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# Domino Sets

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Age 7 to 11

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- Student Solutions
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Well done to everybody who had a go at this activity. This task can be quite fiddly so we were sent several incorrect solutions, as well as a lot of correct ones! One of the solutions we received was more suitable for Amy's Dominoes, so we've uploaded that there instead.

The children from Linkwood Primary in Scotland sent in the following ideas:

We started with the doubles from 6 to 0. Then we thought about all the other dominoes that would have a 6. Then we did the same for 5, 4, 3, 2 and 1. When we had all of the dominoes we counted them all and found we had 28. We also noticed that for the dominoes with 6 there were 7, for 5 there would be 6 and so on.

Good ideas - this method was used by a lot of children!

We were sent in several similar solutions by the children at Brentwood Preparatory School in England. Bella said:

I used a set of 0-6 dominoes and discovered that the number of dominoes was always 1 more than the value of the face. e.g: there are 7 dominoes with a 6 on them, 6 with 5 dots, 5 with 4 dots .......

To check you have all the dominos, you could start with all the doubles and sort them by their faces, eg: 6-6 to 6-0.

In total there are 28 dominoes and I also found that, if you had 0-9, you would have 55 using the pattern that I found.

Chloe said:

You could arrange the dominoes in columns starting with a 0 on the left side of the domino and then 0-6 on the right side. In the next column you would have a 1 on the left and 0-6 on the right side. In every new column the left side would be increased by 1. The final column would have 6 on the left side. If all the dominoes are there you know that you have a full pack, but if there are some missing you know there is not a full pack.

Dhruv from The Glasgow Academy in the UK sent in the following picture, which can be clicked on to make it bigger. This the same arrangement that Chloe described in the previous solution.

Thank you all for sending in these very clear explanations, and well done for noticing that the arrangement will make a triangular shape.

Matilda and Eliza from Rainworth State School in Australia sent in this solution for the 0-9 dominoes:

Our solution is for the 0-9 dominoes. We lay the dominoes out in rows in order of the number on the top of the domino. When we finished laying them out, the pattern looked like a staircase because each one of the dominoes in the row below had been already used in the one above that. We counted the pattern - it went 1+2+3+4 and so on until +10. This equals 55 dominoes. We checked we were correct by counting the dominoes one by one. Our domino set was like the one described in the problem, the box was broken and not all the dominoes were there so we had to use dominoes from another set and we even had to draw one.

This is a very clear solution, and I like the way you've been creative with making a full set of dominoes when you couldn't find one. This is an excellent example of why we might need to be able to check how many dominoes we are expecting to find in a full set!

We received some excellent solutions from the following children at Ganit Kreeda, Vicharvatika, India: Ananya, Shivashree, Shreehari, Kanaa, Ruhi, Mrunmayee, Aarav, Shravani, Pushan, Ishan, Adithya, Aniket, Anirudh, Vikrant and Avyuktth. Take a look at Ganit Kreeda's full solutions to see their ideas.

Shreehari sent in the following video explaining some of these ideas:

Shaunak from Ganit Manthan, Vicharvatika, India also sent in a video to explain how they approached this task:

Thank you both for sending in these clear explanations of how you approached this task.

If you'd like to learn more about Shaunak's method, you might like to have a look at Shaunak's full solution to see an explanation of how the general version of this task can be solved using algebra.

Some children took a different approach and thought about how often each number would come up across a full set of dominoes. Ayden from St John's College in Cardiff, Wales said:

If you bought a pack of 0-6 dominoes, and they came in a bag, and if you wanted to check if you had a full set, you could count the number of dominoes with a particular number on one side, and then see if it adds up to 7, because there are 7 sets of each number. With this method, you can find out exactly what domino you are missing. For example, if you are counting for the number 3, and there are 6 dominoes, and 3 with 0, 1, 2, 3, 5, and 6, then you know that the 3/4 domino is missing. This is a more efficient, and quicker method than counting all the dominoes individually, but if you chose to do that, then you would know if there is a full set by having 28 dominoes in total.

In a set of 0-9 dominoes, you could use the same method mentioned above to find out if you have a full set. However, if you choose to count the dominoes individually, you would know if you are correct by having 55 dominoes in total.

We received a lot of similar solutions from the children at Park Hill Junior School in the UK. Isaac, Felix, Alba and Hugo had a different way of thinking about this task:

There should be 8 of each number so you count 8 sixes, 8 fives, 8 fours, 8 threes, 8 twos, 8 ones and 8 blanks.

The key to finding the missing ones is:

Line them up in rows or columns of its number.

OR:

There should be 28 dominoes in a pack so you count them.

4x7=28

This way of thinking about it is really interesting - I wonder why there are 8 of each number? And I wonder why that means there will be 28 dominoes in a pack?