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Finding numbers (Posted on 2013-02-12) Difficulty: 3 of 5
ABC and CBA are two three-digit decimal numbers, with A and C different non-zero digits. Squares of these numbers are five-digit numbers DEFGH and HGFED respectively. Find all such three-digit numbers.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution | Comment 1 of 2

Attempting to find solutions where each letter represented a different digit, I found none. So assigning the same digit to different letters was allowed.

However, A and C are specified to be different from each other. The solutions which fit that are:


ABC CBA DEFGH HGFED
102  201      10404  40401
103  301      10609  90601
112  211      12544  44521
113  311      12769  96721
122  221      14884  48841
201  102      40401  10404
211  112      44521  12544
221  122      48841  14884
301  103      90601  10609
311  113      96721  12769

were it not for the requirement that A and C be different, the following would have worked:

101  101      10201  10201
111  111      12321  12321
121  121      14641  14641
202  202      40804  40804
212  212      44944  44944
 

DEFDBL A-Z
CLS
FOR a = 1 TO 9
FOR b = 0 TO 9
  FOR c = 1 TO 9
  IF c <> a THEN
    abc = 100 * a + 10 * b + c
    cba = 100 * c + 10 * b + a
    abc2 = abc * abc
    cba2 = cba * cba
    abc2s$ = LTRIM$(STR$(abc2))
    cba2s$ = LTRIM$(STR$(cba2))
    IF LEN(abc2s$) = 5 AND LEN(cba2s$) = 5 THEN
      good = 1
      FOR i = 1 TO 5
       IF MID$(abc2s$, i, 1) <> MID$(cba2s$, 6 - i, 1) THEN good = 0: EXIT FOR
      NEXT
      IF good THEN PRINT abc; cba, abc2; cba2
    END IF
  END IF
  NEXT
NEXT
NEXT

PRINT : PRINT
FOR a = 1 TO 9
FOR b = 0 TO 9
  FOR c = 1 TO 9
  IF c = a THEN
    abc = 100 * a + 10 * b + c
    cba = 100 * c + 10 * b + a
    abc2 = abc * abc
    cba2 = cba * cba
    abc2s$ = LTRIM$(STR$(abc2))
    cba2s$ = LTRIM$(STR$(cba2))
    IF LEN(abc2s$) = 5 AND LEN(cba2s$) = 5 THEN
      good = 1
      FOR i = 1 TO 5
       IF MID$(abc2s$, i, 1) <> MID$(cba2s$, 6 - i, 1) THEN good = 0: EXIT FOR
      NEXT
      IF good THEN PRINT abc; cba, abc2; cba2
    END IF
  END IF
  NEXT
NEXT
NEXT


  Posted by Charlie on 2013-02-12 17:49:03
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