Ordering, odd ones out and liars

Logic puzzles are very old. A book of puzzles for young students was written around the year 800, and it is usually said to be by Alcuin of York, a teacher at the court of King Charlemagne. One of its puzzles is about taking a wolf, a goat and a cabbage across a river in a small boat. Puzzles about people who always tell the truth and people who always lie were made popular in the late 1970s by the American puzzle writer Raymond Smullyan. None of these puzzles needs hard sums. They need careful thinking, one small step at a time, and writing down what you know so you do not have to hold it all in your head.

Putting people in order

You get a few clues like "Ana is taller than Ben" and must say who is the tallest or the shortest. The plain method is to draw a line, like a ladder, and place each person on it.

1. Write every clue the same way round. If one clue says "taller" and another says "shorter", turn the "shorter" ones around: "Eva is shorter than Leo" is the same as "Leo is taller than Eva".
2. Start with one clue and write the two names, the taller one on top.
3. Add the other clues one by one, putting each new name above or below a name you already have.
4. Read the answer from the top or the bottom of the list.

Sometimes the clues are not enough. "Ana is taller than Ben. Ana is also taller than Carla." Ana is clearly the tallest. But Ben and Carla are never compared with each other, so either of them could be the shortest. The right answer is "we can't know".

The odd one out

You see four numbers, and three of them share something the fourth does not. Your job is to find the thing they share. Check the most common ideas in this order:

A prime number is a whole number bigger than 1 that can only be divided exactly by 1 and by itself, like 2, 3, 5, 7, 11 and 13. A square number is a number multiplied by itself, like 9 = 3 × 3 or 16 = 4 × 4.

Truth-tellers and liars

In these puzzles, each person always tells the truth or always lies. A truth-teller never says anything false. A liar never says anything true. The method is to suppose something and follow it until it either works or breaks.

1. Suppose the speaker tells the truth. Then everything they said is true. Does that lead to something impossible?
2. If it does, the speaker must be a liar. Then what they said is false.
3. Use the false (or true) statement to find out about the other person.
4. Check the answer: every truth-teller says only true things, and every liar says only false things.

One more: Ana says "At least one of us lies." Suppose Ana lies. Then her sentence is false, so nobody lies. But we just said Ana lies. Impossible. So Ana tells the truth, and at least one of them really does lie. It is not Ana, so it is Ben.

Three boxes with wrong labels

Three boxes are labeled Apples, Oranges and Mixed. Every label is wrong. You may take one fruit from one box, without looking inside. Which box should you choose so you can fix all three labels?

Choose the box labeled Mixed. Its label is wrong, so it does not hold mixed fruit. It holds only apples or only oranges. One fruit is enough to tell which.

Crossing the river

Four people must cross a river. The boat carries at most 2 people, and someone must row it back each time. What is the fewest crossings, counting trips both ways? The key is that every trip there and back moves only one person across: two go over, and one comes back.

So 4 people need 5 crossings. The last trip does not come back, so it moves 2 people. In general, with a boat for 2, each round trip moves 1 person, until only 2 are left for the final trip. For 2 or more people, the fewest crossings is 2 × (people − 2) + 1. For 4 people: 2 × 2 + 1 = 5.