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Let's all go down to the favourite cafe that sells ice cream which you choose from different tubs.

Three different flavours of ice cream.

Suppose that there's Apricot, Banana and Citrus.

There is just one rule about what you can choose, and here it is:

YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN!

This means that if someone has chosen Banana and Citrus I cannot then go and choose all three but I could choose to have Apricot on its own.

So perhaps something like this happens:-

Sarah, thinking of apricot and citrus!

Sarah, the first child, chooses Apricot and Citrus.

Tim, the second child chooses Banana and Citrus [this obeys the rule because Sarah's choice was not Banana on its own nor was it Citrus on its own].

Raj, the third child, chooses Citrus.

Zoe, the fourth child, chooses Banana.

Matt, has to be the last child because he can only choose Apricot and then there are no other choices left.

In this example, with these children making these choices, only five children can have ice cream [using our rule].

But suppose more children wanted ice cream and so they got together to work out how this could be done.

They might come up with an idea like this:-

[I'm using the short-hand this time of A B C where A is Apricot, B is Banana and C is Citrus.]

1st. choice > A B C
2nd. choice > A B
3rd. choice > A C
4th. choice > B C
5th. choice > A
6th. choice > B
7th. choice > C

So seven children altogether. I think that the children can have different sized scoops so that even if they only have one flavour they have as much ice cream as someone choosing three flavours!

If these children weren't very good at working things out they could come up with the worst way; that would be like this:-

1st. choice > A
2nd. choice > B
3rd. choice > C

AND THAT'S ALL!

Well that's what it's like when there are three flavours. At the most, seven children can go in that order and get those choices of ice cream. At worst, only three children can go and get ice cream.

Mind you I think that there are other ways of getting seven.

Have a go and find all the different ways of having seven children getting ice cream. Remember that when someone goes up and makes their choice they have to obey that rule:

YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN!

So seven is the most, three is the least for three flavours.

What about four flavours now that Damson is available?

What about five ...?

Are you able to go on and on?

Do you know how many different ways of getting the most children to have ice cream for four flavours? For five flavours?

So you can work all these different things out.

Here's the next bit of the challenge - can you find out how many there are for seven flavours, without working out the answers to six flavours first?

And FINALLY ...

I wonder what would happen if ...?

Please send in any results that you get along the way. I have to go - my mouth is watering for some ice cream!