At one period the Ancients recognised four elements:

                Fire
             Air    Earth
                Water

Solve the following base10 alphametric when the elements are in balance according to:
FIRE + EARTH = AIR + WATER,
and AIR wholly divides FIRE.

Note that by ignoring "AIR wholly divides FIRE" there are three other solutions to the alphametric.

Analytical solutions (PnP) can be reasonably achieved.

DECLARE SUB permute (a$)
CLS
s$ = "1234567890": hld$ = s$
DO
  f$ = MID$(s$, 1, 1)
  i$ = MID$(s$, 2, 1)
  r$ = MID$(s$, 3, 1)
  e$ = MID$(s$, 4, 1)
  a$ = MID$(s$, 5, 1)
  t$ = MID$(s$, 6, 1)
  h$ = MID$(s$, 7, 1)
  w$ = MID$(s$, 8, 1)

  fire = VAL(f$ + i$ + r$ + e$)
  air = VAL(a$ + i$ + r$)
  earth = VAL(e$ + a$ + r$ + t$ + h$)
  water = VAL(w$ + a$ + t$ + e$ + r$)
  IF fire + earth = air + water THEN
    IF MID$(s$, 9, 1) > MID$(s$, 10, 1) THEN
      PRINT fire; earth, air; water;
      IF fire MOD air = 0 THEN PRINT "*":  ELSE PRINT
    END IF
  END IF

  permute s$
LOOP UNTIL s$ = hld$

finds

FIRE  EARTH   AIR  WATER
9384  41802   138  51048 *
9485  50821   048  60258
9573  30721   057  40237
9681  14805   468  24018
9684  41832   168  51348
9846  61402   184  71064
9870  03724   387  13207
9873  30751   087  40537

(The leading zeros were inserted manually on the small numbers.)

The real solution is marked with an asterisk. Among the others, some have leading zeros, in either earth or air, as can be seen. As the puzzle states, there are only three alternate alphametic solutions that avoid leading zeros.

Posted by Charlie on 2013-02-15 14:33:37