### Number Detective

Follow the clues to find the mystery number.

### Red Even

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

### Prime Magic

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

# More Carroll Diagrams

## More Carroll Diagrams

Can you put the numbers in the correct place in this Carroll diagram?

You could print off this sheet.pdf.

Can you explain how you know where to place the numbers?

Now have a look at this Carroll diagram. This time, the idea is to fill in the labels for the rows and columns.

You could print off sheet.pdf to complete.

### Why do this problem?

This problem gives children a way of sorting numbers according to different properties and also forces them consider more than one aspect at once. It also provides opportunities for children to explain their placing of the numbers, using appropriate language.

### Possible approach

There are two aspects to this problem: Firstly, it focuses on sorting numbers according to certain properties and secondly, it requires a knowledge of how a Carroll diagram works. If your class has not had much experience of Carroll diagrams, it might be useful to look at the problem Carroll Diagrams which also has suggestions in the notes of how you could go about introducing the relevant ideas.

This problem offers more of a challenge in that children are asked to identify the criteria by which the numbers have been sorted. Encourage them to talk to each other about how they might work out the labels for each row and column. There are many different approaches and sharing some of their ideas with the whole group would be beneficial. Try to focus on the clarity of their arguments, thereby encouraging well-reasoned solutions.

Alan Parr, who has contributed many great ideas to NRICH, has sent in this pdf of further Carroll diagram sorting activities which you may like to use as follow-up to this problem.

### Key questions

Do you think that the number "5" is between 5 and 15 or not?
What can you say about the numbers is this box that is different from the numbers in that box?

### Possible extension

Learners could go on to using these sheets given above as a follow-up, especially the last four questions.