Find an expression that equals 24 using the digits 1, 4, 5, and 6.

• Each digit must be used exactly once.
• You must use only addition, subtraction, multiplication, and division.
• Digits may not be concatenated (one may not put 1 and 4 together to make 14).
• Parentheses (brackets) are allowed.

____________________________

Find at least two solutions!

Two of the solutions posted were also found by computer program.  They are listed here in RPN (Reverse Polish Notation), which avoids the need for parentheses.  In this case as only 1-digit numbers are involved, no commas are needed to separate the numbers:

4156/-/
4156/-/
654/1-/
654/1-/

(each solution was found twice due to the algorithm used)

Explaining the RPN:

In the first solution above, the first / divides the 6 into the 5. Then the - subtracts that result from the 1.  The final / divides that into the initial 4, so it's the equivalent of 4/(1-5/6).

In the same way the last solution is 6/(5/4-1).

The program:

DECLARE SUB permute (a$)
DEFDBL A-Z
source$ = "1456abc"
ops$ = "+-*/"
FOR perm = 1 TO 5040
  FOR a = 1 TO 4
   op(1) = a
   FOR b = a TO 4
    op(2) = b
    FOR c = b TO 4
     op(3) = c
     stSize = 0
     FOR i = 1 TO LEN(source$)
       ch$ = MID$(source$, i, 1)
       oper = INSTR("abc", ch$)
       IF oper THEN
        IF stSize < 2 THEN GOTO nextPerm
        SELECT CASE op(oper)
          CASE 1
           stack(stSize - 1) = stack(stSize - 1) + stack(stSize)
          CASE 2
           stack(stSize - 1) = stack(stSize - 1) - stack(stSize)
          CASE 3
           stack(stSize - 1) = stack(stSize - 1) * stack(stSize)
          CASE 4
           IF stack(stSize) = 0 THEN GOTO nextC
           stack(stSize - 1) = stack(stSize - 1) / stack(stSize)
        END SELECT
        stSize = stSize - 1
       ELSE
        stSize = stSize + 1
        stack(stSize) = VAL(ch$)
       END IF
     NEXT i
     IF stSize <> 1 THEN GOTO nextPerm
     IF ABS(stack(1) - 24) < .0000001 THEN
       FOR i = 1 TO LEN(source$)
        ch$ = MID$(source$, i, 1)
        oper = INSTR("abc", ch$)
        IF oper THEN
         PRINT MID$(ops$, op(oper), 1);
        ELSE
         PRINT MID$(source$, i, 1);
        END IF
       NEXT
       PRINT
     END IF
nextC:
    NEXT c
   NEXT b
  NEXT a
nextPerm:
  permute source$
NEXT

The permute subroutine is left out as it appears elsewhere on the site.

Edited on June 9, 2004, 9:50 am

Posted by Charlie on 2004-06-09 09:49:16