Paradoxes are a great way to get student thinking and talking about thinking. The initial state of confusion, followed by the illusive, enigmatic feeling of understanding is somehow enticing and enjoyable. I spent a little pastoral time discussing the following paradoxes with a group of Year 8 students, and the result was a palpable buzz in the classroom.

They are all taken from the excellent list of paradoxes on Wikipedia, and ordered (roughly) in ascending order of confusion generation:

• Socratic paradox: “I know that I know nothing at all.”
• Liar paradox: “This sentence is false.” This is the canonical self-referential paradox. Also “Is the answer to this question no?” And “I’m lying.”
• Ship of Theseus (a.k.a. George Washington’s axe or Grandfather’s old axe): It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with.
• Sorites paradox (also known as the paradox of the heap): One grain of sand is not a heap. If you don’t have a heap, then adding only one grain of sand won’t give you a heap. Then no number of grains of sand will make a heap.
• Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess what the crocodile will do, how should the crocodile respond in the case that the father correctly guesses that the child will not be returned?
• Barber paradox: A barber (who is a man) shaves all and only those men who do not shave themselves. Does he shave himself? (Russell’s popularization of his set theoretic paradox.)