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.
A jigsaw where pieces only go together if the fractions are
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Can all unit fractions be written as the sum of two unit fractions?
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
What is the smallest number with exactly 14 divisors?
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?
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?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Find the sum of the series.
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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.
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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...
Is it always possible to combine two paints made up in the ratios
1:x and 1:y and turn them into paint made up in the ratio a:b ? Can
you find an efficent way of doing this?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
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 the area of a parallelogram defined by two vectors?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How many different symmetrical shapes can you make by shading triangles or squares?
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
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?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?