Number Tiles
Years 4 - 10

Preparation

  • Tear a piece of paper into nine (9) pieces and number them 1 to 9, or, print and cut this set of digit cards.
    Keep your pieces in an envelope or press-seal bag.
    You will use them in other activities.
    (You don't use zero (0) in this activity.)
  • Write the title of this challenge and today's date on a fresh page in your maths journal.

If the whole family want to join in this activity, use nine
A4 pieces of paper and write big numbers on each page.
Then you can play with the problem on the floor.

Getting Started

  • Open this Number Tiles Starter. You can read it on screen or print it.
  • Question 1 might take you twenty minutes or more. Are you up for it?
After you have tried for at least 15 minutes, if you need a hint click here.
  • When you have answered Question 1, write your solution in your journal.
  • Try Question 2. Record solutions in your journal.

Have fun exploring Number Tiles

Collecting More Data

Mathematicians love problems that have several solutions. It means there is an even bigger challenge to chase by using these questions:
  • How many solutions are there?
  • How do I know when I have found them all?
You need more data to help you find clues that might answer those questions.
These next challenges will help you find solutions more quickly.
Find a few answers each day until you notice something special.

Knowing The Answer Line
__ __ __

+ __ __ __
6 3 9

  • We know there are solutions that have 639 (six hundred and thirty-nine) in the answer line.
  • Place 639 in your answer line.
  • Work backwards to find the two numbers which add to make it.
  • There is more than one way to do it. Write all your solutions in your journal.

Knowing Some Digits
__ 2 __

+ __ 5 __
__ 8 __

  • The 2, 5, and 8 are correct.
  • Build as many solutions as you can from these clues.
  • Write all your solutions in your journal.
  • Check all your solutions so far. Make sure there aren't two or more the same.

1 __ 4

+ __ 8 __
__ __ __

  • Place the 100, 4 and 80.
  • Build as many solutions as you can from these clues.
  • Write all your solutions in your journal.
  • Check all your solutions so far. Make sure there aren't two or more the same.

Note: If your school is a member of Maths300, your teacher can provide software that will help you explore Number Tiles.

Clues In The Data

Now you know there are many solutions to this problem. In fact there are 168 different solutions. Mathematicians found this out by asking:
  • How many solutions are there?
  • How do I know when I have found them all?
But they needed enough data to be able to find clues. If you have about twenty (20) solutions, you have enough data.
If you need more solutions check this board. There might be some you haven't found yet.
First find all the solutions the same. There are at least four the same. One pair is marked.


Do you need to check your solutions again to make sure they are all different?
  • Look back through all your solutions.
  • Make a list of the special things you have learnt about this problem already.
    For example, the total of the hundreds numbers must be 9 or less. Why?
There is one special thing mathematicians noticed that helped them find all the solutions.
  • Have you noticed it yet?
  • Look carefully at all the answer lines and 'play' with the digits.
After you have hunted for yourself for a while, if you need a hint click here.

Final Challenge

So when the mathematician realises that the digits in the answer line must add to 18, they can work backwards to find all the solutions.
  • First they find all the 3 digit numbers whose digits add to 18.
  • Then they put the first one in the answer line (like you already did for 639) and work backwards to find all the solutions for that answer line.
  • Then they do the same for the next one and the next one and the next one ... until they are all found.

When Daniel was in Year 9 he made a list of all the three digit numbers whose digits added to 18. There are 42 of them.

  1. Try to recreate Daniel's list.
  2. Daniel realised some of them wouldn't work. Which ones? Why?
You are not being asked to find all 168 solutions. You are only being asked to learn a bit about how a mathematician works.
But if you want to have a go at finding them all just for fun, go for it.

Just Before You Finish

Read your Working Like A Mathematician page again and put a mark beside everything on the list that you did during this problem.
  • In your journal copy and complete this list.

    In Number Tiles I was working like a mathematician when:
    - I was interested in the problem to start with.
    ...
    ...
    ... (list everything that proves you were working like a mathematician)

  • Noticing a pattern like answer lines adding to 18, doesn't prove that it is always true. It only shows it's true for the ones you try. If you want to know how mathematician's can prove that is always true for this problem, you can look in the link below.

 

Answers & Discussion

These notes were originally written for teachers. We have included them to support parents to help their child learn from Number Tiles.

Send any comments or photos about this activity and we can start a gallery here.

 

Maths At Home is a division of Mathematics Centre

Hint: Question 1

  • What is the special thing that sometimes happens when you do a normal addition like this?
  • Are you letting that happen in this challenge?
Go Back

Hint: Answer line

  • Add up the digits of any answer line.
  • The answer is always the same number (18).
Go Back