How many winning lines can you make in a three-dimensional version of noughts and crosses?
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
Can you describe this route to infinity? Where will the arrows take you next?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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.
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?
Explore the effect of reflecting in two parallel mirror lines.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you maximise the area available to a grazing goat?
If you move the tiles around, can you make squares with different coloured edges?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
Explore the effect of combining enlargements.
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Can all unit fractions be written as the sum of two unit fractions?
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?
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.
How many different symmetrical shapes can you make by shading triangles or squares?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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. . . .
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.
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?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
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. . . .
Can you find the area of a parallelogram defined by two vectors?
Use the differences to find the solution to this Sudoku.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
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?
Is there an efficient way to work out how many factors a large number has?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
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?
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?
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?
An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
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.