Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
What is the smallest number with exactly 14 divisors?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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 solutions can you find to this sum? Each of the different letters stands for a different number.
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 clues for this Sudoku are the product of the numbers in adjacent squares.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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?
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?
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?
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.
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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.
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...
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative 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. . . .
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...
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you work out the dimensions of the three cubes?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
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.
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?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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. . . .
Can you maximise the area available to a grazing goat?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
How many different symmetrical shapes can you make by shading triangles or squares?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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. . . .
Can you describe this route to infinity? Where will the arrows take you next?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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.