Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?
Ben has five coins in his pocket. How much money might he have?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
If the answer's 2010, what could the question be?
What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!
Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?
Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?
These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?
There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?
In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Investigate what happens when you add house numbers along a street in different ways.
We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?
Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?
There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
How many different sets of numbers with at least four members can you find in the numbers in this box?
Investigate the different ways you could split up these rooms so that you have double the number.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?
This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.
Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
In how many ways can you stack these rods, following the rules?
Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?