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?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

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?

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

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

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.

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?

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

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

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.

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

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

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?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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. . . .

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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?

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

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

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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.

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

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

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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.

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

Find out what a "fault-free" rectangle is and try to make some of your own.

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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.

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

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?

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

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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?

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Are these statements relating to odd and even numbers 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.

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?