As you know, the value of pi with some decimals is 3.141592653589...

Write an expression using all digits (0-9) once each, to achieve the closest value to pi. The digits must be in the order 1, 2, 3, 4, ...8, 9, 0. You may also use as many parentheses, +, -, /, and *, as you wish, but only these symbols.

As an example: 123/45 - 6/7 + 8/9 + 0 = 2.765079... gives a difference of approximately 0.376.

You may post the difference or the approximation you achieve in the subject of your comment.

A program to find all possibilities gives (1-2/(3-4/(5/6-7*8)))*(9+0) = 3.141592920353983 (the same 355 / 113 Ady found -- double precision used, so the difference is found more accurately).

A later program, designed to allow a leading negative sign, which the first was not, found an alternate: -1 + 2/((345-6)/(78*(9+0))). This is also equivalent to 355/113.

The table below shows the successively better solutions found by the first program.  The first column shows the Reverse Polish Notation form; the second shows algebraic form; and the third shows the approximation.  Note that some of the algebraic forms involve division by a product; parentheses were not placed around the product, and you must interpret as multiplication taking precedence over division, which is not standard. Just insert the needed parentheses for standard order of operations. This is not involved with the most accurate solution.

123456789,0*                 123456789*0                 0
12345,67890/                 12345/67890                 .1818382677861246
12345,6789,0+/               12345/(6789+0)              1.818382677861246
12345,678,9,0+*/             12345/678*(9+0)             2.023107177974435
12345,67,89,0+*/             12345/67*(89+0)             2.070266644306557
12345,67,8,9,0+**/           12345/67*8*(9+0)            2.55907960199005
12345,6,789,0+*/             12345/6*(789+0)             2.607731305449937
12345,6,78,9,0+**/           12345/6*78*(9+0)            2.930911680911681
12345,6,7,89,0+**/           12345/6*7*(89+0)            3.302568218298555
12345,6,7,8,90+**/           12345/6*7*(8+90)            2.999271137026239
12345,6,7*/,8-,90/           (12345/6*7-8)/90            3.176984126984127
1234,567/,8,9,0+--           1234/567-(8-(9+0))          3.17636684303351
1234,567/,8-,9,0++           1234/567-8+9+0              3.176366843033509
1234,56,7,89,0*+*/           1234/56*(7+89*0)            3.147959183673469
1234,56,7*,8,90/+/           1234/(56*7+8/90)            3.147245522557243
1234,56,7*,8,9,0+--/         1234/(56*7-(8-(9+0)))       3.139949109414758
1234,56,7*,8,9,0+/+/         1234/(56*7+8/(9+0))         3.140837104072398
123,4-,5,6*,7,8,9,0+//+/     (123-4)/(5*6+7/(8/(9+0)))   3.141914191419142
12,34+,5,6,7,8,90/*/+/       (12+34)/(5+6/(7*8/90))      3.141463414634146
12,3,4,5,67,8,90*/+///       12/(3/(4/(5+67/8*90)))      3.14153258794655
12,3,4,5*+,6//,7,8-,90/-     12/((3+4*5)/6)-(7-8)/90     3.141545893719806
12,3,4,5/,6,7+,890/-//       12/(3/(4/5-(6+7)/890))      3.141573033707866
1,23,4,56,7-/-,8,9,0+*//     1/((23-4/(56-7))/8*(9+0))   3.141585040071238
1,2,3,4,5,6/,7,8*-/-/-,9,0+* (1-2/(3-4/(5/6-7*8)))*(9+0) 3.141592920353983

The program actually was two programs--the first to find the RPN form and the second to convert RPN to algebraic notation:

To produce RPN:

DECLARE SUB addon ()
DEFDBL A-Z
DIM SHARED stack(11)
DIM SHARED upTo, stackLevel
DIM SHARED h$, best

best = 9999
stack(1) = 1: stackLevel = 1: upTo = 1
h$ = "1"

addon

SUB addon
 nextOne = upTo + 1
 SELECT CASE nextOne
  CASE 10
   nextOne = 0
  CASE 1
   ' this is the end
   ' evaluate
   IF stackLevel = 1 THEN
     diff = ABS(stack(1) - 3.141592654#)
     IF diff < best THEN
      best = diff
      PRINT h$, stack(1)
     END IF
     EXIT SUB
    ELSE
     nextOne = 999
   END IF
 END SELECT

 'first try concatenation
 IF nextOne < 999 THEN
   lc$ = RIGHT$(h$, 1)
   ix = INSTR("0123456789", lc$)
   IF ix THEN
     h$ = h$ + LTRIM$(STR$(nextOne))
     stack(stackLevel) = 10 * stack(stackLevel) + nextOne
     utSave = upTo
     upTo = nextOne
     addon
     upTo = utSave
     stack(stackLevel) = stack(stackLevel) 10
     h$ = LEFT$(h$, LEN(h$) - 1)
   END IF
  END IF

'try comma
 IF nextOne < 999 THEN
   stackLevel = stackLevel + 1
   stack(stackLevel) = nextOne
   h$ = h$ + "," + LTRIM$(STR$(nextOne))
   utSave = upTo
   upTo = nextOne
   addon
   upTo = utSave
   stackLevel = stackLevel - 1
   h$ = LEFT$(h$, LEN(h$) - 2)
 END IF

 GOSUB tryDiadic

 EXIT SUB

tryDiadic:
  IF stackLevel >= 2 THEN
    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)
    stackLevel = stackLevel - 1

    stack(stackLevel) = s1 + s2
    h$ = h$ + "+"
    addon
    h$ = LEFT$(h$, LEN(h$) - 1)
  
    stack(stackLevel) = s1
    stackLevel = stackLevel + 1
    stack(stackLevel) = s2

    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)
    stackLevel = stackLevel - 1

    stack(stackLevel) = s1 - s2
    h$ = h$ + "-"
    addon
    h$ = LEFT$(h$, LEN(h$) - 1)
 
    stack(stackLevel) = s1
    stackLevel = stackLevel + 1
    stack(stackLevel) = s2

    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)
    stackLevel = stackLevel - 1

    stack(stackLevel) = s1 * s2
    h$ = h$ + "*"
    addon
    h$ = LEFT$(h$, LEN(h$) - 1)
 
    stack(stackLevel) = s1
    stackLevel = stackLevel + 1
    stack(stackLevel) = s2

    s1 = stack(stackLevel - 1)
    s2 = stack(stackLevel)

    IF s2 <> 0 THEN
      stackLevel = stackLevel - 1

      stack(stackLevel) = s1 / s2
      h$ = h$ + "/"
      addon
      h$ = LEFT$(h$, LEN(h$) - 1)
   
      stack(stackLevel) = s1
      stackLevel = stackLevel + 1
      stack(stackLevel) = s2
    END IF

  END IF
RETURN

END SUB

And this one to convert RPN to algebraic:

DECLARE SUB checkParen (s$, ops$)
OPEN "close2pi.txt" FOR INPUT AS #1
OPEN "clsepi.TXT" FOR OUTPUT AS #2
DO
 LINE INPUT #1, l$
 ix = INSTR(l$, " ")
 nOrig$ = LTRIM$(MID$(l$, ix))
 l$ = LEFT$(l$, ix - 1)
 stackLevel = 0
 bld$ = ""
 FOR i = 1 TO LEN(l$)
  c$ = MID$(l$, i, 1)
  ix = INSTR("0123456789", c$)
  IF ix THEN
   bld$ = bld$ + c$
   stkRaise = 1
  ELSE
   IF stkRaise THEN
    stackLevel = stackLevel + 1
    stack$(stackLevel) = bld$
    stkRaise = 0
    bld$ = ""
   END IF
   SELECT CASE c$
    CASE "+"
     stackLevel = stackLevel - 1
     stack$(stackLevel) = stack$(stackLevel) + "+" + stack$(stackLevel + 1)
    CASE "-"
     h2$ = stack$(stackLevel)
     checkParen h2$, "+-"
     stackLevel = stackLevel - 1
     h1$ = stack$(stackLevel)
     stack$(stackLevel) = h1$ + "-" + h2$
    CASE "*"
     h2$ = stack$(stackLevel)
     checkParen h2$, "+-"
     stackLevel = stackLevel - 1
     h1$ = stack$(stackLevel)
     checkParen h1$, "+-/"
     stack$(stackLevel) = h1$ + "*" + h2$
    CASE "/"
     h2$ = stack$(stackLevel)
     checkParen h2$, "+-/"
     stackLevel = stackLevel - 1
     h1$ = stack$(stackLevel)
     checkParen h1$, "+-/"
     stack$(stackLevel) = h1$ + "/" + h2$
    CASE "^"
     h2$ = stack$(stackLevel)
     checkParen h2$, "+-*/"
     stackLevel = stackLevel - 1
     h1$ = stack$(stackLevel)
     checkParen h1$, "+-*/"
     stack$(stackLevel) = h1$ + "^" + h2$
   END SELECT
  END IF
 NEXT
 IF stackLevel <> 1 THEN
      REM
 END IF
 PRINT l$, stack$(1), nOrig$: PRINT #2, l$, stack$(1)
LOOP UNTIL EOF(1)
CLOSE

SUB checkParen (s$, ops$)
 pLevel = 0
 flag = 0
 FOR i = 1 TO LEN(s$)
   SELECT CASE MID$(s$, i, 1)
     CASE "("
      pLevel = pLevel + 1
     CASE ")"
      pLevel = pLevel - 1
     CASE ELSE
      IF pLevel = 0 AND INSTR(ops$, MID$(s$, i, 1)) THEN flag = 1: EXIT FOR
   END SELECT
 NEXT i
 IF flag THEN s$ = "(" + s$ + ")"
END SUB

 

Edited on September 26, 2005, 12:26 am

Posted by Charlie on 2005-09-26 00:14:36