On the first of December, the prison warden informs a mathematician on death row: "We will execute you by the end of the year, on a day you don't expect."
The mathematician is immediately overjoyed! The prison warden is utterly confused by this reaction and leaves him alone, but the mathematician realizes that she's been pardoned! By the power of induction:
There is no day on which the mathematician will be executed.
For suppose it were New Year's Eve (that is, there is one day left in the year). The mathematician would surely be expecting to be executed, if she were still alive then. But then she would surely be expecting to be executed on New Year's Eve, which means she can't be!
So (base step) the mathematician will not be executed on New Year's Eve.
Now suppose the mathematician will not be executed with n days left in the year. By the same reasoning, the mathematician would expect the previous day, when there are n+1 days left, to be her day of execution - but by the same logic it therefore can't be the day of her execution! So if (induction step) she can't be executed on a particular day, she can't be executed on the previous day either!
The mathematician safely concludes that she will not be executed. The prison warden comes for her two days before Christmas and executes her: it certainly was on a day she didn't expect.