On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
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?
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?
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 maximise the area available to a grazing goat?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many different symmetrical shapes can you make by shading triangles or squares?
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.
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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. . . .
How many solutions can you find to this sum? Each of the different letters stands for a different number.
If you are given the mean, median and mode of five positive whole
numbers, can you find the numbers?
Explore the effect of reflecting in two parallel mirror lines.
The area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
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 describe this route to infinity? Where will the arrows take you next?
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.
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
Can all unit fractions be written as the sum of two unit fractions?
Think of two whole numbers under 10. Take one of them and add 1.
Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your
second number. Add 2. Double again. Subtract 8. Halve this. . . .
Explore the effect of combining enlargements.
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
It is known that the area of the largest equilateral triangular
section of a cube is 140sq cm. What is the side length of the cube?
The distances between the centres of two adjacent faces of. . . .
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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...
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
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?
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. . . .
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
In 15 years' time my age will be the square of my age 15 years ago.
Can you work out my age, and when I had other special birthdays?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
A jigsaw where pieces only go together if the fractions are
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.
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
The clues for this Sudoku are the product of the numbers in adjacent squares.