Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you use the diagram to prove the AM-GM inequality?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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?
Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
Can you discover whether this is a fair game?
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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.
Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?
How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.
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?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
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.
Can you describe this route to infinity? Where will the arrows take you next?
Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .
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.
A game for 2 players
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which relates triangular numbers to square numbers?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
Can you find a rule which connects consecutive triangular numbers?
How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?
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?
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
How many different symmetrical shapes can you make by shading triangles or squares?
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
In how many ways can you fit all three pieces together to make shapes with line symmetry?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.