus) + .5)
    IF sr * sr = icus THEN
     mi = m * 6 + i
     fifth = INT(mi ^ (1 / 5) + .5)
     IF fifth * fifth * fifth * fifth * fifth = mi THEN
      us = u * 6 + s
      sr = INT(SQR(us) + .5)
      IF sr * sr = us THEN
       PRINT i; c; u; s, a, m; i, u; s
       PRINT icus, a, mi, us
       PRINT
       amicus = (36 * a + 6 * m + i) * 216 + c * 36 + u * 6 + s
       PRINT a; m; i; c; u; s
       PRINT amicus
      END IF
     END IF
    END IF
  END IF
 
  used(m) = 0
 END IF
NEXT
  used(a) = 0
 END IF
NEXT
  used(s) = 0
 END IF
NEXT
  used(u) = 0
 END IF
NEXT
  used(c) = 0
 END IF
NEXT
  used(i) = 0
NEXT

gives

2  0  1  3    4             5  2          1  3
441           4             32            9

4  5  2  0  1  3
38025

meaning ICUS is 2013 or decimal 441, A is 4, MI is 52 or decimal 32, and US is 13 or decimal 9; AMICUS is 452013 or decimal 38025.

Posted by Charlie on 2009-09-13 14:35:09