By adding 1 to the non leading zeros number N = AAABBBCCC, it becomes a square. Find N.

A, B, and C are different digits.

Note: While a solution is trivial with the aid of a computer program, show how to derive it without one.

Having seen this prior to publication, and realising the futility of a spreadsheet, I compiled the program listing below (QuickBasic) and passed it on.

That we might see how computer power may resolve issues, and it is not to be derided, I am glad that pcbouhid's note reads as it does. 

I trust that those yet without this knowledge can benefit from the insights brought by Dej Mar and K Senguptna from their discipline base.

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CLS
FOR a = 0 TO 9
 FOR b = 0 TO 9
  FOR c = 0 TO 9
     IF (a <> b AND b <> c AND a <> c) THEN
       N = a * 10 ^ 8 + a * 10 ^ 7 + a * 10 ^ 6 + b * 10 ^ 5 + b * 10 ^ 4 + b * 10 ^ 3 + c * 10 ^ 2 + c * 10 + c
       IF SQR(N + 1) = INT(SQR(N + 1)) THEN PRINT a, b, c
     END IF
 NEXT
 NEXT
 NEXT
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This gave me:
 0, 1, 5
 0, 4, 8 and
 1, 9, 8
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I could have gotten just the one result by starting "a" at 1 rather than 0.

Posted by brianjn on 2008-07-18 00:35:17