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## Reach 100

Here is a grid of four "boxes":

You must choose four different digits from $1 - 9$ and put one in each box. For example:

This gives four two-digit numbers:

$52$ |
(reading along the $1$st row) |

$19$ |
(reading along the $2$nd row) |

$51$ |
(reading down the left hand column) |

$29$ |
(reading down the right hand column) |

In this case their sum is $151$.

Try a few examples of your own.

Is there a quick way to tell if the total is going to be even or odd?

Your challenge is to find four different digits that give four two-digit numbers which add to a total of $100$.

How many ways can you find of doing it?

This problem is adapted from Make 200 from 'Mathematical Challenges for Able Pupils Key Stages 1 and 2', published by DfES.

### Why do this problem?

This problem challenges pupils' understanding of place value and is a good way to practise a particular method of written addition.

### Possible approach

It would be good to start with the grid drawn on the board and for you to explain the challenge orally to the group. You could use the example in the problem itself so that the task is clear.

Ask for suggestions as to how they might start and give learners a few minutes to think on their own, then share their ideas with a partner. Open this out to the whole class so that a few pairs share their thoughts with everyone. The suggestions are likely to be quite general at this stage.

Allow children to work together on the problem in their pairs. They will find mini-whiteboards or paper useful for keeping track of their calculations. After some time, bring the group together again and discuss any insights they have gained. Some pairs may have thought about the cells in the grid which make up the units digits of the four numbers, others may have concentrated on the cells
which contribute to the tens digits. In either case, encourage them to explain the restrictions they have noticed, and look out for those pairs who are working systematically through the options.

You may like to leave time (perhaps in a subsequent lesson) for each pair to produce a poster describing how they arrived at the solution(s).

### Key questions

Where could you start?

How do the four digits you choose contribute to the zero in the units column of $100$?

What can you say about the size of the digit in the top left cell?

### Possible extension

What is the smallest total you can make? What is the largest? Can you make all the totals in between?

Two and Two requires similar systematic thinking and could be a good problem for some children to try next.

### Possible support

Having digit cards available for learners to physically manipulate will help those who are reluctant to commit ideas to paper.