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.
There are 90 possible codes. Excluding the following for possible reversals:
16, 91, 18, 81, 19, 61, 68, 89, 86, 98
That brings the total useful to 80. There are nine numbers that have duplicate digits : 11, 22, 33, 44, 55, 66, 77, 88, 99.
69 and 96, however, can be used as both numbers are still viewed the same when they are upside down and are, therefore, not excluded from the list. That leaves the final count of useful codes at 71.
69 and 96
