# Digit Addition

*Digit Addition printable sheet*

Here's a maths trick to try:

- Think of one of the numbers from 1 to 9.
- Add 9 to your number.
- Add the digits of your answer together. What is your new number?

What do you notice?

Try the trick again with a new starting number from 1 to 9. What do you notice now?

**Will this always happen if you start with a number from 1 to 9?**

Jonas tried the trick with a few different numbers. He said:

I think you will always get back to your original number, because adding 9 is the same as adding 10 and subtracting 1.

**Do you agree with Jonas? Can you help him explain what is happening?Can you explain the trick in a different way?**

*If you enjoyed this problem, you may like to take a look at The Number Jumbler.*

Try putting your starting number into a place value grid. How many 10s do you have at the beginning? How many 1s?

Add 9 to your number and then put your answer into the place value grid. How many 10s do you have now? How many 1s?

What do you notice?

Thank you to everybody who sent us their thoughts about this activity. Lots of children agreed with Jonas, and we've focused on the solutions here which explain why Jonas is right and why this makes the trick work. Some children also expanded the trick to work with bigger or different numbers.

Katie from the USA said:

My theory is when you pick a number 1-9 you will always get 1 in the tens place and one less than the number you started with in the ones place. My example is using 3. 3+9=12

This is a really simple way of describing what happens, Katie, and this makes it immediately clear that the digits will add up to the number you started with.

The children at Ms Knotts Girls Maths Club at Thomas's School in the UK had a similar way of thinking about what was happening:

We worked on this together and found it hard to explain. We looked at adding 10 to a number and realised that the ones digit does not change. So 6 becomes 16 and 3 becomes 13.

Our teacher suggested we wrote 6 as 06 because it has a 0 in the tens place. Then it was much easier to explain. 06 + 10 = 16 and 03 + 10 becomes 13.

Then we could easily see that when you add 10 to a single-digit number you end up with a 1 in the 10s place so once you add the digits together it will be one more than the number you first thought of. So adding 9, will give you the number you first thought of.

Well done for investigating adding ten instead and then thinking about what would happen when you changed this back to a nine - this is exactly the sort of thing mathematicians do when they aren't sure how to approach a problem.

Ferdia from Clifton Hill Primary School in Australia sent in the following picture to explain why this trick always works. The picture can be clicked on to make it bigger:

You're right, Ferdia - although the digits represent different amounts at the end of the trick, they still add up to the same total!

Harriet from Thomas Eaton School in the UK sent in this picture to show why the trick works:

Thank you for sending this solution in, Harriet. I like the way that this picture makes it really clear that the sum of the digits hasn't changed after nine ones have been added.

Some children went on to try this trick with bigger or different numbers. James and Austin from Rainworth State School in Australia sent in an explanation of the patterns they had noticed when they tried this with numbers up to 89. You might like to take a look at James and Austin's full solutions - they tried 50 + 9 = 59, but they found that the digits add up to 14, which is different from 5. I wonder if perhaps 14 and 5 are actually quite similar, if we do something to the 14... Can anybody spot what it is?

Shaunak from Ganit Manthan, Vicharvatika sent in this video and wrote an explanation of what happens when we add numbers other than 9 to our starting number:

Here's a maths trick to try:

1. Think of one of the numbers from 1 to 9.

2. Add 9 to your number.

3. Add the digits of your answer together. What is your new number?

What do you notice?

My new number is the same as my original number.

Try the trick again with a new starting number from 1 to 9. What do you notice now?

In both cases, the sum of the digits of my number was the same as my original number.

Will this always happen if you start with a number from 1 to 9?

Yes, this will always happen if we take a number from 1 to 9.

Jonas tried the trick with a few different numbers.

He said: I think you will always get back to your original number, because adding 9 is the same as adding 10 and subtracting 1.

Do you agree with Jonas? Can you help him explain what is happening?

What Jonas has said is correct. I agree that adding 9 is the same as adding 10 and subtracting 1.

The reason for this is when we add 10, then the tens place increases by 1, so the sum of the digits is number + 1, which we do not want. But, if we add 9, which is 10 – 1, then the sum of the digits will be 1 + (number – 1), which is the number itself. That is why the trick will always work.

I think another way of explaining it would be to have a number x, and add 9, which increases the tens place by 1 and decreases the ones place by 1.

x + 1 - 1 = x, so the trick will always work.

We can also expand the trick, say add a greater number instead of 9. We will try the trick by adding 18, which is 20 - 2. This will also work!

Like this, if we add any multiple of 9 to a number from 0 - 9, the sum will always be the original number. There might come such a situation where the sum of the digits is greater than 9. If that happens, we will add the digits in this number again and again till the sum becomes the original number. For example, if I choose 5, and I want to add 9*14326 to 5, then I get 128939. 1 + 2 + 8 + 9 + 3 + 9 = 32. Now, we will add 3 + 2, and we get 5! This is how one can expand the trick.

You've reasoned this really clearly, Shaunak, and the trick is very impressive when you use bigger multiples of nine! Shree from the Netherlands had a similar idea, and used a special term to explain what is happening:

I agree with Jonas.

It is like with normal 9 times tables, your number is the digital root of your ending number.

E.g. 6 x (9) = 54, 5 + 4 = (9)

You're right, Shree, this is exactly like the pattern we see in the nine times table. 'Digital root' is a really interesting phrase here because it means that you keep adding the digits together until the answer you get is a one-digit number, like Shaunak did in the solution above.

Gabe from Wembrook in the UK thought about what might happen if we expanded the trick, writing numbers in different bases instead of base 10. Base 10 is how we normally write numbers, using the ten digits 0-9. Gabe started off with 12 digits instead, using χ and ε to represent the extra two digits:

What if this was in different bases?

Base 12

1 2 3 4 5 6 7 8 9 χ ε

We need to add ε, not 9 because 9 < χ (10) < ε (11) < 10 (12)

So ε is the biggest digit.

Gabe also had a think about what might happen if we wrote numbers in hexadecimal (base 16) - take a look at Gabe's solution if you'd like to see these ideas!

Thank you as well to the following children who sent in similar solutions that we unfortunately don't have space to feature here: the children from Pool-in-Wharfedale School in the UK; Alex, Leonidas and Remy from Daneshill School in the UK; Connor from Sherwood Ridge Public School in Australia; Sasha and Mark from Brentwood Preparatory School in England; Clara and Bella from Brentwood Preparatory School; Arya and Roya from Westridge School in the USA; Cats from CHPS in Australia; Dhruv from The Glasgow Academy in the UK; and Chloe and Maddy from Clifton Hill Primary School in Australia.

### Why do this problem?

'Tricks' tap into children's natural curiosity and can provide the motivation for exploring the underlying mathematics in order to unpick how they are done. This trick provides an engaging context in which to explore place value and in particular 'adding nine' as 'adding ten and subtracting one'.

### Possible approach

*This problem featured in the NRICH Primary webinar in June 2022.*

With the whole class, practise adding 9 to a variety of one-digit numbers. Introduce the trick and discuss what 'digit addition' means. As a class, choose a number between 1 and 9 (inclusive) and work through the steps together on the board. What has happened? Is it magic?

In pairs, ask pupils to choose numbers from 1 to 9, and work through the steps with each number. Working in pairs will allow pupils to check and correct each other's addition, ensuring that the 'trick' always gives back the starting number. You may wish to suggest that particular pairs investigate certain starting numbers so that as a class you have covered all the numbers from 1 to 9.

Once each pair has tried a few numbers, bring the class back together. Ask if anyone has their own ideas about what is happening, and then show the class Jonas' explanation. What does he mean? Why are 10s and 1s interesting?

Choose a starting number and represent that number on the board, either using counters in a place value grid or using 1s counters to represent the number. Work out the answer after 9 has been added to it. Represent this number on the board underneath the first number, either using counters in the place value grid or using a 10s counter and some 1s counters. What can pupils say about the number of
counters? What does the digit addition have to do with the number of counters? (When we add the digits, we add the tens and ones digits together as if all the counters had the same value.)

A different approach would be to write some of the numbers on the board and show the different steps of the trick, e.g.:

4 + 9 = 13, 1 + 3 = 4

5 + 9 = 14, 1 + 4 = 5

Ask the children what they notice. Can they see any patterns? What happens when the starting number gets bigger by one?

### Key questions

What do you notice about your answer?

What happened to the digits when you added 9?

If you add 10 and then subtract 1, what happens to the digits?

### Possible support

Using place value grids or counters will help children see what is happening to each of the digits as they add 9.

### Possible extension

For an extension, ask children to try the trick with numbers from 11 to 19. Does it still work? Why/why not? Is there anything that pupils notice about the answers? (The digit addition should be the same as the digit addition for the original number.)