Brain Teasers and Puzzles

Confusing Zip Codes

The U.S. uses 5-digit zip codes to help determine where mail goes. Since mail can be oriented in all sorts of directions, they avoid assigning zip codes that could be confused with a different zip code when read upside-down. For example, 61666 could be confused with 99919 when upside-down, so mail could be accidentally routed to the wrong zip code if both were actual zip codes.

How many zip codes could be confused with a different zip code when read upside-down?

In the example above, 61666 and 99919 would count as two confusing zip codes. Also, zip codes are allowed to start with 0, such as 00501.


There are 3050 confusing zip codes.

There are only 5 digits that read as valid digits when upside-down: 0, 1, 6, 8, and 9. This means there are 5^5 = 3125 possible 5-digit combinations that can be read upside-down.

However, some of these combinations read as the same zip code when upside-down, which is not a problem since we are looking only for zip codes that will be confused with a different zip code.

Zip codes that read the same when upside-down must have 0, 1, or 8 in the middle, the first digit is the last digit upside-down (i.e., a first digit of 0, 1, 6, 8, or 9 means the 1st digit must be 0, 1, 9, 8, or 6, respectively), and the 2nd digit is the 2nd last digit upside-down. This means there are 5 possibilities for the 1st digit, 5 possibilities for the 2nd digit, 3 possibilities for the middle digit, and only 1 possibility for the last two digits (since they have to match the first two digits): 5 x 5 x 3 x 1 x 1 = 75.

3125 combinations that can be read upside-down minus 75 combinations that read the same upside-down leaves us with 3050 combinations that read as a different zip code upside down.

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