Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Each child in Class 3 took four numbers out of the bag. Who had made the highest even number?

Are these statements always true, sometimes true or never true?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

This article for teachers describes how number arrays can be a useful reprentation for many number concepts.

A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.

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?

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

There are exactly 3 ways to add 4 odd numbers to get 10. Find all the ways of adding 8 odd numbers to get 20. To be sure of getting all the solutions you will need to be systematic. What about. . . .

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

Use the interactivities to complete these Venn diagrams.

How many different sets of numbers with at least four members can you find in the numbers in this box?

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?

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

Read all about Pythagoras' mathematical discoveries in this article written for students.

Explore the relationship between simple linear functions and their graphs.