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Elegant Exponent Exposition (Posted on 2009-02-09) Difficulty: 2 of 5
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?

See The Solution Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 3

    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$,I+2,1))
   70     Tval=Tval+Term
   80    next
   85    if Tval=int(Tval) then Ct=Ct+1
   90    :if Tval<=MinVal then MinVal=Tval:MinStr=D$:gosub *PrIt:endif
  100    :if Tval>=MaxVal then MaxVal=Tval:MaxStr=D$:gosub *PrIt
  110 
  390    gosub *Permute(&D$)
  398    if D$=H$ then goto 400
  399   endloop
  400   print MinStr,MinVal
  410   print MaxStr,MaxVal
  420   print Ct
  690   close
  700   end
  800 
  810   *PrIt
  820   print "(";mid(D$,1,1);"/";mid(D$,2,1);")^";mid(D$,3,1);" + ";
  822   print "(";mid(D$,4,1);"/";mid(D$,5,1);")^";mid(D$,6,1);" + ";
  824   print "(";mid(D$,7,1);"/";mid(D$,8,1);")^";mid(D$,9,1);" = ";
  826   print Tval
  830   return

produces a set of results with decreasing minimum and increasing maximum values. Selected from them are:

The final minimum is 70, given in the six permutations of its terms:

(5/9)^1 + (7/3)^2 + (8/4)^6 =  70
(5/9)^1 + (8/4)^6 + (7/3)^2 =  70
(7/3)^2 + (5/9)^1 + (8/4)^6 =  70
(7/3)^2 + (8/4)^6 + (5/9)^1 =  70
(8/4)^6 + (5/9)^1 + (7/3)^2 =  70
(8/4)^6 + (7/3)^2 + (5/9)^1 =  70

The maximum, 134,217,888, is given by terms two of whose bases amount to 2: 6/3 and 4/2. As a result, these can be interchanged with their powers, 7 and 5, making a total of 12 variations on the theme:

(4/2)^5 + (6/3)^7 + (8/1)^9 =  134217888
(4/2)^5 + (8/1)^9 + (6/3)^7 =  134217888
(4/2)^7 + (6/3)^5 + (8/1)^9 =  134217888
(4/2)^7 + (8/1)^9 + (6/3)^5 =  134217888
(6/3)^5 + (4/2)^7 + (8/1)^9 =  134217888
(6/3)^5 + (8/1)^9 + (4/2)^7 =  134217888
(6/3)^7 + (4/2)^5 + (8/1)^9 =  134217888
(6/3)^7 + (8/1)^9 + (4/2)^5 =  134217888
(8/1)^9 + (4/2)^5 + (6/3)^7 =  134217888
(8/1)^9 + (4/2)^7 + (6/3)^5 =  134217888
(8/1)^9 + (6/3)^5 + (4/2)^7 =  134217888
(8/1)^9 + (6/3)^7 + (4/2)^5 =  134217888

  Posted by Charlie on 2009-02-09 12:51:16
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