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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?
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?
If the answer's 2010, what could the question be?
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?
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.
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
Investigate what happens when you add house numbers along a street in different ways.
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?
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.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.
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?
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
Investigate these hexagons drawn from different sized equilateral triangles.
Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?
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?
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!
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
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?
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
Here are many ideas for you to investigate - all linked with the number 2000.
How many tiles do we need to tile these patios?
Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?
Why does the tower look a different size in each of these pictures?
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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?
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.
"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 ...?
Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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?
Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.
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?
These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.
How many models can you find which obey these rules?
In how many ways can you stack these rods, following the rules?