Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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 maximise the area available to a grazing goat?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Explore the effect of reflecting in two parallel mirror lines.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Can all unit fractions be written as the sum of two unit fractions?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you describe this route to infinity? Where will the arrows take you next?
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?
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?
Explore the effect of combining enlargements.
How many different symmetrical shapes can you make by shading triangles or squares?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
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. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Use the differences to find the solution to this Sudoku.
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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?
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.
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.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Can you find the area of a parallelogram defined by two vectors?
There is a particular value of x, and a value of y to go with it,
which make all five expressions equal in value, can you find that
x, y pair ?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Is there an efficient way to work out how many factors a large number has?
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?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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?
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?
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.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?