The Surprise Exam Paradox

There are several different versions of this particular paradox – like Holli’s Paradox, and The Unexpected Hanging Paradox. Before I present the surprise examination version, it behooves me to mention that no correct (or final) solution to this paradox has been established yet.

Here’s the paradox:

A teacher announces in class that an examination will be held on some day during the following week, and moreover that the examination will be a surprise.

The students argue that a surprise exam cannot occur. For suppose the exam were on the last day of the week. Then on the previous night, the students would be able to predict that the exam would occur on the following day, and the exam would not be a surprise. So it is impossible for a surprise exam to occur on the last day. But then a surprise exam cannot occur on the penultimate day, either, for in that case the students, knowing that the last day is an impossible day for a surprise exam, would be able to predict on the night before the exam that the exam would occur on the following day. Similarly, the students argue that a surprise exam cannot occur on any other day of the week either.

Confident in this conclusion, they are of course totally surprised when the exam occurs (on Wednesday, say). The announcement is vindicated after all. Where did the students’ reasoning go wrong?

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Similar posts: The Monty Hall Paradox, A Mathematical Conundrum, The Voting Paradox, Broken Clocks.

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6 responses to “The Surprise Exam Paradox

  1. An exam on Friday will not be a surprise because of “hindsight” … all the other “deadlines” are based on the assumption that no exam held on Friday will be a surprise. A week has 5 working days so if the exam won’t be done on Friday the latest it can be done is Thursday – which will stop being a surprise at the end of Wednesday and so on …
    The students are assuming that the exam will be held on the last possible day – and this is the problem with their reasoning.IMO.

    • ‘An exam held on Friday will not be a surprise’ is not an assumption, but a logical deduction from the teacher’s statement. The students first consider all 5 days as possible days on which the exam can take place. Now, suppose the exam occurs on Friday, then by Thursday end-of-day they will be able to deduce that exam is going to be on the next day. Hence, an exam that takes place on Friday can not be a surprise exam. So now we are left with 4 possible days on which the exam can take place (Friday is rules out as a possible day). The same logic that we used for Friday is applicable to Thursday, and then to all other days of the week.

  2. Ian

    Simple.
    The students know that if they go to Thursday, then a Friday test would not be a surprise. Thursday is out of the question because Wednesday night is Wing Night at the Black Knight. They also know that since the teacher is secretly cavorting with the blond in the front row each weekend, a Monday test is not very likely. Tuesday is definitely not possible because all the teachers get together on Monday night to drink and watch football. So, Wednesday is the only likely day for the surprise exam. It shouldn’t have been a surprise at all.

  3. only if the exam does not occur on any given day, say Thursday, the exam will not be a surprise to occur on the next day. The possibility of exam to occur on the last day, is based on the assumption that the surprise factor for all other days are negated – that’s amistake on the students’ side

  4. William

    (1) One it depends if the students even conduct the reasoning (they may be retarded) and maybe an omniscient or intelligent teacher knows this and this is why the surprise examination is announced
    (2) The backward regression depends first of all on whether friday is a surprise. If friday cannot be surprise exam as some expect then there is still unsurety as to whether there will still be a non-surprise examination or not on friday (there is a distinction between surprise examination and examination)— but as one cannot know whether there will be such an examination with full knowledge (but can only presume) then obtaining of such an exam, if it does obtain, will be a surprise. Hence if there is a surprise examination on friday, there cannot be a surprise examination and the converse (of course there may be an equivocation here on the nature of how the word surprise is being used in the two instances, morover, the paradox can be strengthed to ensure that it is to be an undoubtable thesis) there will be an examination (surprise or not) .
    (3) The most important problem is the equivocation in the backward regress argument between making deductions at some time and deductions about some time. If you think about it closely, you will see that it is completely circular

  5. William

    it should have read…’Morover, the paradox can be strengthened to ensure that it is to be an undoubtable thesis that there will be an examination (surprise or not)’ .

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