An intellingence agency wants to have codes. For this it uses two digit natural numbers such that the two digits are different. Each of these codes are written on different sheets of paper so as to be used. However, the director of the agency soon realizes that many codes are not uniquely recognisable.
For example, 61 and 19 is one such pair because when the sheet of paper is read upside down, a different number may be read. However, 01 is invalid (no leading zeroes).

How many useful codes are there that the agency can use?
Note: The only digits that make sense when inverted are 0,1,6,8 and 9.

(In reply to solution by gary wallravin)

I guess 22, 33, 44, 55 and 77 won't be interpreted as different number when reversed. So, the count is 74.

My approach was like this:

Out of the 90 possibilities (10 to 99), multiples of 10 can all be used because they all have zero.

So, what is left to choose from is, a two digit combination of numbers - 1 to 9. That leaves us with 81 numbers. The probability that the first digit is reversible is 4/9 (1,6,8,9 out of 1 to 9). And for the second digit it is again 4/9. In order for a number to be disqualified, both digits have to be reversible. So, the probability for this is (4/9)*(4/9)  = 16/81.

That means 65 out of 81 numbers can be used as a 2 digit code + the 9 numbers which are multiples of 10. So, the total count is 74.

Posted by venkat v on 2005-04-04 19:24:13