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Sorting by one rule

Logic and puzzles · level 1, lesson 1 of 6

A question to start

Sam had a big jar of buttons. He tipped them out and sorted them into two trays. Here they are.

left tray right tray

Look closely. Sam did not sort them by colour: there are orange, blue and green buttons in both trays. He did not sort them by size either: there are big and small buttons in both trays.

So what was Sam's rule? What is the same about every button in the left tray, and different about every button in the right tray? Look before you read on.

(Here is a hint: count the little holes.)

The idea

What sorting means

To sort means to put things into groups, so that things in the same group are alike in some way. You sort when you tidy up: socks in one drawer, shirts in another. You sort when you play: red blocks here, blue blocks there.

The way you decide which group a thing goes in is called the rule. Sam's rule was:

Buttons with 2 holes go in the left tray. Buttons with 4 holes go in the right tray.

Every button has 2 holes or 4 holes. So every button has exactly one place to go. That is what makes it a good rule.

A yes group and a no group

The simplest kind of rule is a question with a yes or no answer.

  • "Can it fly?"
  • "Is it blue?"
  • "Does it have corners?"

You ask the question about each thing, one at a time. If the answer is yes, the thing goes in the yes group. If the answer is no, it goes in the no group.

Here are some animals, sorted by the question "Can it fly?"

yes: it can fly no: it cannot fly bird bee butterfly bat dog fish snail penguin

Look at the bat. A bat is not a bird. It has fur, not feathers. But a bat can fly, so it goes in the yes group. The rule asks only one thing: can it fly? It does not ask "is it a bird?"

Now look at the penguin. A penguin is a bird. It has feathers and wings. But a penguin cannot fly. It uses its wings to swim! So the penguin goes in the no group.

That is the big lesson of sorting: follow the rule, and only the rule. Do not let other things trick you.

Every thing goes in one group

With a yes or no rule, each thing goes in exactly one group. Why?

  • It cannot go in both groups, because the answer cannot be yes and no at the same time.
  • It cannot go in no group, because the answer has to be yes or no.

So if you count the yes group and the no group and add them, you get every thing. In the animal picture there are 4 in the yes group and 4 in the no group. That is 8 animals, and there are 8 animal cards. Nothing is missing. Nothing is counted twice.

A rule can make more than two groups

Some rules have more than two answers. "What colour is it?" can have the answer orange, blue or green. Sort Sam's buttons by colour and you get three groups.

orange blue green

The same buttons. A different rule. Different groups! There is not just one right way to sort things. There are many, and each way shows you something different about them.

A good rule is clear

A good rule is one that everybody can use and get the same groups.

  • "Is it blue?" is a clear rule. Everybody can look and agree.
  • "Has it got 4 holes?" is a clear rule. Everybody can count.
  • "Is it pretty?" is not a clear rule. You might think a button is pretty and your friend might not. You would make different groups.

In math we like clear rules, because then nobody has to argue. You can check.

Saying your rule

When you sort, try to say your rule out loud, in a whole sentence: "I put the ones that can fly here, and the ones that cannot fly there." When you see someone else's groups, try to find their rule, like you did with Sam's buttons. That is detective work: you look for the thing that is the same inside a group and different between the groups.

Worked examples

Example 1. Sort the shapes. Sort these shapes by the rule "Does it have corners?"

circle triangle square oval star rectangle

Go one shape at a time.

  1. The circle: no corners. It goes in the no group.
  2. The triangle: 3 corners. Yes.
  3. The square: 4 corners. Yes.
  4. The oval (an egg shape): no corners. No.
  5. The star: it has lots of points, and each point is a corner. Yes.
  6. The rectangle: 4 corners. Yes.

Yes group: triangle, square, star, rectangle (4 shapes). No group: circle, oval (2 shapes). Check: 4 and 2 make 6, and there are 6 shapes.

Example 2. Find the rule. Mia sorted some numbers into two groups.

left right 1 3 2 4 7 9 6 8 10

What is Mia's rule?

  1. Look at the left group: 1, 3, 2, 4. They are all small numbers.
  2. Look at the right group: 7, 9, 6, 8, 10. They are all bigger.
  3. Where does the change happen? The biggest number on the left is 4. The smallest on the right is 6. Where is 5? Mia does not have a 5 card, so the rule could be "less than 5" or "less than 6". Both fit!
  4. A good guess for Mia's rule: "numbers smaller than 5 go left, bigger ones go right".

If you ask Mia where she would put a 5, she can tell you which rule she meant. Sometimes more than one rule fits what you see. That is fine, as long as you say so.

Example 3. Find the mistake. Leo sorted toys by "Does it have wheels?" Here are his groups.

  • Yes group: car, bike, train, ball.
  • No group: kite, doll, drum.

Check each toy against the rule.

  1. Car: wheels. Yes. Correct.
  2. Bike: two wheels. Yes. Correct.
  3. Train: wheels. Yes. Correct.
  4. Ball: no wheels! A ball rolls, but it does not have wheels. It should be in the no group.
  5. Kite, doll, drum: no wheels. Correct.

Leo made one mistake: the ball. Maybe he was thinking "Does it roll?" instead of "Does it have wheels?" Those are two different rules.

Example 4. One set of things, two ways. Here are four fruits: a green apple, a red apple, a banana, a lime.

  • Sort by "Is it green?": yes group is green apple and lime; no group is red apple and banana.
  • Sort by "Is it an apple?": yes group is green apple and red apple; no group is banana and lime.

The same four fruits, two good rules, two different ways to split them.

Try it

  1. Sort these by "Does it live in water?": fish, cow, whale, cat, crab, dog.
  2. Sort these numbers by "Is it more than 5?": 2, 8, 5, 6, 1, 9.
  3. Ann sorted some clothes. Group 1: hat, scarf, mittens. Group 2: shorts, sun hat, sandals. What could her rule be?
  4. Is "Is it big?" a clear rule for sorting animals? Why or why not?
  5. Sort the letters of the word BANANA by "Is it the letter A?" How many are in each group?
  6. Dan sorted shapes by "Does it have 4 sides?" His yes group: square, rectangle, triangle. His no group: circle, oval. Is there a mistake?
  7. Think of a rule that puts a duck and a fish in the same group, and a cat in the other group.
Answers
  1. Lives in water (yes): fish, whale, crab. Does not (no): cow, cat, dog. (A crab can walk on the beach too, but most crabs live in water, so most people put it in yes. If you said "no" and gave a reason, that is fine.)
  2. More than 5 (yes): 8, 6, 9. Not more than 5 (no): 2, 5, 1. Look at 5: 5 is not more than 5, it is the same as 5. So 5 goes in the no group.
  3. A good rule: "clothes for cold weather" and "clothes for hot weather". Group 1 keeps you warm. Group 2 is for summer.
  4. Not really. Is a dog big? Next to a mouse, yes. Next to an elephant, no. People will not agree, so the groups will come out different. A clearer rule would be "Is it bigger than a cat?"
  5. B, A, N, A, N, A: the A's are the 2nd, 4th and 6th letters. Yes group: 3 letters (A, A, A). No group: 3 letters (B, N, N). Check: 3 and 3 make 6, and BANANA has 6 letters.
  6. Yes, there is a mistake. A triangle has 3 sides, not 4. It belongs in the no group.
  7. Many rules work. "Can it swim?" (duck yes, fish yes, cat not really). "Does it like water?" "Is it often in a pond?" Any rule where the duck and fish both say yes and the cat says no.

Common mistake

Changing the rule halfway. Sometimes people start sorting by colour, and then a big button comes along, and they put it with the big ones. Now the groups do not make sense: some are together because of colour and some because of size. Pick one rule, and use it for every single thing.

Letting looks trick you. A penguin looks like it should fly, because it is a bird with wings. A ball looks like it belongs with the wheels, because it rolls. Always go back and ask the rule's question, word for word.

Where it leads