A Cute Paradox (Do Not Apply)

A maths teacher informs her class that the following week she will slap a surprise test on them. This will ensure that they keep up with the work and study every night rather than the night before the test. All of the students gasp in shock horror, except for some young smart-ass up the front who chortles that “there is no such thing as a surprise test“.

The student’s reasoning is as follows:

“Well, you definitely can’t give us a surprise test next Friday because if we got to Thursday night and we hadn’t been given a test yet, we would know that it would be on the next day. So it is impossible for you to give us a surprise test on the Friday.

Furthermore, let’s think about what would happen if it got to Wednesday night and we hadn’t been given a test yet. We would know that the surprise test would have to be on either Thursday or Friday. But we know that there can’t be a surprise test on the Friday, and so we would automatically know that the surprise test would have to be on Thursday, thus making it not a surprise. So there can’t be a surprise test on Thursday or Friday.

So if we got to Tuesday night without being surprise-tested, then we would know that the test would be on either Wednesday, Thursday or Friday. Of course, we have already established that the surprise test can not be on Thursday or Friday, and so must be on Wednesday. But once again, this cancels the property of surprise for the Wednesday.

We can keep going with this reasoning until we establish that you can’t give us a surprise test on any of the days.”

The smart-ass leans back in his chair proudly as the other kids smile and give him “mad props”. The teacher simply goes to her desk, pulls out a set of test papers and hands them out. “Surprise!” she says.

This is a well known paradox, as though the logic may seem airtight, we all know that surprise tests do indeed exist. So what’s the problem here? What assumptions is the smart-ass making in his reasoning?