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Sorting circles

Logic and puzzles · level 2, lesson 2 of 6

A question to start

Ten children in a class have a pet. Every one of them has a dog, or a cat, or both.

The teacher asks: "Who has a dog?" 8 hands go up. "Who has a cat?" 5 hands go up.

But 8 and 5 make 13. There are only 10 children! Did somebody put their hand up twice? Is the teacher's counting wrong? Or is something else going on? How many children have both a dog and a cat?

The idea

Two hoops on the floor

Put two hoops on the floor so that they overlap, like this. One hoop is for things that are even. The other is for things that are more than 10. Now sort the numbers 1 to 20.

even more than 10 2 4 6 8 10 12 14 16 18 20 11 13 15 17 19 1 3 5 7 9

Where does each number go?

  • 12 is even, and it is more than 10. It belongs in both hoops. So it goes in the part where the hoops overlap: the middle.
  • 4 is even, but it is not more than 10. It goes in the "even" hoop only, in the part that is not shared.
  • 15 is more than 10, but it is not even. It goes in the "more than 10" hoop only.
  • 7 is not even and not more than 10. It goes outside both hoops.

A picture like this is called a Venn diagram, after John Venn, an English mathematician who wrote about these pictures in 1880. Many people just call them sorting circles.

Four places to go

Two overlapping hoops make four places on the floor:

  1. inside the left hoop only;
  2. inside both hoops (the overlap in the middle);
  3. inside the right hoop only;
  4. outside both hoops.

Every number goes in exactly one of these four places. In the picture, each place got 5 numbers:

  • even, not more than 10: 2, 4, 6, 8, 10;
  • both: 12, 14, 16, 18, 20;
  • more than 10, not even: 11, 13, 15, 17, 19;
  • neither: 1, 3, 5, 7, 9.

5 and 5 and 5 and 5 make 20. Every number from 1 to 20 is there, once.

It is the sorting mat again

Does "four places" sound familiar? In level 1 you used a sorting mat with two questions and four boxes. Sorting circles are the same idea, drawn a different way.

yes no yes no both left only right only neither left? right? left right left only both right only neither

The overlap is the "yes, yes" box. "Left only" is "yes, no". "Right only" is "no, yes". Outside is "no, no". Same four boxes, same answers. The hoops just make it easier to see which things share both properties: they sit together in the middle.

The whole hoop, or just one part?

This is the most important thing to be careful about. "How many numbers are in the even hoop?" means the whole hoop, including the overlap.

  • In the even hoop: 2, 4, 6, 8, 10 (left part) and 12, 14, 16, 18, 20 (overlap). That is 10 numbers.
  • In the even hoop only: just the left part, 5 numbers.

The overlap belongs to both hoops at once. It is counted when you count the left hoop, and counted again when you count the right hoop.

Solving the pet puzzle

Now we can understand the question at the start. Draw two hoops: "has a dog" and "has a cat". Every child has at least one pet, so nobody is outside.

The 8 children with a dog fill the whole dog hoop. The 5 children with a cat fill the whole cat hoop. If nobody had both, that would be 8 and 5, which is 13 children. But there are only 10. So 3 children got counted twice: once in the dog hoop and once in the cat hoop. Those 3 are the ones in the overlap. They have both pets.

has a dog has a cat 5 3 2

Check: dog only 5, both 3, cat only 2.

  • Dog hoop: 5 and 3 make 8. ✓
  • Cat hoop: 3 and 2 make 5. ✓
  • Everyone: 5 and 3 and 2 make 10. ✓

So nobody was cheating. The 3 children who have both pets put their hand up both times, and that was correct!

Hoops that do not overlap

Sometimes nothing can be in both hoops. "Odd" and "even": no number is both. Then you can draw the hoops apart, with no overlap.

odd even 1 3 5 7 2 4 6 8

(If you still draw them overlapping, that is fine too. The middle will just be empty, like an empty box on the sorting mat.)

A hoop inside a hoop

Sometimes everything in one hoop is also in the other. Every square has 4 sides. So the "squares" hoop sits completely inside the "has 4 sides" hoop.

has 4 sides squares rectangle

A rectangle that is not a square goes in the big hoop, but not in the small one. A triangle and a circle go outside both. And nothing can be in the "squares" hoop but outside the "4 sides" hoop: there is no square without 4 sides.

Worked examples

Example 1. Sort 1 to 12. Hoops: "even" and "in the count by 3s" (3, 6, 9, 12, ...).

  1. Even numbers: 2, 4, 6, 8, 10, 12.
  2. Count by 3s: 3, 6, 9, 12.
  3. In both lists: 6 and 12. They go in the overlap.
  4. Even only: 2, 4, 8, 10. Threes only: 3, 9.
  5. Neither: 1, 5, 7, 11.
even count by 3s 2 4 8 10 6 12 3 9 1 5 7 11

Check: 4 and 2 and 2 and 4 make 12. ✓

Example 2. Find the labels. Someone sorted some numbers, but the labels fell off.

? ? 1 3 7 9 5 15 10 20 2 4 6
  1. Look at the left hoop (both parts): 1, 3, 7, 9, 5, 15. All odd.
  2. Look at the right hoop (both parts): 5, 15, 10, 20. All in the count by 5s.
  3. Check the overlap: 5 and 15 are odd and in the count by 5s. ✓
  4. Check the outside: 2, 4, 6 are not odd and not in the count by 5s. ✓

Labels: left "odd", right "in the count by 5s".

Example 3. Counting with an overlap. In a group of 12 friends, 7 can swim and 6 can ride a bike. 2 friends can do neither. How many can do both?

  1. 12 friends, 2 outside both hoops. So 10 friends are inside at least one hoop.
  2. Count the two hoops: 7 and 6 make 13.
  3. But only 10 friends are inside. So 13 take away 10 is 3 friends who were counted twice. They can do both.
  4. Fill in: swim only 7 take away 3 is 4. Bike only 6 take away 3 is 3. Both 3. Neither 2.

Check: 4 and 3 and 3 and 2 make 12. ✓ Swim hoop: 4 and 3 make 7. ✓ Bike hoop: 3 and 3 make 6. ✓

Try it

  1. Hoops: "has wheels" and "has an engine". Where do these go: car, bike, boat, skateboard, plane, horse?
  2. Hoops: "even" and "less than 6". Sort the numbers 1 to 10.
  3. In question 2, how many numbers are in the "even" hoop (the whole hoop)? How many are in the "even" hoop only?
  4. Hoops: "has 3 sides" and "has 4 sides". Can anything go in the middle? How would you draw the hoops?
  5. In a class of 20, 12 like apples and 10 like bananas. Everybody likes at least one. How many like both?
  6. Draw hoops for "animals" and "dogs". Which hoop goes inside which?
  7. Hoops: "in the count by 2s" and "in the count by 5s". Sort 1 to 20. What do the numbers in the middle have in common?
Answers
  1. Both (wheels and engine): car, plane. Wheels only: bike, skateboard. Engine only: boat (most boats with an engine have no wheels). Neither: horse.
  2. Both (even and less than 6): 2, 4. Even only: 6, 8, 10. Less than 6 only: 1, 3, 5. Neither: 7, 9. Check: 2 and 3 and 3 and 2 make 10. ✓
  3. The whole even hoop: 2, 4, 6, 8, 10, so 5. Even only: 6, 8, 10, so 3.
  4. Nothing can have 3 sides and 4 sides. The middle is empty, so you can draw the two hoops apart.
  5. 12 and 10 make 22, but there are only 20 children. So 22 take away 20 is 2 children counted twice. 2 like both. (Apples only 10, bananas only 8, both 2: 10 and 8 and 2 make 20. ✓)
  6. The "dogs" hoop goes inside the "animals" hoop, because every dog is an animal.
  7. Count by 2s: 2, 4, 6, ..., 20. Count by 5s: 5, 10, 15, 20. Middle: 10 and 20. 2s only: 2, 4, 6, 8, 12, 14, 16, 18. 5s only: 5, 15. Neither: 1, 3, 7, 9, 11, 13, 17, 19. The middle numbers are in the count by 10s.

Common mistake

Putting a thing in both hoops, but not in the middle. If 12 is even and more than 10, some people write 12 in the left hoop and write it again in the right hoop. Write it once, in the overlap. The overlap is inside both hoops at the same time.

Forgetting the overlap when you count. "How many are in the even hoop?" includes the middle. "How many are even only?" does not. Read the question carefully.

Where it leads