Teddy bear line-up

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

Age

5 to 7

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Lachlan was playing with his bear counters.

He had four blue, then four red followed by four yellow and finally four green bears.

"What are you doing now?" Jenni asked.

"I want to arrange them so that no two bears of the same colour are next to each other", he said.

"That's easy!" cried Jenni.

"Ah, but you have to do it in the least number of moves possible", replied Lachlan.

What's the least number of moves you can take to rearrange the bears?

Printable NRICH Roadshow resource.

Student Solutions

We received lots and lots of very well-described solutions to this Teddy Bear problem - I'm sorry that we can't mention everyone. It was interesting to see what you called a 'move'. Jessica decided that a move was a 'swap':

Four steps:

Start: BBBB RRRR YYYY GGGG

Step 1, Switch the second blue and the second red:

BRBB RBRR YYYY GGGG

Step 2, Switch the fourth blue and the fourth yellow:

BRBY RBRR YYYB GGGG

The fourth red is not used because that would result in two reds being side by side. Switching fourth with fourth makes it easier to keep track of what has been done.

Step 3, Switch the fourth red and the fourth green:

BRBY RBRG YYYB GGGR

Step 4, Switch the second yellow and the second green:

BRBY RBRG YGYB GYGR

The bears are now fully mixed-up. This is not a unique solution.

Jessica says this is not a unique solution, meaning that it can be done in four moves, but in different ways. Will, Milly and Emily from Swiss Gardens Primary School agreed with Jessica and sent in this picture of the four moves they made, which are slightly different from Jessica's:

Gemma and Maggie, also from Swiss Gardens, looked at simpler cases with smaller numbers of bears. They said:

We did $2\times2$ which was 1 move also $3\times3$ which was 2 moves.

$4\times4$ was 4 moves.

It can be useful to do this to see if you can identify and explain the pattern. I wonder whether you could predict the least number of moves for five lots of five bears?

Jodie thought about a 'move' in a different way and here is what she did:

If you call the blue bears B, the red bears R, the yellow bears Y, and the green bears G, you get:

B,B,B,B,R,R,R,R,Y,Y,Y,Y,G,G,G,G

1. Move one of the green bears to separate the first two blue bears:

B,G,B,B,B,R,R,R,R,Y,Y,Y,Y,G,G,G

2. Move one of the yellow bears to separate the second two blue bears:

B,G,B,Y,B,B,R,R,R,R,Y,Y,Y,G,G,G

3. Move one of the red bears to separate the last two blue bears:

B,G,B,Y,B,R,B,R,R,R,Y,Y,Y,G,G,G

4. Move one of the green bears to separate the second two red bears:

B,G,B,Y,B,R,B,R,G,R,R,Y,Y,Y,G,G

5. Move one of the yellow bears to separate the final two red bears:

B,G,B,Y,B,R,B,R,G,R,Y,R,Y,Y,G,G

6. Move one of the green bears to separate the final two yellow bears:

B,G,B,Y,B,R,B,R,G,R,Y,R,Y,G,Y,G

Elijah also used this idea and, like Gemma and Maggie, he investigated simpler cases. He even tried five lots of five bears as well:

I found that the smallest number of moves to mix up the four lots of four colours was $6$.

Call the bears a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4, d1, d2, d3, d4 (the letters mean different colours).

Move a1 between b1 and b2

Move a2 between b2 and b3

Move a3 between b3 and b4

Now you have a4, b1, a1, b2, a2, b3, a3, b4

You can't do it in less than 3 moves because in 4 bears coloured 'b' there are 3 pairs that need to be split up.

Now do the same for the c's and d's.

That is $6$ moves.

You can do other moves and get a different mixture at the end.

I tried it for two lots of two colours and could do that in $1$ move.

Then I tried it for three lots of three colours. This was a bit harder because it doesn't split into two halves and you have to mix the colours up. I did it like this:

Start with a1, a2, a3, b1, b2, b3, c1, c2, c3

Move a1 between b1 and b2

Move c1 between b2 and b3

Move a2 between c2 and c3

Now you have a3, b1, a1, b2, c1, b3, c2, a2, c3

I noticed that the sequence $1, 3, 6 ... $ is the start of the triangle numbers.

So then I thought that for five lots of five colours the answer would be $10$. I think this is right - I couldn't do it in less than $10$ moves.

I think these are right for the smallest numbers of moves because you can imagine the bears as pairs of two colours (like the two lots of two colours problem) and see how many moves it must take to mix them up.

Fantastic reasoning, Elijah, thank you for your detailed solution. We can really understand how you were thinking.

We also received the following message from a mum which I couldn't resist including!

My daughter was trying this as we came back from Science Festival today when her little brother found the solution: he picked all the sets of buttons she was using (for counters) and threw them in the air ... as they fell they all scattered ..."Mum! It's just ONE move when HE does it!" Not sure this is what you intended, though...

Teachers' Resources

Teddy Bear Line-up

Lachlan was playing with his bear counters.

He had four blue, then four red followed by four yellow and finally four green bears.

"What are you doing now?" Jenni asked.

"I want to arrange them so that no two bears of the same colour are next to each other", he said.

"That's easy!" cried Jenni.

"Ah, but you have to do it in the least number of moves possible", replied Lachlan.

What's the least number of moves you can take to rearrange the bears?

Why do this problem?

This problem requires children to develop a logical way of moving the teddy bears and in doing this they may well use visualisation to plan ahead. Therefore, it offers an opportunity to talk about visualisation with young children.

Possible approach

Many opportunities arise in the classroom for you to invite children to explore this task, for example after a story or poem relating to bears or as an extension to another mathematical activity when children are using counters. The initial part of the activity, when the arrangement of bears is set up, involves counting, sorting and pattern, and so is valuable in itself.

It would be a good idea to begin by introducing the problem with just four bears, two of one colour and two of another colour. If you don't have any plastic bears, you could use any other suitable toys which are differently coloured. (Or counters as a last resort!) Line the bears up so that those of the same colour are next to each other and ask children to think on their own which bear they would move first, and then to talk to a partner. Share ideas, and encourage learners to give a reason for their suggestions. Some children will be picturing the new line-up of bears and this is a good opportunity to draw attention to how useful 'seeing something in your mind' can be for planning what to do in this problem. Of course there are different ways to arrive at a solution, so it would be useful to record them (the children may have good ideas about how to do this). Make sure the group can see that in this case, we can do it in just one move. You could then introduce a third colour bear so that you have nine bears altogether, before going on to the main challenge of sixteen bears.

Ask the children to work together so they have to engage in mathematical conversation as they explain their plans and negotiate which strategy requires the least number of moves. When you bring the whole group together, you can ask for suggestions as to how to solve the challenges, but you may also want to encourage children to notice the similarities between the strategies for two, three and four bears of each colour and so to try and make some general observations.

Key questions

Have you got a good idea about how to move the bears?

Which bear could you move next?

How do you know how many moves you have made?

How will you remember the moves you have made?

Possible extension

Some children will be able to extend the activity by using sets of five or more bears. To extend their thinking and use their prior experience, before carrying out the follow up activity children should be encouraged to predict how many moves it will take to rearrange sets of five or more bears and to explain what evidence they are basing their prediction on.

Possible support

It might help some children to label the bears with numbers or letters to help with recording. Alternatively, taking pictures of each move would be a nice way to display a strategy.

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