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Palindromic and Tautonymic III (Posted on 2010-03-23) Difficulty: 3 of 5
Make a list of distinct positive integers that are obtained by assigning a different base ten digit from 1 to 9 to each of the capital letters in this expression.

(A+B)*C + (DE)/F + (GH)*I

What are the respective minimum and maximum positive palindromes from amongst the elements that correspond to the foregoing list?

As a bonus, what are the respective minimum and maximum positive tautonymic numbers that are included in the list? How about the respective maximum and minimum prime numbers?

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution computer solution | Comment 5 of 11 |

Overall the values extend from 24 to 939,524,151, with 13,253 different values altogether, and so too voluminous to list here.

For palindromes, the lowest is 33:

 A B  C  D E  F  G H  I
(2+6)*3+(9-4)/5+(1^7)*8                 =33
(2+6)*3+(9-5)/4+(1^7)*8                 =33
(4+5)*3+(2-8)/6+(1^9)*7                 =33
(4+5)*3+(6-8)/2+(1^9)*7                 =33
(4+9)*2+(8-3)/5+(1^7)*6                 =33
(4+9)*2+(8-5)/3+(1^7)*6                 =33
(5+4)*3+(2-8)/6+(1^9)*7                 =33
(5+4)*3+(6-8)/2+(1^9)*7                 =33
(5+7)*2+(9-3)/6+(1^4)*8                 =33
(5+7)*2+(9-6)/3+(1^4)*8                 =33
(5+8)*2+(7-3)/4+(1^9)*6                 =33
(5+8)*2+(7-4)/3+(1^9)*6                 =33
(5+9)*2+(3-7)/4+(1^8)*6                 =33
(5+9)*2+(4-7)/3+(1^8)*6                 =33
(6+2)*3+(9-4)/5+(1^7)*8                 =33
(6+2)*3+(9-5)/4+(1^7)*8                 =33
(6+7)*2+(4-9)/5+(1^3)*8                 =33
(6+7)*2+(5-9)/4+(1^3)*8                 =33
(6+9)*2+(3-8)/5+(1^7)*4                 =33
(6+9)*2+(5-8)/3+(1^7)*4                 =33
(7+5)*2+(9-3)/6+(1^4)*8                 =33
(7+5)*2+(9-6)/3+(1^4)*8                 =33
(7+6)*2+(4-9)/5+(1^3)*8                 =33
(7+6)*2+(5-9)/4+(1^3)*8                 =33
(7+8)*2+(3-9)/6+(1^5)*4                 =33
(7+8)*2+(6-9)/3+(1^5)*4                 =33
(8+5)*2+(7-3)/4+(1^9)*6                 =33
(8+5)*2+(7-4)/3+(1^9)*6                 =33
(8+7)*2+(3-9)/6+(1^5)*4                 =33
(8+7)*2+(6-9)/3+(1^5)*4                 =33
(9+4)*2+(8-3)/5+(1^7)*6                 =33
(9+4)*2+(8-5)/3+(1^7)*6                 =33
(9+5)*2+(3-7)/4+(1^8)*6                 =33
(9+5)*2+(4-7)/3+(1^8)*6                 =33
(9+6)*2+(3-8)/5+(1^7)*4                 =33
(9+6)*2+(5-8)/3+(1^7)*4                 =33

or if that's too simple a palindrome, then 101:

(those with A>B have been eliminated from this list, for brevity; just reverse them for the alternative.)

(2+6)*9+(8-3)/5+(4^1)*7                =101
(2+6)*9+(8-3)/5+(7^1)*4                =101
(2+6)*9+(8-5)/3+(4^1)*7                =101
(2+6)*9+(8-5)/3+(7^1)*4                =101
(3+5)*9+(8-2)/6+(4^1)*7                =101
(3+5)*9+(8-2)/6+(7^1)*4                =101
(3+5)*9+(8-6)/2+(4^1)*7                =101
(3+5)*9+(8-6)/2+(7^1)*4                =101
(3+8)*6+(2-7)/5+(4^1)*9                =101
(3+8)*6+(2-7)/5+(9^1)*4                =101
(3+8)*6+(5-7)/2+(4^1)*9                =101
(3+8)*6+(5-7)/2+(9^1)*4                =101
(3+9)*8+(2-7)/5+(1^4)*6                =101
(3+9)*8+(5-7)/2+(1^4)*6                =101
(3+9)*8+(7-2)/5+(1^6)*4                =101
(3+9)*8+(7-5)/2+(1^6)*4                =101
(4+5)*9+(2-8)/6+(3^1)*7                =101
(4+5)*9+(2-8)/6+(7^1)*3                =101
(4+5)*9+(6-8)/2+(3^1)*7                =101
(4+5)*9+(6-8)/2+(7^1)*3                =101
(4+6)*3+(2-7)/5+(8^1)*9                =101
(4+6)*3+(2-7)/5+(9^1)*8                =101
(4+6)*3+(5-7)/2+(8^1)*9                =101
(4+6)*3+(5-7)/2+(9^1)*8                =101
(4+7)*9+(2-8)/6+(1^5)*3                =101
(4+7)*9+(6-8)/2+(1^5)*3                =101
(4+9)*6+(2-7)/5+(3^1)*8                =101
(4+9)*6+(2-7)/5+(8^1)*3                =101
(4+9)*6+(5-7)/2+(3^1)*8                =101
(4+9)*6+(5-7)/2+(8^1)*3                =101
(5+6)*4+(9-1)/8+(2^3)*7                =101
(5+6)*4+(9-8)/1+(2^3)*7                =101
(5+6)*8+(9-2)/7+(3^1)*4                =101
(5+6)*8+(9-2)/7+(4^1)*3                =101
(5+6)*8+(9-7)/2+(3^1)*4                =101
(5+6)*8+(9-7)/2+(4^1)*3                =101
(5+6)*9+(2-8)/3+(1^7)*4                =101
(5+6)*9+(3-7)/2+(1^8)*4                =101
(5+7)*4+(1-9)/8+(3^2)*6                =101
(5+7)*4+(8-9)/1+(3^2)*6                =101
(5+7)*6+(3-9)/2+(4^1)*8                =101
(5+7)*6+(3-9)/2+(8^1)*4                =101
(5+7)*6+(8-4)/2+(3^1)*9                =101
(5+7)*6+(8-4)/2+(9^1)*3                =101
(5+7)*8+(9-3)/6+(1^2)*4                =101
(5+7)*8+(9-6)/3+(1^2)*4                =101
(5+8)*3+(2-6)/4+(7^1)*9                =101
(5+8)*3+(2-6)/4+(9^1)*7                =101
(5+8)*3+(4-6)/2+(7^1)*9                =101
(5+8)*3+(4-6)/2+(9^1)*7                =101
(5+8)*7+(6-2)/4+(1^3)*9                =101
(5+8)*7+(6-4)/2+(1^3)*9                =101
(5+9)*6+(7-3)/4+(2^1)*8                =101
(5+9)*6+(7-3)/4+(8^1)*2                =101
(5+9)*6+(7-4)/3+(2^1)*8                =101
(5+9)*6+(7-4)/3+(8^1)*2                =101
(5+9)*7+(2-8)/6+(1^3)*4                =101
(5+9)*7+(6-8)/2+(1^3)*4                =101
(6+8)*5+(9-3)/2+(4^1)*7                =101
(6+8)*5+(9-3)/2+(7^1)*4                =101
(6+8)*7+(2-5)/3+(1^9)*4                =101
(6+8)*7+(3-5)/2+(1^9)*4                =101
(6+8)*7+(9-4)/5+(1^3)*2                =101
(6+8)*7+(9-5)/4+(1^3)*2                =101
(6+9)*4+(8-1)/7+(2^3)*5                =101
(6+9)*4+(8-7)/1+(2^3)*5                =101
(6+9)*5+(2-8)/3+(4^1)*7                =101
(6+9)*5+(2-8)/3+(7^1)*4                =101
(7+8)*5+(2-6)/4+(3^1)*9                =101
(7+8)*5+(2-6)/4+(9^1)*3                =101
(7+8)*5+(4-6)/2+(3^1)*9                =101
(7+8)*5+(4-6)/2+(9^1)*3                =101
(7+9)*4+(8-3)/5+(6^2)*1                =101
(7+9)*4+(8-5)/3+(6^2)*1                =101
(7+9)*6+(5-2)/3+(1^8)*4                =101
(7+9)*6+(5-3)/2+(1^8)*4                =101
(7+9)*6+(8-3)/5+(1^2)*4                =101
(7+9)*6+(8-4)/2+(1^5)*3                =101
(7+9)*6+(8-5)/3+(1^2)*4                =101

The highest is 327723:

(6+7)*3+(9-1)/2+(4^8)*5             =327723
(7+6)*3+(9-1)/2+(4^8)*5             =327723

For tautonyms, the lowest is 1313:

(1+7)*4+(9-3)/6+(2^8)*5               =1313
(1+7)*4+(9-6)/3+(2^8)*5               =1313
(1+7)*8+(3-9)/6+(5^4)*2               =1313
(1+7)*8+(6-9)/3+(5^4)*2               =1313
(7+1)*4+(9-3)/6+(2^8)*5               =1313
(7+1)*4+(9-6)/3+(2^8)*5               =1313
(7+1)*8+(3-9)/6+(5^4)*2               =1313
(7+1)*8+(6-9)/3+(5^4)*2               =1313

and the highest tautonym is 19131913:

(1+6)*5+(8-2)/3+(9^7)*4           =19131913
(2+5)*6+(3-8)/1+(9^7)*4           =19131913
(5+2)*6+(3-8)/1+(9^7)*4           =19131913
(6+1)*5+(8-2)/3+(9^7)*4           =19131913
    5   dim Used(9)
    6   kill "paltaut3.txt":open "paltaut3.txt" for output as #2
   10   for G=1 to 9
   20     Used(G)=1
   30     for H=1 to 9
   40       if Used(H)=0 then
   50         :Used(H)=1
   60         :Gh=G^H
   70     :for I=1 to 9
   80       :if Used(I)=0 then
   90         :Used(I)=1
  100         :Ghi=Gh*I
  110     :for A=1 to 9
  120       :if Used(A)=0 then
  130         :Used(A)=1
  140     :for B=1 to 9
  150       :if Used(B)=0 then
  160         :Used(B)=1
  170     :for C=1 to 9
  180       :if Used(C)=0 then
  190         :Used(C)=1
  200     :for D=1 to 9
  210       :if Used(D)=0 then
  220         :Used(D)=1
  230     :for E=1 to 9
  240       :if Used(E)=0 then
  250         :Used(E)=1
  260     :for F=1 to 9
  270       :if Used(F)=0 then
  280         :Used(F)=1
  290         :V=(A+B)*C+(D-E)//F+Ghi
  300         :if V=int(V) then
  310            :print #2,A;B;C;D;E;F;G;H;I;using(19,0),V
  320            :print A;B;C;D;E;F;G;H;I;using(19,0),V
  330         :endif
  340         :Used(F)=0
  350       :endif
  360     :next
  370         :Used(E)=0
  380       :endif
  390     :next
  400         :Used(D)=0
  410       :endif
  420     :next
  430         :Used(C)=0
  440       :endif
  450     :next
  460         :Used(B)=0
  470       :endif
  480     :next
  490         :Used(A)=0
  500       :endif
  510     :next
  520         :Used(I)=0
  530       :endif
  540     :next
  550         :Used(H)=0
  560       :endif
  570     next H
  580     Used(G)=0
  590   next G
  600   close #2
The results were then sorted and then analysed with:
OPEN "paltaut3.txt" FOR INPUT AS #1
DO
  LINE INPUT #1, l$
  l$ = LTRIM$(RTRIM$(MID$(l$, 29)))
  pal = 1
  FOR i = 1 TO LEN(l$) / 2
    IF MID$(l$, i, 1) <> MID$(l$, LEN(l$) + 1 - i, 1) THEN pal = 0: EXIT FOR
  NEXT
  IF pal THEN p$ = l$
  IF LEN(l$) MOD 2 = 0 THEN
    taut = 1
    FOR i = 1 TO LEN(l$) / 2
     IF MID$(l$, i, 1) <> MID$(l$, i + LEN(l$) / 2, 1) THEN taut = 0: EXIT FOR
    NEXT
    IF taut THEN
     IF hadTaut = 0 AND pal = 0 THEN hadTaut = 1: PRINT l$
     t$ = l$
    END IF
  END IF
LOOP UNTIL EOF(1)
PRINT p$
PRINT t$

Operation symbols were added manually.

BTW, the highest without consideration of palindrome/tautonym, 939,524,151, is (5+4)*6 + (3-2)/1 + (8^9)*7

Primes:

There are 1198 primes on the list and the minimum prime is 27 and the maximum prime is 939,524,147, from

list
    1   Had=0
    5   open "PALTAU~1.TXT" for input as #1
   10   while not eof(1)
   20    input #1,A:N=val(cutspc(A))
   30    if prmdiv(N)=N then if Had=0 then print N:endif:Ct=Ct+1:Big=N:Had=1
   40    if prmdiv(N)=0 then print "***";N:end
   50   wend
   60   print Big
   70   print Ct
OK
run
 29
 939524147
 1198
OK
(3+7)*2+(9-4)*5+(1^6)*8                 =29
(3+7)*2+(9-5)*4+(1^6)*8                 =29
(3+8)*2+(9-4)*5+(1^7)*6                 =29
(3+8)*2+(9-5)*4+(1^7)*6                 =29
(4+7)*2+(3-9)*6+(1^5)*8                 =29
(4+7)*2+(6-9)*3+(1^5)*8                 =29
(4+7)*2+(8-3)*5+(1^9)*6                 =29
(4+7)*2+(8-5)*3+(1^9)*6                 =29
(5+6)*2+(3-7)*4+(1^9)*8                 =29
(5+6)*2+(4-7)*3+(1^9)*8                 =29
(5+7)*2+(9-3)*6+(1^8)*4                 =29
(5+7)*2+(9-6)*3+(1^8)*4                 =29
(5+8)*2+(3-9)*6+(1^7)*4                 =29
(5+8)*2+(6-9)*3+(1^7)*4                 =29
(6+5)*2+(3-7)*4+(1^9)*8                 =29
(6+5)*2+(4-7)*3+(1^9)*8                 =29
(6+7)*2+(3-8)*5+(1^9)*4                 =29
(6+7)*2+(5-8)*3+(1^9)*4                 =29
(7+3)*2+(9-4)*5+(1^6)*8                 =29
(7+3)*2+(9-5)*4+(1^6)*8                 =29
(7+4)*2+(3-9)*6+(1^5)*8                 =29
(7+4)*2+(6-9)*3+(1^5)*8                 =29
(7+4)*2+(8-3)*5+(1^9)*6                 =29
(7+4)*2+(8-5)*3+(1^9)*6                 =29
(7+5)*2+(9-3)*6+(1^8)*4                 =29
(7+5)*2+(9-6)*3+(1^8)*4                 =29
(7+6)*2+(3-8)*5+(1^9)*4                 =29
(7+6)*2+(5-8)*3+(1^9)*4                 =29
(8+3)*2+(9-4)*5+(1^7)*6                 =29
(8+3)*2+(9-5)*4+(1^7)*6                 =29
(8+5)*2+(3-9)*6+(1^7)*4                 =29
(8+5)*2+(6-9)*3+(1^7)*4                 =29
(4+6)*5+(3-1)*2+(8^9)*7          =939524147
(4+6)*5+(3-2)*1+(8^9)*7          =939524147
(6+4)*5+(3-1)*2+(8^9)*7          =939524147
(6+4)*5+(3-2)*1+(8^9)*7          =939524147

  Posted by Charlie on 2010-03-23 15:33:34
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