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.

Before anything is eliminated, there are 100 possible combonations.
The numbers 01-09 are invalid, so that takes 9 off.  Out of the 91 remaining, 00, 11, 22, 33, 44, 55, 66, 77, 88, and 99 do not use different numbers, so they are invalid as well.  There are now 81 numbers left.  The numbers that aren't possible because they can be read upside down, not including any number already eliminated, are 16, 18, 19, 61, 68, 81, 86, 89, 91, and 98 (Note: 10, 60, 80, and 90 are good, even though their upside down versions are not, and 69 and 96 look the same even if they are viewed incorrectly).  Those ten numbers, once removed, leave 71 possible useful codes.

Thus, 71 is the answer if 0 is viewed as a natural number.  If it is not, 10, 20, 30, 40, 50, 60, 70, 80, and 90 must be subtracted from the the 71 codes, and only 62 remain.  (Note: In the problem itself, it says that the only DIGITS that appear correct when inverted are 0, 1, 6, 8, and 9, not NATURAL NUMBERS)

So, to show the mathematics:

    100         Original Number Set
    -  9         01-09
      91
    - 10        Non-unique digits
      81
    - 10        Combonations that appear the same upside down
      71

And, if 0 is not included as a natural number:

      71
    -  9         Numbers with 0
      61

Posted by Rick on 2005-05-02 03:30:56