Once the tiles were shown to have the wrong numbers on them, the problem becomes easier.
All perfect squares mod 9 are either 0, 1, 4, or 7. For example, all perfect squares who are integers from 0 to 9 squared mod 9 are 0, 1, 4, or 7.
00 mod 9 is 0
01 mod 9 is 1
04 mod 9 is 4
09 mod 9 is 0
16 mod 9 is 7
25 mod 9 is 7
36 mod 9 is 0
49 mod 9 is 4
64 mod 9 is 1
81 mod 9 is 0
Then it can be proved that all perfect squares are mod 9 by expressing integers as another integer times 9 plus an integer. For integers a and b, where b is less than 9, any integer can be expressed as (9a + b), so any perfect square can be expressed as (9a + b)^2 which equals 81a^2 + 18ab + b^2 or 9(9a^2 + 2ab) + b^2. Since b is 9 or less, b^2 mod 9 is 0, 1, 4, or 7. When this is added on to the left term (which obviously is 0 mod 9) it can only be 0, 1, 4, or 7.
Not using the 6, the sum of the digits in the number is 71 which is 8 mod 9.
Not using the 3, the sum of the digits in the number is 74 which is 2 mod 9.
Not using the 2, the sum of the digits in the number is 75 which is 3 mod 9.
Not using the 8, the sum of the digits in the number is 69 which is 6 mod 9.
Not using the 9, the sum of the digits in the number is 68 which is 5 mod 9.
Not using the 3, the sum of the digits in the number is 74 which is 2 mod 9.
None of these includes 0, 1, 4, or 7, so the answer is 0.

Posted by Gamer
on 20040720 16:23:28 