Substitute each of the capital letters in bold by a different decimal digit from 1 to 9 and determine the minimum positive integer value that the following cryptarithmetic expression can assume.

(A/B)^C + (D/E)^F + (G/H)^I

How about the maximum positive integer value that this expression can assume?

(In reply to computer solution by Charlie)

If the result could be a rational, rather than an integer, we get:

    5   MinVal=9999999999999:MaxVal=0
   10   D$="123456789":H$=D$
   20   loop
   40    Tval=0
   50    for I=1 to 7 step 3
   60     Term=(val(mid(D$,I,1))//val(mid(D$,I+1,1)))^val(mid(D$,I+2,1))
   70     Tval=Tval+Term
   80    next
   90    if Tval<MinVal then MinVal=Tval:MinStr=D$
  100    if Tval>MaxVal then MaxVal=Tval:MaxStr=D$
  110  
  390    gosub *Permute(&D$)
  398    if D$=H$ then goto 400
  399   endloop
  400   print MinStr,MinVal,MinVal/1.0
  410   print MaxStr,MaxVal,MaxVal/1.0
  700   end
giving (after manual reformatting):

(3/9)^7 + (2/5)^8 + (1/4)^6 = 4747532587/3499200000000 ~= 0.0013567479958276177

(8/1)^9 + (6/2)^7 + (5/3)^4 = 10871813740/81 ~= 134,219,922.7160493827160493827

The maximum is not much larger, due to most of it coming from the (8/1)^9.

Edited on February 9, 2009, 1:02 pm

Posted by Charlie on 2009-02-09 13:01:14