Determine all possible positive integer base(s) T such that:

(111)_T = (aabbcc)₆, where each of a, b and c denotes a different base 6 digit from 0 to 5 and, a is not zero.

Note: Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.

The following program does not check that a, b and c are all different, but no cases of duplication were found among them:

t = 2
DO
  n = t * t + t + 1
  n6$ = ""
  DO
    r = n MOD 6
    n = n \ 6
    n6$ = LTRIM$(STR$(r)) + n6$
  LOOP UNTIL n = 0
  IF LEN(n6$) = 6 THEN
   IF MID$(n6$, 1, 1) = MID$(n6$, 2, 1) THEN
   IF MID$(n6$, 3, 1) = MID$(n6$, 4, 1) THEN
   IF MID$(n6$, 5, 1) = MID$(n6$, 6, 1) THEN
    PRINT t, n6$
   END IF
   END IF
   END IF
  END IF
  t = t + 1
LOOP UNTIL LEN(n6$) > 6

resulting in:

base     base-6 representation 
 100          114433
 137          223311

Posted by Charlie on 2008-10-07 13:03:41