There are 83 NRICH Mathematical resources connected to Exploring patterns and noticing structures, you may find related items under Thinking mathematically.
Broad Topics > Thinking mathematically > Exploring patterns and noticing structures
These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
"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 ...?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Can you find a way to identify times tables after they have been shifted up or down?
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?
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Can all unit fractions be written as the sum of two unit fractions?
This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
Can you describe this route to infinity? Where will the arrows take you next?
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.
Can you find the values at the vertices when you know the values on the edges?
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
These pictures show squares split into halves. Can you find other ways?
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?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Can you explain the strategy for winning this game with any target?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
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?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
How many faces can you see when you arrange these three cubes in different ways?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
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?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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.
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.
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?
Six new homes are being built! They can be detached, semi-detached or terraced houses. How many different combinations of these can you find?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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.