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Palindromic and Tautonymic II (Posted on 2009-12-20) 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 + (C-D)E + (F-G)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 negative tautonymic numbers?

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

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Solution computer solutions | Comment 2 of 4 |

If 1 is considered a palindrome, then there are various ways of producing it, such as

(1-2)^3 + (3-4)^6 + (8-7)^9 = 1

and 3128 other ways.

If, however, you need at least a couple of digits then

(6-9)^2 + (2-1)^7 + (5-4)^3 = 11

as well as many other ways of producing 11.

But if you really need diversity of digits, then the ways of getting 101 are sufficiently few to list them all:

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

For the maximum, there's no doubt:

(4-6)^7 + (7-5)^1 + (2-9)^8 = 5764675
(4-6)^7 + (7-5)^1 + (9-2)^8 = 5764675
(6-4)^1 + (1-3)^7 + (2-9)^8 = 5764675
(2-9)^8 + (8-6)^1 + (3-5)^7 = 5764675
(6-4)^1 + (1-3)^7 + (9-2)^8 = 5764675
(4-5)^1 + (1-6)^3 + (9-2)^8 = 5764675
(4-5)^1 + (1-6)^3 + (2-9)^8 = 5764675
(9-2)^8 + (8-6)^1 + (3-5)^7 = 5764675
(1-6)^3 + (3-4)^5 + (2-9)^8 = 5764675
(4-5)^9 + (9-2)^8 + (1-6)^3 = 5764675
(1-6)^3 + (3-4)^7 + (9-2)^8 = 5764675
(1-6)^3 + (3-4)^7 + (2-9)^8 = 5764675
(1-6)^3 + (3-4)^5 + (9-2)^8 = 5764675

For the tautonyms, as defined, we need at least 4 digits, so 1010 wins the low spot (pun intended):

(9-4)^2 + (2-5)^6 + (3-1)^8 = 1010
(5-9)^4 + (4-7)^6 + (3-8)^2 = 1010
(5-9)^4 + (4-1)^6 + (8-3)^2 = 1010
(5-9)^4 + (4-1)^6 + (3-8)^2 = 1010
(5-1)^4 + (4-7)^6 + (8-3)^2 = 1010
(5-1)^4 + (4-7)^6 + (3-8)^2 = 1010
(7-5)^8 + (8-3)^2 + (1-4)^6 = 1010
(5-7)^8 + (8-3)^2 + (1-4)^6 = 1010
(3-7)^4 + (4-9)^2 + (5-8)^6 = 1010
(3-7)^4 + (4-9)^2 + (8-5)^6 = 1010
(4-9)^2 + (2-5)^6 + (3-1)^8 = 1010
(4-9)^2 + (2-5)^6 + (1-3)^8 = 1010
(9-5)^4 + (4-7)^6 + (8-3)^2 = 1010
(9-5)^4 + (4-7)^6 + (3-8)^2 = 1010
(4-7)^6 + (6-1)^2 + (5-3)^8 = 1010
(9-5)^4 + (4-1)^6 + (8-3)^2 = 1010
(1-5)^4 + (4-7)^6 + (8-3)^2 = 1010
(9-5)^4 + (4-1)^6 + (3-8)^2 = 1010
(7-3)^4 + (4-9)^2 + (5-8)^6 = 1010
(9-4)^2 + (2-5)^6 + (1-3)^8 = 1010
(7-5)^8 + (8-3)^2 + (4-1)^6 = 1010
(7-3)^4 + (4-9)^2 + (8-5)^6 = 1010
(8-5)^6 + (6-1)^2 + (3-7)^4 = 1010
(4-7)^6 + (6-1)^2 + (3-5)^8 = 1010
(8-5)^6 + (6-1)^2 + (7-3)^4 = 1010
(5-8)^6 + (6-1)^2 + (7-3)^4 = 1010
(5-8)^6 + (6-1)^2 + (3-7)^4 = 1010
(1-3)^8 + (8-5)^6 + (9-4)^2 = 1010
(7-9)^8 + (8-3)^2 + (1-4)^6 = 1010
(5-9)^4 + (4-7)^6 + (8-3)^2 = 1010
(7-4)^6 + (6-1)^2 + (3-5)^8 = 1010
(1-5)^4 + (4-7)^6 + (3-8)^2 = 1010
(9-7)^8 + (8-3)^2 + (4-1)^6 = 1010
(9-7)^8 + (8-3)^2 + (1-4)^6 = 1010
(3-1)^8 + (8-5)^6 + (4-9)^2 = 1010
(3-1)^8 + (8-5)^6 + (9-4)^2 = 1010
(7-4)^6 + (6-1)^2 + (5-3)^8 = 1010
(7-9)^8 + (8-3)^2 + (4-1)^6 = 1010
(1-3)^8 + (8-5)^6 + (4-9)^2 = 1010
(5-7)^8 + (8-3)^2 + (4-1)^6 = 1010

and the largest:

(4-2)^9 + (9-3)^5 + (8-1)^7 = 831831
(6-4)^9 + (9-3)^5 + (8-1)^7 = 831831

For palindromes:

    5   kill "paltaut2.txt":open "paltaut2.txt" for output as #2
   10   L="123456789":H=L
   20   repeat
   30        A=val(mid(L,1,1))
   40        B=val(mid(L,2,1))
   50        C=val(mid(L,3,1))
   60        D=val(mid(L,4,1))
   70        E=val(mid(L,5,1))
   80        F=val(mid(L,6,1))
   90        G=val(mid(L,7,1))
  100        I=val(mid(L,9,1))
  200         V=cutspc(str((A-B)^C+(C-D)^E+(F-G)^I))
  210         V1=val(V)
  220         if V1>0 then
  230            :Good=1
  240            :for I=1 to int(len(V)/2)
  250              :if mid(V,I,1)<>mid(V,len(V)-I+1,1) then Good=0:endif
  260            :next
  270            :if Good then print #2,L,:print #2,using(17,0),V1
  280            :print L,:print using(17,0),V1
  930     gosub *Permute(&L)
  950   until L=H
 9999   end

For tautonyms:

    5   kill "paltaut3.txt":open "paltaut3.txt" for output as #2
   10   L="123456789":H=L
   20   repeat
   30        A=val(mid(L,1,1))
   40        B=val(mid(L,2,1))
   50        C=val(mid(L,3,1))
   60        D=val(mid(L,4,1))
   70        E=val(mid(L,5,1))
   80        F=val(mid(L,6,1))
   90        G=val(mid(L,7,1))
  100        I=val(mid(L,9,1))
  200         V=cutspc(str((A-B)^C+(C-D)^E+(F-G)^I))
  210         V1=val(V)
  220         if V1>0 and len(V)@2=0 then
  230            :Good=1
  240            :for I=1 to int(len(V)/2)
  250              :if mid(V,I,1)<>mid(V,int(len(V)/2)+I,1) then Good=0:endif
  260            :next
  270            :if Good then print #2,L,:print #2,using(17,0),V1
  280            :print L,:print using(17,0),V1
  930     gosub *Permute(&L)
  950   until L=H
 9999   end

Reformatted (after sorting) by:

OPEN "paltaut2.txt" FOR INPUT AS #1
OPEN "paltau2.txt" FOR OUTPUT AS #2
DO
  LINE INPUT #1, l$
  a$ = MID$(l$, 1, 1)
  b$ = MID$(l$, 2, 1)
  c$ = MID$(l$, 3, 1)
  d$ = MID$(l$, 4, 1)
  e$ = MID$(l$, 5, 1)
  f$ = MID$(l$, 6, 1)
  g$ = MID$(l$, 7, 1)
  h$ = MID$(l$, 8, 1)
  i$ = MID$(l$, 9, 1)
  PRINT #2, "("; a$; "-"; b$; ")^"; c$; " + ("; c$; "-"; d$; ")^"; e$; " + ("; f$; "-"; g$; ")^"; i$; " = "; LTRIM$(MID$(l$, 11))
LOOP UNTIL EOF(1)
CLOSE

 


 


  Posted by Charlie on 2009-12-20 16:31:54
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