On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
If you move the tiles around, can you make squares with different coloured edges?
How many different symmetrical shapes can you make by shading triangles or squares?
Explore the effect of reflecting in two parallel mirror lines.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you describe this route to infinity? Where will the arrows take you next?
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?
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 maximise the area available to a grazing goat?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Explore the effect of combining enlargements.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
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. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
Which set of numbers that add to 10 have the largest product?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can all unit fractions be written as the sum of two unit fractions?
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.
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?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
A circle of radius r touches two sides of a right angled triangle,
sides x and y, and has its centre on the hypotenuse. Can you prove
the formula linking x, y and r?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Can you work out the dimensions of the three cubes?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
If a sum invested gains 10% each year how long before it has
doubled its value?
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.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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...