The Wednesday Morning Math Challenge: Week 4 Answers

Below, you’ll find the answers to last week’s puzzles. For those who didn’t get a chance, if you would like to take a crack at the Week 4 challenge, just click here.

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1. Simon is a knight, and Bob is a knave. To verify this, let’s first suppose that Bob is a knight. If Bob were a knight, then he would be telling the truth when he said that Simon was a knave (because knights always tell the truth). But that would mean that Simon was lying when he said that either he or Bob, but not both, was a knight (because knaves always lie). But here we have a contradiction, because if Bob is a knight, and Simon is a knave, then one of them is a knight, and Simon is telling the truth — which means that Simon is not a knave. Therefore, we know that Bob is not a knight but a knave. That means that we know he’s lying when he says that Simon is a knave. Therefore, Simon must be a knight. Simon’s statement checks out: he’s telling the truth, as a knight should, when he says that one of them is a knight and one a knave.

This problem is a part of a classic type of logic puzzle where one character always answers the truth and the other always lies. If you’ve seen the movie Labyrinth, you’ve seen this at work.

2. This is a trick question! A response is that this kind of barber doesn’t exist. The barber can’t shave himself, because he does not shave anyone who shaves himself. But if he doesn’t shave himself, then he belongs to the set of men that the barber shaves. This puzzle is a common representation of a paradox proposed by the philosopher and mathematician Bertrand Russell involving the set of sets that aren’t members of themselves. The barber question might be a little silly (and you could try to come up with all sorts of refutations involving hairless men or gender fluidity), but Russell’s more rigorous exploration of the paradox in set theory shook the foundations of mathematics at the turn of the twentieth century.

3. There isn’t a positive integer that is not definable in fewer than twelve words. “The smallest positive integer not definable in fewer than twelve words” has eleven words — which means that it is definable in fewer than twelve words. This is another trick question that we can trace back to Bertrand Russell, who attributed it to a librarian named G.G. Berry. (It’s known as the Berry paradox.)