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Nowadays the calculator is very familiar to many of us. What did people do to save time working out more difficult problems before the calculator existed?

### How Odd

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

### Count the Digits

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

# Jumping Squares

## Jumping Squares

In this problem it is not the squares that jump, you do the jumping!
It is not a race, but a game of skill. You have to be able to look ahead.
The idea is to go round the track in as few jumps as possible, keeping to the rules.

The first line of the track looks like this:

You start on the green square which tells you that you can jump forward either one or three squares. Here is the whole track:

You make your way round the track and finish on the red square with 'end' on it.

If you land on a square which has 2 and 3 on it, you can jump forward - or back - either 2 or 3 squares.

If the square has 2 and 0 on it, you can jump forward - or back - only 2 squares.

If the square has 0 and 0 on it, you cannot jump at all. You have to go right back to the beginning and start again!

You can do this on your own or with a friend.

You can count your jumps by making a note on paper whenever you jump or by counting out twenty counters and taking one from the pile every move.

What is the least number of jumps you can make to get round the whole track?

Which squares do you need to land on?

### Why do this problem?

This problem will encourage children to work systematically, and think and plan ahead. The activity can be done by one child working alone, but might be better if two work together.

### Possible approach

You could start by looking at the first line of the track with the whole group. Explain the rules of the challenge and invite children to talk to a partner about what their first move might be.  Ask for some suggestions and encourage good explanations of their choices.  This initial discussion will allow you to reinforce the rules and make sure that learners do not count the square they are on when jumping forward (or back).

Ask the group for suggestions as to how they will keep track of the number of moves they've made.  Allow them to choose a way that suits them and have available all sorts of equipment that might help, for example whiteboards, paper, counters, number lines, number squares, digit cards ...

Once learners have got the idea of the task, suggest that they work either on their own or in pairs to try and find the smallest number of jumps that are needed to get to the centre. This sheet gives the full track which can be printed and copied (perhaps even laminated).

After some time, you may wish to draw the group together for a brief discussion about progress so far. It might be helpful to ask some pairs to share what they've done, for example you may notice that they have recorded which squares they have jumped on in a good way.  Keeping a record not just of the number of jumps but where the jumps are to and from might help children tweak and improve their total number of jumps.

At the end of the lesson bring the group together again. What is the least number of jumps that were made to get round the whole track? Is that the very best way to jump round or does the group think that there might be a better way still to find?

### Key questions

Where could you land next?
Which move might be better?  Why?
Have you thought of jumping backwards?
How are you keeping count of your moves?
What is the least number of jumps that you made to get round the whole track?
Do you think that is the very best way to jump round?  How do you know?

### Possible extension

Learners could make their own 'jumping squares' track for others to try or perhaps they could introduce a different rule using the same track.

### Possible support

It might be that an adult could keep count of the number of moves made so this takes out one level of detail for the children to attend to.