# Butterfly Cards

Four children were sharing a set of twenty-four butterfly cards.
Are there any cards they all want? Are there any that none of them
want?

## Problem

Four children were sharing a set of twenty-four butterfly cards. It looked easy! They could have six each.

But it was far from easy!

Image

First Kim said, "I like the ones with curly antennae. I want those!"

"I don't like the ones with oval heads," complained Wim, "But want all the others!"

Jim only wanted the ones with yellow spots.

Tim wanted butterflies which only had dark wings and blue spots.

"You can't have two things," complained Jim, "That's greedy! You'll get more cards!"

Do you think he was right?

Are there any cards that nobody wants?

Are there any cards that all the children want?

Are there any cards that just one child wants?

Can all the children have the cards they want?

Can you think of a fair way for them to share out the cards?

You might like to print off an A4 copy of the cards.

## Getting Started

## Student Solutions

Three groups of children put a lot of thought into this problem. They were: Ella, Luke, Richard and Sebastian from St Nicolas CE Junior School; Jessica and Emily from Aldermaston CE Primary and Burcu from FMV Ozel Erenkoy Isik Ilkogretim Okulu in Turkey.

The pupils from St Nicolas and Burcu started in the same way by numbering the cards as the hint suggests:

Image

Burcu said:

According to the wishes of Kim, Wim, Jim and Tim they can have the following cards:Kim: $2, 8, 14, 20, 4, 10, 16, 22, 6, 12, 18, 24.$

Wim: $7, 8, 9, 10, 11, 12, 19, 20, 21,
22, 23, 24.$

Jim: $5, 11, 17, 23, 6, 12, 18,
24.$

Tim: $13, 14, 19, 20.$

Jim's complaint is not right, because Jim
wants to have two properties, it does not mean that, he'll get more
cards.

Nobody wants the cards numbered as $1,
3$, and $15$.

There is no card that all the children
want.

Kim wants just the cards $16, 2,
4$.

Jim wants just the cards $5, 17$.

Wim wants just the cards $7, 9,
21$.

Tim wants just the card $13$.

All children can not have the cards they
want.

Pupils from St Nicolas School wrote:

We found it easier to cut out the cards to find out how to share them out fairly. We had two ways which we thought were fair.1) Everyone gets four cards they want and two cards they don't
want e.g.

Kim - $2, 6, 8, 10$ she wants and $9$ and
$15$ she doesn't want.

Wim - $7, 12, 21, 22$ she wants and $5$
and $16$ she doesn't want.

Tim - $13, 14, 19, 20$ he wants and $4$ and $23$ he doesn't want.

Tim - $13, 14, 19, 20$ he wants and $4$ and $23$ he doesn't want.

Jim - $11, 17, 18, 24$ he wants and $1$
and $3$ he doesn't want.

2) Kim, Wim and Jim get five cards they
like and Tim has four cards he likes and two cards he half likes
which would make one he likes and one he doesn't. e.g.

Kim - $2, 4, 6, 16, 18$ and $3$ she
doesn't like.

Wim - $7, 9, 10, 21, 22$ and $5$ she
doesn't like.

Tim - $13, 14, 19, 20$ and $1$ and $15$
he half likes.

Jim - $11, 12, 17, 23, 24$ and $2$ he
doesn't like.

Jessica and Emily used a similar method to try to share out the butterflies. They said:

Our final choice was to give them the
closest to what they wanted.

We tried to be systematic and give the
people at least four of what they wanted first, then five then
six.

We did Tim first because there are only
four of his choice anyway.

Next we did Jim, because there are only
eight of his choice and we tried not to choose curly antennae
because Kim wanted those.

Next we did Wim because he didn't want
the ones with oval heads.

Finally, we did Kim and we gave her any
curly antennae we could find.

Another possibility would be to change the features that the children want. Burcu suggested that the butterflies could be shared in the following way:

If someone wants the butterflies which have oval heads and
light blue wings (six butterflies) ($1, 2, 3, 4, 5, 6$)

If someone wants the butterflies which
have not oval heads and light blue wings (six butterflies) ($7, 8,
9, 10, 11, 12$)

If someone wants the butterflies which
have oval heads and dark blue wings (six butterflies) ($13, 14, 15,
16, 17, 18$)

If someone wants the butterflies which
have not oval heads and dark blue wings (six butterflies) ($19, 20,
21, 22, 23, 24$)

Such a sharing can be a fair way, and
everyone can take the butterflies she/he wants.

## Teachers' Resources

### Why do this problem?

This problem gives learners some simple data to organise. As in many cases, there is no obvious solution. It can be done practically with the interactivity or the cards, but it can also be opened up as it can be seen from the Possible extension. It could be a good introduction into the idea of sets and Venn
diagrams.

### Possible approach

This sheet gives small copies of the cards (not numbered) for individual use and these sheets have two sets of much larger cards, one set
plain and the other numbered.

You might find this interactivity useful for illustrating different solutions at the end of the lesson.

### Key questions

What can you tell me about this butterfly?

Which attribute are you collecting there?

How many can you find that have Curly Antenna [Round Heads/Yellow Spots/Dark Wings/Blue Spots]?

Have you found any cards that nobody wants?

Have you found any cards that all the children want?

Have you found any cards that just one child wants?

Can all the children have the cards they want?

Can you think of a fair way for them to share out the cards?

How are you going to record what you have done?