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?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
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?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?
If you move the tiles around, can you make squares with different coloured edges?
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?
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?
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?
Can you maximise the area available to a grazing goat?
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?
Can all unit fractions be written as the sum of two unit fractions?
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.
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?
Is there an efficient way to work out how many factors a large number has?
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Explore the effect of combining enlargements.
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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 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.
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Different combinations of the weights available allow you to make different totals. Which totals can you make?
How many different symmetrical shapes can you make by shading triangles or squares?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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. . . .
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?
Here's a chance to work with large numbers...
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
How many solutions can you find to this sum? Each of the different letters stands for a different number.