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 Elegant Exponent Exposition (Posted on 2009-02-09)
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 No Rating

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 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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