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Ice-cream towers

Counting and combinations · level 1, lesson 2 of 6

A question to start

Mia gets an ice-cream cone with two scoops: one chocolate scoop and one vanilla scoop. Her brother Leo gets one too, with the same two flavours. Mia looks at Leo's cone and says: "That's not the same as mine!"

CV VC MiaLeo

Is Mia right? They have the same two flavours. But Mia's chocolate is on top, and Leo's chocolate is on the bottom. If you eat a cone from the top, Mia tastes chocolate first and Leo tastes vanilla first.

So are these the same, or different? The surprising answer is: it depends on the question you are asking. This lesson is about one of the biggest ideas in counting: sometimes the order matters, and sometimes it does not.

The idea

A tower has a top and a bottom

Think of a two-scoop cone as a little tower. A tower has a top place and a bottom place. When we say "a tower", we care about which flavour is in which place.

  • Chocolate on top, vanilla on the bottom: that is one tower.
  • Vanilla on top, chocolate on the bottom: that is a different tower.

So with one chocolate scoop and one vanilla scoop, there are 2 towers. When order matters, swapping the two things gives something new.

A cup has no top and bottom

Now put the same two scoops in a cup and stir them a little. It does not matter which went in first: the cup just has "chocolate and vanilla" in it. Swapping does not change anything.

So with chocolate and vanilla, there is only 1 cup. When order does not matter, swapping the two things gives the same thing again.

Same scoops, different question, different answer: 2 towers, but 1 cup.

Three flavours

Now the shop has three flavours: chocolate (C), vanilla (V) and mint (M). You want two different flavours.

Towers first. Be tidy: choose the top scoop first and keep it still, then try every bottom scoop that is different.

  • Chocolate on top: vanilla below, or mint below. 2 towers.
  • Vanilla on top: chocolate below, or mint below. 2 towers.
  • Mint on top: chocolate below, or vanilla below. 2 towers.
CVCM VCVM MCMV C on topV on topM on top

2 and 2 and 2 make 6 towers.

Now cups. In a cup, "chocolate and vanilla" is one cup, whichever is on top. Look at the 6 towers and find the partners: towers with the same two flavours, just swapped.

  • C on top of V, and V on top of C: both are the cup chocolate and vanilla.
  • C on top of M, and M on top of C: both are the cup chocolate and mint.
  • V on top of M, and M on top of V: both are the cup vanilla and mint.

The 6 towers pair up into 3 cups.

Why towers are always double the cups

Here is the reason, and it works for any flavours. Take any cup with two different flavours. Tip it out and build a tower. You can put either flavour on top, so every cup makes exactly 2 towers. And every tower comes from exactly one cup (just squash it into a cup). So there are always twice as many towers as cups.

3 cups make 3 and 3 towers, which is 6. ✓

What about two scoops of the same flavour?

Some shops let you have double chocolate: two chocolate scoops. If you swap the two scoops of a double chocolate tower, nothing changes. Both are chocolate! So a double tower has no partner, and it is a cup and a tower at the same time.

With chocolate and vanilla, and doubles allowed, the towers are:

  • chocolate on top: chocolate below (double), or vanilla below;
  • vanilla on top: chocolate below, or vanilla below (double).

That is 4 towers. Notice it is a box, just like Teddy's outfits in the last lesson: 2 rows (top flavour) and 2 columns (bottom flavour), and 2 and 2 make 4.

Worked examples

1. Three flavours, doubles allowed, towers. Chocolate, vanilla and mint, and you may pick the same flavour twice. How many towers?

Make a box. Each row is a top flavour, each column is a bottom flavour.

top \ bottom C V M
C C on C C on V C on M
V V on C V on V V on M
M M on C M on V M on M

3 rows of 3 make 9 towers. The 3 doubles sit on the slanting line from the top-left corner to the bottom-right corner. The other 6 are the towers with two different flavours, which we found before. 9 is 6 and 3. ✓

2. Three flavours, doubles allowed, cups. Now the same, but in cups. The 6 towers with different flavours make 3 cups (they pair up). The 3 doubles make 3 cups (they have no partner). So there are 3 and 3, which makes 6 cups:

chocolate and vanilla, chocolate and mint, vanilla and mint, double chocolate, double vanilla, double mint.

3. Is it a tower or a cup? For each one, does the order matter?

  • Two children sit on a seesaw, one on each end. Order matters if you care who is on the left: Ana on the left and Ben on the right is a different picture from Ben on the left and Ana on the right.
  • Two children are chosen to carry the class ball. Order does not matter: "Ana and Ben" is the same pair as "Ben and Ana".
  • Two digits make a number, like 12 or 21. Order matters: twelve and twenty-one are different numbers.
  • Two coins go in a piggy bank. Order does not matter: the bank has the same money either way.

4. Four flavours, towers with different scoops. Add blueberry. Keep the top scoop still and try each different bottom: each top flavour has 3 partners below. 4 top flavours, 3 each: 3 and 3 and 3 and 3 make 12 towers. And the cups are half of that: 6 cups.

Try it

  1. You have a strawberry scoop and a lemon scoop. How many different two-scoop towers can you make? How many cups?
  2. Write all the towers you can make with two different scoops from banana, cherry and grape.
  3. How many cups (order does not matter) can you make from the flavours in question 2?
  4. Peanut butter and jam: does it matter which is on top in a sandwich? (There is no wrong answer if you explain it, but say what you are counting.)
  5. Is the order important? (a) the two digits on a two-digit lock; (b) two friends you invite to play; (c) the first and second place in a race.
  6. With chocolate and vanilla, doubles allowed, how many cups are there?
  7. A shop has 4 flavours. You want a two-scoop tower, doubles allowed. How many towers? (Hint: draw a box with 4 rows and 4 columns.)
Answers
  1. 2 towers (strawberry on top, or lemon on top) and 1 cup (strawberry and lemon).
  2. Banana on top: cherry or grape below. Cherry on top: banana or grape below. Grape on top: banana or cherry below. That is 2 and 2 and 2, 6 towers.
  3. 3 cups: banana and cherry, banana and grape, cherry and grape. Half of the 6 towers, because each cup makes 2 towers.
  4. If you only care what is in the sandwich, order does not matter: peanut butter and jam is one sandwich. If you care which is on top (some people say the jam must touch the top bread!), then it matters and there are 2 ways.
  5. (a) Yes: 1 then 2 is a different code from 2 then 1. (b) No: inviting Ana and Ben is the same as inviting Ben and Ana. (c) Yes: winning and coming second are very different.
  6. 3 cups: chocolate and vanilla, double chocolate, double vanilla.
  7. 16 towers. 4 rows of 4: 4 and 4 and 4 and 4 make 16. (Of these, 4 are doubles and 12 have two different flavours.)

Common mistake

Forgetting to ask about order. Before you count, ask: if I swap these two, do I get something new? For towers, codes and races the answer is yes. For cups, pairs of friends and handshakes the answer is no. The same scoops can give 6 or 3 depending on this one question.

Pairing up the doubles. A double chocolate tower has no partner, because swapping chocolate with chocolate changes nothing. Do not halve it.

Where it leads