Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

This activity investigates how you might make squares and pentominoes from Polydron.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Can you draw a square in which the perimeter is numerically equal to the area?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

Can you find all the different triangles on these peg boards, and find their angles?

How many different triangles can you make on a circular pegboard that has nine pegs?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

These practical challenges are all about making a 'tray' and covering it with paper.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

How many ways can you find of tiling the square patio, using square tiles of different sizes?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

What is the best way to shunt these carriages so that each train can continue its journey?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

How many trapeziums, of various sizes, are hidden in this picture?

The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?