Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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.
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?
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?
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?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
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 see how to build a harmonic triangle? Can you work out the next two rows?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this. . . .
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?
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?
Which set of numbers that add to 10 have the largest product?
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Explore the effect of combining enlargements.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
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?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
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.
Can you find the area of a parallelogram defined by two vectors?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
If a sum invested gains 10% each year how long before it has doubled its value?
The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?
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. . . .
According to an old Indian myth, Sissa ben Dahir was a courtier for a king. The king decided to reward Sissa for his dedication and Sissa asked for one grain of rice to be put on the first square. . . .
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
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?
Explore the effect of reflecting in two parallel mirror lines.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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?
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?
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.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
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?
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.
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?
Can all unit fractions be written as the sum of two unit fractions?
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
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?