This article for primary teachers suggests ways in which to help children become better at working systematically.
In this article, the NRICH team describe the process of selecting solutions for publication on the site.
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?
What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.
Investigate the different ways you could split up these rooms so that you have double the number.
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
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.
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Can you substitute numbers for the letters in these sums?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
How many models can you find which obey these rules?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Moira is late for school. What is the shortest route she can take from the school gates to the entrance?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
My coat has three buttons. How many ways can you find to do up all the buttons?
10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
How many triangles can you make on the 3 by 3 pegboard?
This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?
How many trains can you make which are the same length as Matt's, using rods that are identical?
Number problems at primary level that require careful consideration.
Can you find the chosen number from the grid using the clues?
Can you find all the different ways of lining up these Cuisenaire rods?