Make a list of distinct positive integers that are obtained by assigning a different decimal digit from 1 to 9 to each of the capital letters in bold in this expression.

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

Determine the smallest and the largest positive palindromes. What are the smallest and the largest prime number? How many amongst the abovementioned list are tautonymic numbers?

   10   LgPal=0:LgTau=0:LgPr=0
   20   SmPal=9999999999999999999999999
   30   SmTau=9999999999999999999999999
   40   SmPr=9999999999999999999999999
  110   V$="123456789":H$=V$
  115   repeat
  120     gosub *Permute(&V$)
  130     A=val(mid(V$,1,1))
  140     B=val(mid(V$,2,1))
  150     C=val(mid(V$,3,1))
  160     D=val(mid(V$,4,1))
  170     E=val(mid(V$,5,1))
  180     F=val(mid(V$,6,1))
  190     G=val(mid(V$,7,1))
  200     H=val(mid(V$,8,1))
  210     I=val(mid(V$,9,1))
  220     V=(A+B)^C+(D+E)^F+(G+H)^I
  230     S=cutspc(str(V))
  240     Pal=1
  250     for J=1 to int(len(S)/2)
  260       if mid(S,J,1)<>mid(S,len(S)+1-J,1) then Pal=0
  270     next
  280     if Pal=1 and V>LgPal then LgPal=V:LgPalS=V$
  290     if Pal=1 and V<SmPal then SmPal=V:SmPalS=V$
  300     if prmdiv(V)=V or prmdiv(V)=0 then Pr=1:else Pr=0
  310     if Pr=1 and V>LgPr then LgPr=V:LgPrS=V$
  320     if Pr=1 and V<SmPr then SmPr=V:SmPrS=V$
  330     if len(S) @ 2=0 then Tau=1:else Tau=0
  331     Diff=0
  332     for J=2 to len(S)
  333      if mid(S,J,1)<>mid(S,1,1) then Diff=1
  334     next
  335     if Diff=0 then Tau=0
  340     for J=1 to int(len(S)/2)
  350       if mid(S,J,1)<>mid(S,int(len(S)/2)+J,1) then Tau=0
  360     next
  370     if Tau=1 and V>LgTau then LgTau=V:LgTauS=V$
  380     if Tau=1 and V<SmTau then SmTau=V:SmTauS=V$
  700   until V$=H$
  710   print "("+mid(SmPalS,1,1)+"+"+mid(SmPalS,2,1)+")^"+mid(SmPalS,3,1)+"+("+mid(SmPalS,4,1)+"+"+mid(SmPalS,5,1)+")^"+mid(SmPalS,6,1)+"+("+mid(SmPalS,7,1)+"+"+mid(SmPalS,8,1)+")^"+mid(SmPalS,9,1),SmPal
  720    print "("+mid(LgPalS,1,1)+"+"+mid(LgPalS,2,1)+")^"+mid(LgPalS,3,1)+"+("+mid(LgPalS,4,1)+"+"+mid(LgPalS,5,1)+")^"+mid(LgPalS,6,1)+"+("+mid(LgPalS,7,1)+"+"+mid(LgPalS,8,1)+")^"+mid(LgPalS,9,1),LgPal
  730   print "("+mid(SmPrS,1,1)+"+"+mid(SmPrS,2,1)+")^"+mid(SmPrS,3,1)+"+("+mid(SmPrS,4,1)+"+"+mid(SmPrS,5,1)+")^"+mid(SmPrS,6,1)+"+("+mid(SmPrS,7,1)+"+"+mid(SmPrS,8,1)+")^"+mid(SmPrS,9,1),SmPr
  740    print "("+mid(LgPrS,1,1)+"+"+mid(LgPrS,2,1)+")^"+mid(LgPrS,3,1)+"+("+mid(LgPrS,4,1)+"+"+mid(LgPrS,5,1)+")^"+mid(LgPrS,6,1)+"+("+mid(LgPrS,7,1)+"+"+mid(LgPrS,8,1)+")^"+mid(LgPrS,9,1),LgPr
  750   print "("+mid(SmTauS,1,1)+"+"+mid(SmTauS,2,1)+")^"+mid(SmTauS,3,1)+"+("+mid(SmTauS,4,1)+"+"+mid(SmTauS,5,1)+")^"+mid(SmTauS,6,1)+"+("+mid(SmTauS,7,1)+"+"+mid(SmTauS,8,1)+")^"+mid(SmTauS,9,1),SmTau
  760    print "("+mid(LgTauS,1,1)+"+"+mid(LgTauS,2,1)+")^"+mid(LgTauS,3,1)+"+("+mid(LgTauS,4,1)+"+"+mid(LgTauS,5,1)+")^"+mid(LgTauS,6,1)+"+("+mid(LgTauS,7,1)+"+"+mid(LgTauS,8,1)+")^"+mid(LgTauS,9,1),LgTau
  790   end

 

finds

 

Palindromic
(4+5)^3+(6+9)^1+(7+8)^2 = 969
(1+3)^9+(2+5)^8+(6+7)^4 = 6055506

Prime
(4+5)^3+(6+8)^2+(7+9)^1 = 941
(1+4)^2+(3+6)^5+(7+8)^9 = 38443418449

Tautonymic
(4+7)^3+(5+8)^2+(6+9)^1 = 1515
(4+7)^3+(5+8)^2+(6+9)^1 = 1515

(apparently there's only one tautonym, serving as both smallest and largest)

Posted by Charlie on 2010-12-27 16:35:06