(abc)^d=(efghbghfbe)

What two digits are not represented in the above equation?

Rem: (xyz) defines a numerical value of the string of the integers x,y,z - not their product;
i.e. (xyz)=100*x+10*y+z.

(abc)^d=(efghbghfbe)

What two digits are not represented in the above equation?

Rem: (xyz) defines a numerical value of the string of the integers x,y,z - not their product;
i.e. (xyz)=100*x+10*y+z.

At this point I've changed to uppercase.

Firstly A and E may not be 0.
Now for ABC to produce a 10 digit numeral we need to satisfy this expression:
180^4 < (ABC)^4 < 399^4 , ie D=4  [Note too that A is not given the value of 4 in the expression].
 
Next consider the digit C raised to the 4th power to yield the digit E:
C 1 2 3 4 5 6 7 8 9
E 1 6 1 6 5 6 1 6 1
C<>E so we immediately have some eliminations to indicate that E is either 1 or 6.
From the above expression A = 1, 2 or 3. If A is 3 then the 10 digit number begins with either 8 or 9 which cannot be E. If A is 1 then the 10 digit number will be either 1 or 2.

Having decided that ABC was 2#7 or 2#9, # being the tens digit, and not wanting to do several PnP calculations I began to set up an Excel spreadsheet.  My first entry into 2#7 yielded  1836036801!

The digits missing from the calculation are 5 and 9.


To ensure that I hadn’t miscalculated I ran the following program which incidentally exposes all characters to 0.  I don’t understand but the program prints 207 and 1836036801 three times!
CLS
DEFDBL A-Z
FOR a = 0 TO 9
 IF used(a) = 0 THEN
  used(a) = 1
  FOR b = 0 TO 9
   IF used(b) = 0 THEN
    used(b) = 1
    FOR c = 0 TO 9
     IF used(c) = 0 THEN
      used(c) = 1
      FOR d = 0 TO 9
       IF used(d) = 0 THEN
        used(d) = 1
        FOR e = 0 TO 9
         IF used(e) = 0 THEN
          used(e) = 1
          FOR f = 0 TO 9
           IF used(f) = 0 THEN
            used(f) = 1
            FOR g = 0 TO 9
             IF used(g) = 0 THEN
              used(g) = 1
              FOR h = 0 TO 9
               IF used(h) = 0 THEN
                used(h) = 1
                left = (a * 100 + b * 10 + c) ^ 4
                right = e * 10 ^ 9 + f * 10 ^ 8 + g * 10 ^ 7 + h * 10 ^ 6 + b * 10 ^ 5 + g * 10 ^ 4 + h * 10 ^ 3 + f * 10 ^ 2 + b * 10 + e
                IF left = right THEN PRINT a * 100 + b * 10 + c; right
                used(h) = 0
               END IF
              NEXT
              used(g) = 0
             END IF
            NEXT
            used(f) = 0
           END IF
          NEXT
          used(e) = 0
         END IF
        NEXT
        used(d) = 0
       END IF
      NEXT
      used(c) = 0
     END IF
    NEXT
    used(b) = 0
   END IF
  NEXT
  used(a) = 0
 END IF
NEXT


 

Posted by brianjn on 2013-02-20 18:12:19