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Being organised

Counting and combinations · level 1, lesson 6 of 6

A question to start

Kofi is making little flags with two stripes, one on top and one below. He has three colours: orange, blue and green. Each flag uses two different colours. He writes down the flags as he thinks of them:

blue over green, orange over blue, green over orange, blue over orange, orange over green, green over blue, blue over green.

Did he find them all? Did he write any twice? It is very hard to tell from a messy list like that! (He did find them all, but he wrote "blue over green" twice, so his list has 7 flags when there are only 6.) In this lesson you will learn how to make a list so tidy that you can see at a glance that nothing is missing and nothing is twice.

The idea

Messy lists and tidy lists

All through this level you have been counting things: outfits, towers, line-ups, handshakes and paths. Every time, the answer came from a tidy list: a list made in a planned order. Now let's look at the plan itself, because it works for almost any counting question.

A messy list writes things down in the order you happen to think of them. It is fine for finding a few, but you can never be sure you have them all, and it is easy to write the same one twice.

A tidy list (grown-ups say an organised list or a systematic list) follows a rule, like this:

  1. Choose an order for your colours (or flavours, or friends). For example: orange, then blue, then green.
  2. Fix the first thing. Write down everything that starts with orange, then everything that starts with blue, then everything that starts with green.
  3. Inside each group, use the same order again for the next thing.

Kofi's flags, done tidily

Use the order orange, blue, green. Fix the top stripe first.

  • Orange on top: blue below, green below.
  • Blue on top: orange below, green below.
  • Green on top: orange below, blue below.
orange on topblue on topgreen on top

There are 3 groups of 2, which makes 6 flags.

How a tidy list proves it has everything

The tidy list does more than find the answer. It shows why the answer is right.

  • Nothing is missing: every flag has some colour on top, so it belongs to one of the three groups. Inside a group, the bottom stripe can be any colour except the top one, and we wrote them all in order.
  • Nothing is twice: flags in different groups have different top stripes. Flags in the same group have different bottom stripes.
  • The groups look alike: every group has the same number of flags (2). When the groups all look the same, it is a good sign that the list is right. If one group had 3 and another had 2, you would know to look again.

Now go back to Kofi's messy list. Sort it into the tidy groups, and you will find "blue over green" sitting there twice. The tidy list catches the mistake.

Towers of three blocks

Here is a bigger one. You build a tower of 3 blocks, one on top of another, and each block can be orange or blue (you can use a colour more than once). How many different towers?

Fix the bottom block first.

  • Orange at the bottom. Now fix the middle block. Orange in the middle: the top can be orange or blue. Blue in the middle: the top can be orange or blue. That is 4 towers.
  • Blue at the bottom. The same thinking gives 4 towers.
orange at the bottomblue at the bottom

4 and 4 make 8 towers. Look at the picture: the right-hand group is the left-hand group with just the bottom block changed to blue. That is why the two groups are the same size.

Counting numbers in order too

Tidy lists also work for numbers. Ella and Sam share 5 sweets. Each gets some, or even none. How many ways can they share?

Make a tidy list by how many Ella gets, going up from 0:

Ella 0 1 2 3 4 5
Sam 5 4 3 2 1 0

6 ways. As Ella's number goes up by 1, Sam's goes down by 1, and you stop when Ella has them all. Going up in order (0, 1, 2, ...) is the number version of "orange, then blue, then green".

Ways to check your count

Even with a tidy list, a good mathematician checks. Here are three checks you now know:

  1. Do the groups look alike? In the flags, every group had 2. In the towers, both groups had 4.
  2. Count another way. Handshakes: the tidy list gave 6, and halving 12 also gave 6.
  3. Try a smaller one first. If you are not sure about towers of 3 blocks, count towers of 2 blocks (4) and of 1 block (2). Then 3 blocks should be double 4, which is 8. ✓

Worked examples

1. Lunch box. You choose one sandwich (cheese or egg) and one fruit (apple, banana or pear). Make a tidy list.

  • Cheese: with apple, with banana, with pear.
  • Egg: with apple, with banana, with pear.

6 lunches. Two groups of 3, just like Teddy's outfits in lesson 1.

2. Find the missing one. Rosa wrote the towers of 2 blocks using orange and blue (bottom first): orange-orange, blue-orange, blue-blue. One is missing. Which one?

Make a tidy list: orange at the bottom: orange-orange, orange-blue. Blue at the bottom: blue-orange, blue-blue. Rosa is missing orange-blue (orange at the bottom, blue on top).

3. Two-letter words. Using the letters A, B and C, how many two-letter "words" can you make if the letters must be different? (They don't have to be real words.)

Fix the first letter, in alphabet order.

  • A first: AB, AC.
  • B first: BA, BC.
  • C first: CA, CB.

6 words. Using the alphabet as the order is a very common way to keep a list tidy. Dictionaries do it!

4. Sharing 4 marbles among 3 children? This is harder, but the tidy list still works. Fix how many the first child gets, from 4 down to 0, then share the rest between the other two in order:

  • First child 4: the others share 0 in 1 way (0 and 0).
  • First child 3: the others share 1 in 2 ways (1 and 0, 0 and 1).
  • First child 2: the others share 2 in 3 ways.
  • First child 1: the others share 3 in 4 ways.
  • First child 0: the others share 4 in 5 ways.

1 and 2 and 3 and 4 and 5 make 15 ways. Here the groups are not all the same size, and that is fine: each group is a smaller sharing problem, and we know how many ways two people can share (one more than the number of things).

Try it

  1. Make a tidy list of the two-stripe flags using only orange and blue, with different colours. How many?
  2. Using the letters A and B, make a tidy list of all two-letter "words". Letters may repeat (AA is allowed). How many?
  3. Kofi gets a fourth colour, yellow. How many two-stripe flags with different colours now? (Hint: how many groups, and how many in each?)
  4. Ella and Sam share 3 sweets. Make a tidy table. How many ways?
  5. Here is a messy list of towers of 3 blocks (bottom first, O for orange, B for blue): OBO, BBB, OOB, BOB, OBB, BBO, OOO, BOB. Which one is written twice? Which one is missing?
  6. A pizza can have mushrooms or not, and olives or not. Make a tidy list of all the pizzas. How many?
  7. How many towers of 4 blocks can you make with orange and blue? (Hint: try a smaller one first.)
Answers
  1. Orange on top: blue below. Blue on top: orange below. 2 flags.
  2. A first: AA, AB. B first: BA, BB. 4 words.
  3. 12. There are 4 groups (one for each top colour), and in each group the bottom stripe can be any of the other 3 colours. 3 and 3 and 3 and 3 make 12.
  4. Ella 0, 1, 2, 3 and Sam 3, 2, 1, 0: 4 ways.
  5. BOB is written twice. The tidy list is OOO, OOB, OBO, OBB, BOO, BOB, BBO, BBB, so BOO is missing.
  6. No mushrooms, no olives; no mushrooms, olives; mushrooms, no olives; mushrooms and olives. 4 pizzas.
  7. 16. Towers of 3 blocks: 8. A tower of 4 blocks is a bottom block (orange or blue) with a tower of 3 on top. So there are 8 with orange at the bottom and 8 with blue at the bottom: 8 and 8 make 16.

Common mistake

Changing the order halfway. If you use orange, blue, green for the first group and then green, orange, blue for the second, it is easy to skip one. Choose your order once and stick to it all the way through.

Stopping too soon. In a tidy list you know when you are done: you have gone through every first choice. If you stop when "it feels like enough", you might miss a whole group.

Where it leads