You have created a 19 digit number with your 20 digit tiles as follows:

7_340_46_2010_51_49

Unfortunately someone knocked out 5 of the number tiles and placed them with the remaining number tile. The 6 tiles that are out are 6 3 2 8 9 3.

Without using any calculators, programs, or similar devices, what is the easiest way to figure out what the probability that my original 19-digit number will be a perfect square and what is this probability?

Note: The pesky "someone" also renumbered the number tiles so they were wrong and you couldn't tell what the right numbers are. This has been fixed to make the problem easier.

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 2004-07-20 16:23:28