15 minute parking sign. Latitude 42.871466 S, Longitude 147.31518 E. Photo: Mark Diamond

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

Word-reversed road signage. Image: Mark Diamond

Here in Canberra, the words “LANE ONE FORM” can frequently be seen painted on the road at the point where two lanes of traffic merge. The words are painted in a column, rather than a straight line so that they appear as
LANE
ONE
FORM
Similarly configured signs say things like “ONLY LANE TAXI BUS”.

No doubt the engineers who designed the road signs had convinced themselves that a driver would see, and read, the word “FORM” before seeing the word “ONE”, and that that word would, in turn, be seen and read prior to the word “FORM”. The words never appear to me to read like this. Instead I have learned simply to associate the nonsensical “LANE ONE FORM” with the idea that two lanes of traffic merge at this point.

If the driver of a vehicle were travelling at a speed which was appropriately related to the distance separating the words along the length of the road, then it is obvious that the words would flash into view separately, and in succession, and would be seen in the order “FORM ONE LANE”. However, the necessary relationship between speed and separation does not appear to have been described anywhere that I have looked, and the distance that separates the words as they actually appear on Canberra roads appears, to me, to be absurdly short. Indeed, at the distance at which the words are separated, it would appear to be more sensible to write the instructions in the natural order, namely “FORM ONE LANE”. Nonetheless, my speculations are just that, and I have no empirical evidence to support my view. However, the problems posed by presenting stationary words to moving drivers raise some interesting research questions that might well be within reach of a psychology or engineering student, perhaps at honours or masters level.

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