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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:


butterflies numbered

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.
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.