The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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?
To avoid losing think of another very well known game where the patterns of play are similar.
A game for 2 players
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.
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Can you describe this route to infinity? Where will the arrows take you next?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
It would be nice to have a strategy for disentangling any tangled ropes...
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.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
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.
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you use the diagram to prove the AM-GM inequality?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
What is the total number of squares that can be made on a 5 by 5 geoboard?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
What's the largest volume of box you can make from a square of paper?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
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?
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
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
It starts quite simple but great opportunities for number discoveries and patterns!
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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.
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .