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

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?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

This activity involves rounding four-digit numbers to the nearest thousand.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Benâ€™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?

Got It game for an adult and child. How can you play so that you know you will always win?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

What happens when you round these three-digit numbers to the nearest 100?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

This task follows on from Build it Up and takes the ideas into three dimensions!

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Can you find all the ways to get 15 at the top of this triangle of numbers?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

An investigation that gives you the opportunity to make and justify predictions.

Nim-7 game for an adult and child. Who will be the one to take the last counter?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Here are two kinds of spirals for you to explore. What do you notice?

Can you explain the strategy for winning this game with any target?

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

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.