Logic and Casework

On this island knights always tell the truth and knaves always lie. A says B is a knave. B says we are the same type. Try each case. Suppose A is a knight. Then A tells the truth, so B is a knave. They are different types, so B's statement is false, and that is fine because knaves lie. Now suppose A is a knave. Then A's statement is false, so B is a knight. They are still different types, so B's statement is false, but a knight cannot say something false. That case fails. Only one case survives: A is a knight and B is a knave. Always test every case, because a puzzle with one answer has exactly one survivor.

Ordering puzzles fall apart once you find the clue that pins someone down. Here, ask who can finish last. Not Amy, by clue one. Not Ben, because Cal finished right after him. Not Dee, because she beat Ben. So Cal is last, which puts Ben third. Amy was not first, so she is second, and Dee is first. Check every clue against Dee, Amy, Ben, Cal and they all hold.

When a question asks how many ways, choose one thing to split on and count each case. For three-digit numbers whose digits add to five, split on the hundreds digit, which can be one to five but not zero. If it is one, the other two digits add to four: zero four, one three, two two, three one, four zero, five ways. If it is two, they add to three, four ways. Then three ways, two ways, and one way. The cases do not overlap and they cover everything, so the total is fifteen.

If you put more objects than boxes, some box gets two. That is the pigeonhole principle. It answers questions like how many socks must you take in the dark to be sure of a pair. Imagine the unluckiest draw: one red, one blue, one black, no pair yet. The fourth sock must match one of them, so four is enough. To be sure of two blue socks, the unluckiest draw is every red and every black first, fourteen socks, then you still need two blue, so sixteen. Always build the worst case, then add what you need.

When you know the end and the steps, work backwards. At each shop you spend half your money plus two dollars, and after three shops you have one dollar. Leaving a shop with one dollar means that half your money, minus two, was one dollar, so you had one plus two, doubled, which is six. Undo the next shop: six plus two, doubled, is sixteen. Undo the first: sixteen plus two, doubled, is thirty-six. Check forwards: thirty-six, spend twenty, leaves sixteen; spend ten, leaves six; spend five, leaves one. Quiz and exercises now.