It would be nice to have a strategy for disentangling any tangled ropes...

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Can all unit fractions be written as the sum of two unit fractions?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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

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?

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?

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

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?

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

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.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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

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

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

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

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

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?

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

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?

Can you find the values at the vertices when you know the values on the edges?

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

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?

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?

What happens when you round these numbers to the nearest whole number?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

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?

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

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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.

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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?