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

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

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.

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?

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?

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

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?

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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

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

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.

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

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

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?

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?

Think of a number, square it and subtract your starting number. Is the number youâ€™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?

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

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?

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

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.

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

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

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

A package contains a set of resources designed to develop pupils’ mathematical thinking. This package places a particular emphasis on “generalising” and is designed to meet the. . . .

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

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

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?

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

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

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?

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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.

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

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

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

Can you describe this route to infinity? Where will the arrows take you next?

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