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There are **83** NRICH Mathematical resources connected to **Exploring patterns and noticing structures**, you may find related items under Thinking mathematically.

Problem
Primary curriculum
Secondary curriculum
### Shape Products

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Division Rules

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seven Squares

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magic Letters

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Steel Cables

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Vector Journeys

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ip Dip

"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 ...?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Charlie's Delightful Machine

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shifting Times Tables

Can you find a way to identify times tables after they have been shifted up or down?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Much Can We Spend?

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cyclic Quadrilaterals

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Keep it Simple

Can all unit fractions be written as the sum of two unit fractions?

Age 11 to 14

Challenge Level

General
Primary curriculum
Secondary curriculum
### Interactive Spinners

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Repeating Patterns

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Caterpillars

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Route to Infinity

Can you describe this route to infinity? Where will the arrows take you next?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Brush Loads

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Arithmagons

Can you find the values at the vertices when you know the values on the edges?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painted Cube

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tilted Squares

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Pyramids

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pair Products

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pick's Theorem

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Halving

These pictures show squares split into halves. Can you find other ways?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 1 Step 2 Step

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Attractive Tablecloths

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Marbles in a Box

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What's Possible?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Summing Consecutive Numbers

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Have You Got It?

Can you explain the strategy for winning this game with any target?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Let's Investigate Triangles

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?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sticky Triangles

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round and Round the Circle

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Roll These Dice

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pebbles

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes

How many faces can you see when you arrange these three cubes in different ways?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Presents

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sending Cards

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### My New Patio

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Stairs

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.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magazines

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.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Squares

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?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Homes

Six new homes are being built! They can be detached, semi-detached or terraced houses. How many different combinations of these can you find?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tea Cups

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.

Age 7 to 14

Challenge Level