(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 AZ
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 20130220 18:12:19 