Determine all possible value(s) of a positive integer N, such that N and N⁴ together contain precisely nine digits from 0 to 9 which are all different. Neither N nor N⁴ can contain any leading zero.

Note: Try to solve this problem analytically, although computer program/spreadsheet solutions are welcome.

The fourth root of the lowest possible 7-digit number,1023456, is 31.8..., so the lowest N to try would be 32. The highest would be 56 ([4th root of 9876543]}. By happenstance, the lowest, 32, raised to the fourth power is 1048576, meeting the conditions of the problem.

The following program verifies not only that the answer is unique as posed, but also that no full-palindrome fourth power of an integer exists (all 10 digits accounted for).

DEFDBL A-Z
FOR n = 1 TO 100
  ns$ = LTRIM$(STR$(n))
  n4s$ = LTRIM$(STR$(n * n * n * n))
  t$ = ns$ + n4s$
  IF LEN(t$) = 9 OR LEN(t$) = 10 THEN
    good = 1
    FOR i = 1 TO LEN(t$) - 1
      IF INSTR(i + 1, t$, MID$(t$, i, 1)) > 0 THEN good = 0: EXIT FOR
    NEXT
    IF good THEN PRINT n; n4s$
  END IF
NEXT

 

Posted by Charlie on 2008-11-01 19:21:28