There exists a number oddity with three different 4-digit duodecimal (base 12) positive integers. One is BA01, where: (BA+01)²= BA01.

It also works with 3630 as: (36+30)² = 3630

What is the other number?

What is the smallest 6-digit duodecimal positive integer that would work? (in other words, in a 6-digit number pqrstu with pqrstu=(pqr+stu)², where each of the letters denote a base 12 digit whether same or different.)

Note: None of the numbers can contain leading zeroes.

CLS
FOR a = 1 TO 11
FOR b = 0 TO 11
 ab = 12 * a + b
FOR c = 0 TO 11
FOR d = 0 TO 11
  cd = 12 * c + d
  abcd = ab * 144 + cd
  IF (ab + cd) * (ab + cd) = abcd THEN
     PRINT a; b; c; d
  END IF
NEXT
NEXT
NEXT
NEXT

PRINT

FOR p = 1 TO 11
FOR q = 0 TO 11
FOR r = 0 TO 11
  pqr = p * 144 + q * 12 + r
FOR s = 0 TO 11
FOR t = 0 TO 11
FOR u = 0 TO 11
  stu = s * 144 + t * 12 + u
  pqrstu = pqr * 1728 + stu
  IF (pqr + stu) * (pqr + stu) = pqrstu THEN
    PRINT p; q; r; s; t; u
  END IF

NEXT
NEXT
NEXT
NEXT
NEXT
NEXT

finds

 2  6  3  0
 3  6  3  0
 11  10  0  1
 
 1  7  0  2  9  4
 4  10  4  2  9  4
 11  11  10  0  0  1
 
indicating that the third of the 4-digit numbers is 2630, and that the smallest 6-digit number that would work is 170294.  The other two 6-digit numbers that would work are 4A4294 and BBA001.
 
 

Posted by Charlie on 2012-10-28 16:54:45