Unwanted Zip Codes
In a certain country, they make use of ZIP (Zoning Improvement Plan) codes to help direct mail. Each city/town makes use of a 5-digit code (which could start with 0). For example, Rose Hill's ZIP is 18619.
Because certain codes could be interpreted differently when read upside down, they have been marked as unwanted. For example, Rose Hill's ZIP code becomes 61981 when read upside down, hence it is not used.
How many 5-digit ZIP codes are unwanted?
Note: This not an original problem.
The answer is 3050.
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1 solution
Consider the digits 0 through 9. 0,1,8 have rotational symmetry; 6,9 are rotationally symmetric with each other.
This means that all possible unwanted zip codes must contain only 0,1,6,8,9. Otherwise, the code would not be a number when read upside down. Since we have 5 possible digits in {0,1,6,8,9}, and the zip codes contain 5 digits, we have a preliminary total of 5^5 unwanted codes.
A code containing only 0,1,6,8,9 is still good if it reads the same upside down as right side up. Suppose the code ABCDE reads the same upside down. This means
The pairs (A,E) and (B,D) could be (0,0), (1,1), (8,8), (6,9), (9,6). C can be 0,1, or 8. The number of these good codes is #(A,E) * #(B,D) * #C = 5 * 5 * 3.
Hence, the number of unwanted zip codes is 5^5 - 5 * 5 * 3 = 3050.