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

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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?

Delight your friends with this cunning trick! Can you explain how it works?

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?

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

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

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

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

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?

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

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

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.

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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

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?

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?

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

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

Explore the effect of reflecting in two intersecting mirror lines.

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.

How many centimetres of rope will I need to make another mat just like the one I have here?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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?

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.

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

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

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

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

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.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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.