How many winning lines can you make in a three-dimensional version of noughts and crosses?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

If you move the tiles around, can you make squares with different coloured edges?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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.

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

Explore the effect of reflecting in two parallel mirror lines.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

How many different symmetrical shapes can you make by shading triangles or squares?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Explore the effect of combining enlargements.

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

The clues for this Sudoku are the product of the numbers in adjacent squares.

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Which set of numbers that add to 10 have the largest product?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

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