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?

I added together some of my neighbours house numbers. Can you explain the patterns I noticed?

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

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?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

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

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

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

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

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?

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.

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.

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

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

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

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

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

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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.

This challenge asks you to imagine a snake coiling on itself.

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?

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

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.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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?

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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

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

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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?

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

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

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

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

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

Try out this number trick. What happens with different starting numbers? What do you notice?

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

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?