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Logic puzzles

A logic puzzle gives you clues instead of numbers, and the answer is already fixed before you start — your job is to find the one arrangement the clues allow. There is no arithmetic in it at all. What it trains is the thing arithmetic sits on top of: reading carefully, holding several facts at once, and being able to say why the answer must be that one and not another.

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Putting three people in order

Kit is taller than Hugo. Elle is taller than Kit. Who is the tallest?

  1. Take the first clue: Kit is above Hugo. Write Kit, then Hugo underneath.
  2. Take the second: Elle is above Kit. So Elle goes above the pair you already have.
  3. The order from tallest down is Elle, Kit, Hugo.
  4. The tallest is Elle.

Worth knowing. The clues do not arrive in order, and they never will. Place the first pair, then fit each new clue onto what you already have.

Matching people to things

Ana, Lena and Finn each have a different pet: cat, dog or fish. Ana does not have the dog. Lena has the fish. Which pet does Finn have?

  1. Start with what you are told outright: Lena has the fish.
  2. That means neither Ana nor Finn has the fish — each pet belongs to one person.
  3. Ana does not have the dog, and cannot have the fish, so Ana has the cat.
  4. Only the dog is left, so Finn has the dog.

Worth knowing. "Ana does not have the dog" looks like it tells you nothing. It tells you a great deal once you have crossed the fish off her list too.

Finding a middle place

Hugo finished ahead of Elle. Kit finished ahead of Hugo. Elle finished ahead of Jonah. Who finished 3rd?

  1. Kit is ahead of Hugo, and Hugo is ahead of Elle, so far: Kit, Hugo, Elle.
  2. Elle is ahead of Jonah, so Jonah goes on the end: Kit, Hugo, Elle, Jonah.
  3. Count from the front: Kit is 1st, Hugo is 2nd, Elle is 3rd.
  4. Elle finished 3rd.

Worth knowing. Building the whole order is usually quicker than trying to jump straight to the place you were asked about.

Where it usually goes wrong

  • Answering from one clue. Every puzzle here needs at least two put together — if one clue seemed to be enough, something has been misread.
  • Forgetting that everybody has something different. That fact is doing half the work in a matching puzzle.
  • Skipping the "does not" clues because they feel empty. They are usually what finishes the puzzle.
  • Stopping at the first arrangement that fits. Check it against every clue, not just the one you used last.

The Professor reads every clue before writing anything down. Half of them, he says, are only useful once you know something else — and the ones that seem useless first time round are usually the ones that finish the puzzle.

Practise it

All level paths

Questions teachers and parents ask

Is there always exactly one answer?

Yes, and that is checked rather than hoped for. Every puzzle on this site is solved by the computer from its own clues before it is printed, and any puzzle that turns out to have two possible answers is thrown away instead. A child who reasons perfectly should never be marked wrong.

There is no arithmetic here. Is this really maths?

It is the part of maths that everything else rests on. The Common Core lists it separately as the Standards for Mathematical Practice — making sense of a problem, sticking with it, and building an argument you can defend. That is exactly what a logic puzzle asks for, which is why these pages tag MP standards rather than pretending to teach a number skill.

My child gets the right answer but cannot explain it. Does that matter?

It is worth working on, gently. Ask which clue they used first and what it ruled out. The explaining is the skill; the answer is just the evidence they have it. A puzzle solved by lucky guess and one solved by reasoning look identical on the page.

Are these taken from somewhere?

No. Ranking people and matching them to things are old puzzle forms that belong to nobody, but every individual puzzle here is built fresh by the site — the people, the order and the clues are generated, and then the whole thing is solved to prove it has one answer. The names are invented.

Where this shows up

Standards

  • 3.OA · Operations and algebraic thinking
  • 4.OA · Operations and algebraic thinking
  • 5.OA · Operations and algebraic thinking
  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them: understand what is being asked, plan a route to the answer, and check that it works.
  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others: follow a chain of clues and justify why the conclusion must hold.
  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure: find the rule or relationship underneath a set of facts, and use it to settle what has to follow.

Formats