Shawn from Lanesborough School sent in this solution:

 

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He added: 

Step 1/ Put all the triangles in pairs to ten

Step 2/ Put all the red numbers together to make 10, do the same with blue and yellow

Step 3/ Make a hexagon, then the last three put them on with the coloured pairs.

Step 4/ Celebrate and write your maths down.

Ryan from Hayton Primary, Cumbria sent the same solution and wrote:

I started working from the top to the bottom with a small triangle.

I worked downwards to test out which was the right triangle to go at the top.

It took me about five tries to get the right triangle at the top.

When I got the right triangle at the top, all the other triangles were easy to fit in, working top to bottom.

Rowling Class from Manor School sent in the same picture and wrote:

Some of our class found it easy to start with but harder as you had to match two sides of the triangles at once. One group got stuck and had to start again with a different triangle.

It was really important to know your number bonds to 10. We used 10 - 7 to find out that we needed a triangle with a 3 to match with 7.

We were not surprised by the fact we found lots of different solutions.

Sophie managed to match one with colours as well as number bonds!

We also received our first solution from China in Chinese! Ma Xueying sent in this image which is the same solution as Shawn, Ryan and Rowling Class above:

 

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The translation of the Chinese shows a very interesting approach and can be found here.docx or here.pdf.

Cohen, Nate, Isla and Alex from Burke Ward Public School in Australia sent the following different solution:

 

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We worked together and figured out how to make the numbers equal ten and make them into a triangle.

- we remembered our friends of 10

- we put the numbers together

- when it didn't make a triangle we pulled the triangles apart and started again.

When we finished we checked we had made a triangle and that the numbers equalled ten.

 

Cruz, Maddix, Lillah and Zak from Burke Ward Public School in Australia found the same solution and wrote:

 

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We saw that they all make ten and it looks like a triangle. We didn't worry about the numbers, we focused on the triangle shape first. Then we sorted out the numbers by ten and making a triangle. We turned the little triangles around to make it. Sometimes we had to try again.

We knew when we were done because then we checked that we had a big triangle and the numbers touching were making their friends of ten.

Aidan, Alicia and Jimmie from Millfield Primary School found a third solution and said:

 

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Aidan started at the top. I worked my way down until it made a big triangle. When you're making the triangle, if one piece has 8, you need to find another piece with a 2. Sometimes there are multiple pieces with 2 on them so I looked carefully at the other numbers on that piece to see which would fit best. I kept going until there were no pieces left. Jimmie helped to work out the number bonds using his fingers to make sure the number bonds were correct.

Alicia started at the bottom. I worked my way up and I found the right numbers for the number bonds. Sometimes I didn't have a piece that would fit, so I had to swap it with a piece already on the triangle.

 

Victoria, a teacher from Broadfield Academy in the United Kingdom, described her approach:

- First, I lined all the triangles in a long line playing with different combinations. I cut the line at various points to attempt to stack the triangle from bottom to top. Unsuccessful.

- Then, I looked for tiles with duplicate numbers and used those as a starting point in the middle of the triangle, then on the edge to exclude them from use. Unsuccessful.

- I noted which numbers appeared least frequently (e.g. 5). I used this starting point and moved tiles around the pair of fives. As it began to take shape the colour pattern confirmed I was on the right lines.

I found the solution after an hour of trial and error. 

She felt it was great for revising number bonds to 10 and developing problem-solving and reasoning skills, and resilience!

Thank you all for sharing these different solutions. I wonder if there are more different solutions?

The year 2 children in Laurel Class at Colman Infant School Norfolk worked on this problem and the teacher sent in the following:-

We decided to personalise the problem to fit in with our recent trip to Hickling Broad nature reserve In Norfolk. The children were amused by the idea that Noah might have visited Hickling and were all eager to getstarted.

As a class we looked at how there could be 2 creatures with 12 legs. We quickly established that there could either be 2 ladybirds or a spider and a mouse. We decided that this meant there would probably be more than one way of having three, four or five creatures etc. From then on the children worked with a partner. Some children worked on 12 legs and others on a more ambitious 18. They began by drawing pictures of the animals and making a zig-zag book. This was a way into the problem that made sense to everyone. All the children worked enthusiastically. Having pictures also meant that the children didn't use the same combination of animals twice as is often the case when ust using numbers. (E.g. 6+4+2=12 and 2+6+4=12)

The idea was to work systematically to find as many different ideas as possible. Some children began to realise that they could use one answer to inform another. For example, if you had 1 Iadybird, 1 mouse and 1 duck you could work out 4 creatures by exchanging a mouse for 2 ducks or a duck for 2 snails.

Later in the week the children looked at their booklets and realised that there were still a lot more answers to find. They were encouraged to work in a more abstract way but could still refer to their numbers as spiders, ladybirds, mice, ducks or snails. This is where they really began to work systematically. They were keen to find as many answers as possible.

Thank you all at Laurel, sounds good

Rhea from Bablake Junior School sent in the following 

There are lots of different answers for this. NRICH have put up a very open ended question!

Firstly, lets pretend that all the animals that Noah saw were all 4 legged animals. This would mean that Noah saw 3 animals. You do this by dividing 12 by 4, which is 3.

Secondly, lets pretend that all the animals saw were all 2 legged animals. This would mean that Noah saw 6 animals. You find this out by dividing 12 by 2, which is 6.

Thirdly, lets pretend that all the animals were 3 legged animals. You do this by dividing 12 by 3, which is 4.

Fourthly, lets pretend that that half of the animals Noah saw were 2 legged animals, and the other half were 3 legged animals. To find out the half that were 2 legged, you need to find out what half of 12 is, which is 6. You then have to divide 6 by 2, which is 3! 3 of the animals Noah saw where 2 legged animals. Then, you need to find out how many animals were 3

legged. You do this by dividing 6 (half of twelve: 12 divided by 2) by 3, which is 2. 2 of the animals Noah saw were 3 legged animals.

This is all I have got time for, so make sure you come up with some more solutions for NRICH! Hope you understood 4 of my solutions, thank you for reading this! Bye!

She entered a second submission saying:-

Well, I just entered a solution but I forgot to give examples of some animals Noah could have seen.

For a two legged, maybe birds, like: robins, pigeons, blackbirds, finches, blue tits, doves or swans.

For 4 legged: lion, dogs, cats, tiger, hedgehog, fox, wolf and more. These are examples, but there could be a lot more! Read my other solution on how to work on the 4 legged and two legged animals. It will be very helpful, hopefully you will read it and even make a solution your self! Good luck, hope you have fun on NRICH!

Olivia also from  Bablake Junior sent in this solution:-

There are definitely lots of different ways to solve this! Lets start and pretend that all the animals Noah saw were 1 legged animals. In this very easy case, all you need to do is divide 12 by 1, which is of course 12. This means that if all the animals Noah saw were 1 legged animals, there would be 12 of them. I'm sorry about this, but maybe you know some 1 legged animals? Now lets pretend that all the animals Noah saw were 2 legged animals. In

this case, all you need to do is divide 12 by 2, which is 6. If all the animals Noah saw were 2 legged animals, there would be 6 of them. Now lets think what types of animals they might be. Probably birds- robins, blackbirds, pigeons- maybe even humans- there are lots more!

Now lets pretend that all the animals Noah saw were 3 legged animals (I know there might not be 3 legged animals, but lets pretend there is!) Now all you have to do is divide 12 by 3 which is 4. If all the animals Noah saw were 3 legged animals, there would be 4 of them. Now lets think what type of animals they might be. I'm sorry about this, but I don't know any 3 legged animals, maybe you do? Now lets pretend that all the animals Noah saw were 4 legged animals. In this case, all you need to do is divide 12 by 4, which is definitely 3. So,

if all the animals Noah saw were 4 legged animals, there would be 3 of those animals. Different types of 4 legged animals are dogs, cats, hedgehogs, horses, pigs, cows and lots, lots more! Now lets pretend that all the animals Noah saw were 6 legged animals. (I

know these are probably not real, but lets just pretend, okay?) All you have to do now is divide 12 by 6, which is 2. If all the animal Noah saw were 6 legged animals, there would be 2 of them. They are all the basic solutions, but see if you can mix 2 legged animals and 4 legged animals. There are loads of answers but these are the basics. Hope I've helped, thank you for reading this! Hope you have fun on NRICH!

The teacher of Year 1 at Bute House Prep School sent in the following:-

Year 1 had a lot of fun investigating how many animals could have gone into the Ark if Noah had counted 12 legs.  They thought of their own line of inquiry within this problem and so some looked at whether they could have every number of animal from 1-12 with 12 legs, others looked at how many ways they could have groups of animals with the same number of legs making 12, while others investigated how they could best record their findings

systematically.  Here are a few of their discoveries :-) 

Amelia and Olivia found a number of ways of counting 2,3 and 6 animals, although they realised they couldn't have a rabbit and a fly as this wouldn't be enough legs

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Children at Thorpe Primary in Peterborough created a display of their work. Thank you to their teacher Mr Condon for sending in these photos of the display:

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Thank you all for these very detailed accounts of your work. maybe some would go further and consider a larger number of legs seen by Noah. What is there would be only two types of animals and the same numbers of each, which Rhea used.

Thank you to those of you who submitted comments about this game.

Anika from Holy Family Catholic Primary School in Boothstown, Manchester suggested the following rules:

• The answer from the previous calculation should be the answer to the next one
• You can only use each number once
• Player A uses addition and Player B uses subtraction, and this pattern follows throughout the game
• The aim of the game is to stop the other player from getting a calculation from the remaining numbers

Thank you, Anika! Jake said:

• Player A does a calculation using numbers 0-20. They cross off the numbers they use but circle the answer.
• Player B then uses the answer of player A's calculation to start their number sentence.
• Each player can decide whether they would like to add or subtract.
• The player who cannot make a calculation from the numbers remaining loses the game.

So, it seems that Anika and Jake were largely in agreement. Anika, it does look like the order of calculations goes addition, subtraction, addition, subtraction etc from what we see in the video, you're right. However, I would suggest that the order doesn't matter. (You can read our suggested rules by clicking on the 'Show' button underneath the video on the problem page itself.) 

Eleanor from Cottenham Primary School told us:

Good ways to win are to try and use up the numbers 1, 2, 3, 4 and possibly 5 and 6, because you can get to many different numbers by adding those whereas with the larger numbers you cannot get to so many because you can only go up to the number 20.

Eleanor also went on to explore the cooperative challenges. She says:

We can't use all of the numbers because:

For the number line going from 0 to 20, you cannot use zero as you wouldn't be adding or taking away anything so you would stay on the same number.

Yes, Eleanor, if you add zero, or take away zero, from a number, the answer to your calculation is the same number, so you would need to be able to use a number more than once and the rules don't allow this.

For the number line going from 1 to 20, you cannot use all the numbers because there will always be one number left over.

Each new calculation uses two new numbers apart from the first calculation which uses three so if you keep adding two to three you will always be on odd numbers and 20 is an even number, not odd.

Thank you, Eleanor, what great reasoning. Can you follow Eleanor's argument?