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Play: Riddles, clues and Sudoku

Logic and puzzles · step 1 of 5

A question to start

Here are three boxes. One holds only apples, one holds only oranges, and one holds both. Someone has put a label on each box, but every label is wrong.

Apples Oranges Apples and oranges ???

You may reach into one box, without looking, and pull out one fruit. After that you must say what is really in every box. Which box should you reach into?

Think about it for a minute before you read on. The answer is in the worked example below, and it uses the most important trick in this whole topic.

The idea

A clue rules things out

A clue is a piece of information you know is true. Most clues do not tell you the answer straight away. Instead, they tell you what the answer cannot be. When you have ruled out everything except one choice, that choice must be the answer.

Detectives work like this. So do doctors, mechanics and scientists. So does anyone who solves a Sudoku.

Make a list of what is still possible

When a puzzle has several people or boxes or squares, keep a list of what each one could still be. Every time a clue rules something out, cross it off. When a list is down to one thing, you have found an answer, and that answer often rules something out somewhere else. One small step leads to the next.

Try a guess, and see if it breaks

Sometimes no clue gives you a straight answer. Then you can say: "Suppose it is this. What would happen?" Follow the clues. If you reach something impossible, like one box holding two different things, or a person being in two places at once, then your guess was wrong and you can cross it off. This is called trying a case. It is how mathematicians prove things, and you will meet it again at every step of this topic.

Watch out for "every", "only" and "not"

Small words matter. "Every label is wrong" is a very strong clue: it tells you something about all three boxes at once. "Not" turns a clue around. Read each clue slowly and ask: what does this tell me, and what does it not tell me?

Worked examples

1. The three boxes. Reach into the box labelled "Apples and oranges". Here is why.

  1. That label is wrong, so this box is not the mixed box. It holds only apples or only oranges.
  2. So whatever fruit you pull out tells you the whole box. Say you pull out an apple. Then this box is the apples box.
  3. Now look at the box labelled "Oranges". Its label is wrong, so it is not oranges. It is not apples either, because we just found the apples. So it must be the mixed box.
  4. Only one box is left, the one labelled "Apples", and only one thing is left: oranges.

If you had pulled out an orange instead, the same steps work with apples and oranges swapped. Why not reach into another box? If you reach into "Apples" and get an orange, that box could be the oranges box or the mixed box, and you cannot tell which. The mixed-label box is the only one where one fruit tells you everything.

2. A small Sudoku. In a 4 by 4 Sudoku, each row, each column and each of the four 2 by 2 boxes (the thick squares) must hold the numbers 1, 2, 3 and 4, each exactly once.

124 41 14 41
  1. Row 1 has 1, 2 and 4. Only 3 is missing, so the empty square is 3.
  2. Row 2 is missing 2 and 3. Look at its first square. The top-left box already has 1, 2 and 4, so this square cannot be 2. It must be 3, and the last square of row 2 is 2.
  3. Row 3 is missing 2 and 3. Its first square is in column 1, which now has 1, 3 and 4. So it is 2, and the last square of row 3 is 3.
  4. Row 4 is missing 2 and 3. Its second square is in column 2, which has 2, 4 and 1. So it is 3, and the other empty square is 2.

The finished grid, row by row: 1 2 3 4, then 3 4 1 2, then 2 1 4 3, then 4 3 2 1. Check every column and box: each has 1, 2, 3 and 4 once. Checking at the end is part of solving.

Notice that no step needed a guess. Each one asked: what is still possible here? and found only one answer.

Try it

  1. Ann, Ben and Sam each have one pet: a cat, a dog or a fish, all different. Ann's pet has no legs. Ben's pet does not bark. Who has which pet?
  2. A farmer must take a wolf, a goat and a cabbage across a river. The boat holds the farmer and one other thing. If left alone together, the wolf eats the goat, and the goat eats the cabbage. How can the farmer get everything across?
  3. A drawer holds 10 red socks and 10 blue socks, all mixed up. The room is dark. How many socks must you take out to be sure you have two of the same colour?
  4. One row of a 4 by 4 Sudoku reads 2, empty, empty, 4. The column of the first empty square already has a 1 in it. Fill in the row.
  5. On a puzzle island, some people always tell the truth and the rest always lie. Can anyone on the island say "I am a liar"?
Answers
  1. Ann's pet has no legs, so Ann has the fish. Ben's pet does not bark, so it is not the dog, and the fish is taken, so Ben has the cat. Sam has the dog.
  2. Take the goat over first (the wolf will not eat the cabbage). Go back alone. Take the wolf over, and bring the goat back. Take the cabbage over and leave it with the wolf. Go back alone and fetch the goat. That is 7 trips across the river. The clever step is bringing something back.
  3. 3 socks. With 2 socks you might be unlucky and get one of each colour. A 3rd sock must match one of them, because there are only two colours.
  4. The row is missing 1 and 3. The first empty square cannot be 1, so it is 3, and the other is 1. The row is 2, 3, 1, 4.
  5. No. A truth-teller cannot say it, because it would be a lie. A liar cannot say it either, because it would be true, and liars never say true things. Some sentences are impossible for everyone on the island. You will use this idea in the next step.

Common mistake

Stopping as soon as an answer fits. In the box puzzle, it is easy to think "the 'Apples' box has oranges" and stop. But a good solver checks that the answer works with every clue, and also asks whether some other answer might fit just as well. If two answers fit, you have not finished yet; you need another clue or another step.

Where it leads