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. . . .
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Can you describe this route to infinity? Where will the arrows take you next?
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?
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?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Explore the effect of reflecting in two parallel mirror lines.
Can you find the area of a parallelogram defined by two vectors?
Can all unit fractions be written as the sum of two unit fractions?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Explore the effect of combining enlargements.
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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.
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
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?
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 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?
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?
Can you maximise the area available to a grazing goat?
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 ?
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.
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?
All CD Heaven stores were given the same number of a popular CD to
sell for £24. In their two week sale each store reduces the
price of the CD by 25% ... How many CDs did the store sell at. . . .
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. . . .
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
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 work out the dimensions of the three cubes?
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?