# Six is the Sum

What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?

## Problem

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What do the digits in the number fifteen add up to?

How many other numbers have digits with the same total if we only include numbers without zeros?

## Getting Started

How will you know when you have found all the numbers?

It might help to have some digit cards to 'play' with.

What is the smallest number whose digits add to six?

What is the largest number whose digits add to six?

It might help to have some digit cards to 'play' with.

What is the smallest number whose digits add to six?

What is the largest number whose digits add to six?

## Student Solutions

We received some excellent solutions to this problem. Pupils at Ysgol Aberdyfi Gwynedd wrote to tell us:

The answer we got was that there were thirty-one different numbers with digits that total the number $6$.

We started off in groups to try and find out the answer, and the answer varied between twelve and eighteen therefore we knew that there were obviously more and that we needed to be more organised.

Therefore we started with two-digit numbers, then three-digit numbers, four-digit numbers, five-digit and then six. We saw that the numbers were reverse of each other and we also saw some palindrome numbers.

The smallest number was $15$ and the largest was $111 111$.

We knew that we got them all because of the way we worked it out.

Two-digit numbers:

$15$; $24$; $33$; $42$; $51$

Three-digit numbers:

$114$; $123$; $132$; $141$; $213$; $222$; $231$; $312$; $321$; $411$

Four-digit numbers:

$1113$; $1122$; $1131$; $1212$; $1221$; $1311$; $2112$; $2121$; $2211$; $3111$

Five-digit numbers:

$11112$; $11121$; $11211$; $12111$; $21111$

Six-digit numbers:

$111111$

This is a very careful approach, well done.

Ben and Charlie from Brewood Middle School sent in exactly the same list of numbers and explained how to make sure you find all the solutions:

Work systematically.

Start with a two-digit number (that adds up to six), when you have finished writing all the two-digit numbers go on to three-digit, four-digit, five-digit and six-digit numbers.

For all the digits that you make, start with the smallest number and then the second smallest and then the third smallest and then the fourth smallest etc. This way you will not miss out any numbers.

Swap the numbers around, for example: $11112$ then $11121$ then $11211$ then $12111$ then $21111$.

Make sure no numbers are repeated.

Well done too to Ha Young from Wesley College who also found these solutions. Children from Oakhampton Primary School decided that there are in fact thirty-two solutions because they included the single-digit number $6$ as well. Good thinking!

## Teachers' Resources

### Why do this problem?

This activity allows pupils to explore numbers in what might be a new and unusual way. It encourages them to work systematically, and different approaches can then be discussed.

### Possible approach

You could introduce this challenge simply by asking children to write down a number whose digits add to six, perhaps on a mini-whiteboard. Tell them to keep their number hidden from everyone else and then ask them to consider whether there might be any other numbers whose digits add to six. Give them time to think and write down any others that come to mind.

You can then set up the task and you could start by inviting children to compare their numbers with a neighbour. At this stage you may need to clarify whether numbers with a zero in can be included or not. Encourage the children themselves to justify why we should leave out numbers with a zero. Pairs could then work together to find other numbers.

After some time, stop the group and ask how they will know when they have found all the possibilities. Draw on suggestions that focus on finding numbers in a particular order or by using a particular system, and then give more time for paired work.

You could encourage pairs to record each number they find on a strip of paper. Then, in the plenary you could attach strips to the board, each displaying a different number. By ordering the numbers the group can then work out whether any are missing. Different pupils will have different ways of doing this ordering, so encourage pairs to explain their own way rather than only focusing on one
approach.

### Key questions

What numbers have you found?

How did you find these answers?

How do you know that you have found all the numbers?

### Possible extension

Pupils could use a similar systematic approach to try other numbers whose digits have a different sum.

### Possible support

By writing each number on a different piece of paper, children are not expected to be systematic straight away. Having digit cards might help some learners.