Paradoxical parking

I had occasion recently to park my car within the zone defined by the sign shown in the accompanying photograph. By chance, it was around 5.50 pm when I parked. My musing on how long I could safely leave the car in the space I’d found led me to consider the following (apparent) paradox, which I have not previously seen documented …

The sign says that parking between 8 am and 6 pm is limited to a quarter of an hour. Yet, if one parks after 5.45 pm then by the time the stated quarter hour limit has elapsed, parking will have become unrestricted, meaning that the true limit of quarter of an hour is actually only applicable from 8 am to 5.45 pm. So the sign is wrong. Why not paint it correctly?

One can see where this line of thinking is headed. Repainting the sign, followed by induction (or, equivalently, recursion) leads one to the conclusion that parking is completely unrestricted—which it clearly is not, since I have seen cars parked in the same spot but decorated with violation tickets.

The paradox seems to me to be closely related to a well known paradox, first described, without a name, by O’Connor [1], but better known as the Unexpected Hanging [2] or the Surprise Examination [3]. That paradox has been addressed by many different people over the years, primarily with a focus on what it means for something to be a “surprise” or to be “unexpected”. The point at which the argument (both with the Unexpected Hanging and the Paradoxical Parking) goes awry is clearly at the point of attempting to apply recursion but I know of no literature specifically addressing that issue.

References

[1] O’Connor, D. J. (1948). Pragmatic paradoxes. Mind, 57(227), 358–9.

[2] Gardner, M. (1991). The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: University of Chicago Press.

[3] Smullyan, R. M. (2000). Forever Undecided: A Puzzle Guide to Gödel. Oxford: Oxford Paperbacks.

Contributors: Mark R. Diamond, Angela O’Brien-Malone