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?
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?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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.
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?
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?
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.
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Use the differences to find the solution to this Sudoku.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
What is the smallest number with exactly 14 divisors?
How many different symmetrical shapes can you make by shading triangles or squares?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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. . . .
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n
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?
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 to. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Explore the effect of combining enlargements.
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?
Here's a chance to work with large numbers...
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Can all unit fractions be written as the sum of two unit fractions?
The Egyptians expressed all fractions as the sum of different unit
fractions. The Greedy Algorithm might provide us with an efficient
way of doing this.
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
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.