Butterfly Cards
1 out of 3
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!
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:
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: $5, 11, 17, 23, 6, 12, 18, 24.$
Kim wants just the cards $16, 2, 4$.
Jim wants just the cards $5, 17$.
Wim wants just the cards $7, 9, 21$.
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.
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.
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.
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) ($19, 20, 21, 22, 23, 24$)
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
Possible approach
Key questions
Possible extension
Learners could sort the cards using Venn diagrams as described on this sheet.
Possible support
Suggest sorting copies of the cards. These cards are quite small, while these are much larger and include the numbered cards.