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# Being Curious - Secondary Students

*Being Curious is part of our Developing Mathematical Habits of Mind collection.*

Good thinkers are curious and ask good questions. They are excited by new ideas and are keen to explore and investigate them.

Want to become a more curious mathematician?

We hope these problems will provoke you to ask good mathematical questions. Take a look, we think you'll get hooked on them!

*You can browse through the Number, Algebra, Geometry or Statistics collections, or scroll down to see the full set of problems below.*
### Five Steps to 50

### Next-door Numbers

### Eightness of Eight

### Light the Lights

### Little Man

### Ring a Ring of Numbers

### Colouring Triangles

### Chain of Changes

### More Numbers in the Ring

### Digit AdditionLive

### If the World Were a Village

### Shaping It

### Brush Loads

### Tumbling Down

### Number Differences

### Two Clocks

### Curious Number

### Your Number Is...

### Statement Snap

### The Number Jumbler

### Consecutive Numbers

### Subtraction Surprise

### Triangle Transformation

### Nice or Nasty

### Three Neighbours

### Fruity Totals

### Dicey Operations

### Satisfying Statements

### Elevenses

### Your Number Was...

### How Much Can We Spend?

### Summing Consecutive Numbers

### Can They Be Equal?

### Tilted SquaresLive

### Shifting Times Tables

### Special Numbers

### Largest Product

### What Numbers Can We Make?

### Blue and White

### Number Pyramids

### Perimeter Possibilities

### Square Coordinates

### More Number Pyramids

### Two's Company

### Stars

### Non-transitive Dice

### Cosy Corner

### Who's the Best?

### Estimating Time

### Opposite Vertices

### Think of Two Numbers

### On the Edge

### What Numbers Can We Make Now?

### Sending a Parcel

### Reversals

### Unequal Averages

### Right Angles

### Which Solids Can We Make?

### A Chance to Win?

### Cola Can

### Litov's Mean Value Theorem

### Arithmagons

### What's it Worth?

### Semi-regular Tessellations

### Charlie's Delightful Machine

### Marbles in a Box

### Cuboid Challenge

### Wipeout

### Take Three from Five

### Searching for Mean(ing)

### How Old Am I?

### Last One Standing

### Beelines

### Curvy Areas

### A Little Light Thinking

### Pair Products

### Circles in Quadrilaterals

### Arclets

### Triangle Midpoints

### Trapezium Four

### Triangles and Petals

### Partly Painted Cube

### Same Number!

### What's Possible?

### Odds and Evens Made Fair

### Where to Land

### Multiplication Arithmagons

### Pick's Theorem

### Hexy-metry

### Fit for Photocopying

### Mathsland National Lottery

### Vector Journeys

### Data Matching

### Which Spinners?

### Three by One

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Good thinkers are curious and ask good questions. They are excited by new ideas and are keen to explore and investigate them.

Want to become a more curious mathematician?

We hope these problems will provoke you to ask good mathematical questions. Take a look, we think you'll get hooked on them!

Age 5 to 7

Challenge Level

Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?

Age 5 to 7

Challenge Level

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

Age 5 to 7

Challenge Level

What do you see as you watch this video? Can you create a similar video for the number 12?

Age 5 to 7

Challenge Level

Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

Age 5 to 7

Challenge Level

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

Age 5 to 7

Challenge Level

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Age 5 to 7

Challenge Level

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Age 5 to 7

Challenge Level

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Age 5 to 7

Challenge Level

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Age 5 to 11

Challenge Level

Try out this number trick. What happens with different starting numbers? What do you notice?

Age 5 to 11

Challenge Level

This activity is based on data in the book 'If the World Were a Village'. How will you represent your chosen data for maximum effect?

Age 5 to 11

Challenge Level

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.

Age 7 to 11

Challenge Level

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

Watch this animation. What do you see? Can you explain why this happens?

Age 7 to 11

Challenge Level

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Age 7 to 11

Challenge Level

These clocks have only one hand, but can you work out what time they are showing from the information?

Age 7 to 11

Challenge Level

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

Age 7 to 14

Challenge Level

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Age 7 to 14

Challenge Level

You'll need to know your number properties to win a game of Statement Snap...

Age 7 to 14

Challenge Level

The Number Jumbler can always work out your chosen symbol. Can you work out how?

Age 7 to 14

Challenge Level

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

Age 7 to 14

Challenge Level

Try out some calculations. Are you surprised by the results?

Age 7 to 14

Challenge Level

Start with a triangle. Can you cut it up to make a rectangle?

Age 7 to 14

Challenge Level

There are nasty versions of this dice game but we'll start with the nice ones...

Age 7 to 14

Challenge Level

Take three consecutive numbers and add them together. What do you notice?

Age 7 to 16

Challenge Level

In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?

Age 11 to 14

Challenge Level

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

Age 11 to 14

Challenge Level

Can you find any two-digit numbers that satisfy all of these statements?

Age 11 to 14

Challenge Level

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Age 11 to 14

Challenge Level

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Age 11 to 14

Challenge Level

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

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Age 11 to 14

Challenge Level

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

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

Age 11 to 14

Challenge Level

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Age 11 to 14

Challenge Level

Which set of numbers that add to 100 have the largest product?

Age 11 to 14

Challenge Level

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

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Age 11 to 14

Challenge Level

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

Age 11 to 14

Challenge Level

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Age 11 to 14

Challenge Level

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Age 11 to 14

Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Age 11 to 14

Challenge Level

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

Age 11 to 14

Challenge Level

Can you work out what step size to take to ensure you visit all the dots on the circle?

Age 11 to 14

Challenge Level

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

Age 11 to 14

Challenge Level

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

Age 11 to 14

Challenge Level

Which countries have the most naturally athletic populations?

Age 11 to 14

Challenge Level

How well can you estimate 10 seconds? Investigate with our timing tool.

Age 11 to 14

Challenge Level

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Age 11 to 14

Challenge Level

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

Age 11 to 14

Challenge Level

If you move the tiles around, can you make squares with different coloured edges?

Age 11 to 14

Challenge Level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Age 11 to 14

Challenge Level

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Age 11 to 14

Challenge Level

Where should you start, if you want to finish back where you started?

Age 11 to 14

Challenge Level

Play around with sets of five numbers and see what you can discover about different types of average...

Age 11 to 14

Challenge Level

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Age 11 to 14

Challenge Level

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Age 11 to 14

Challenge Level

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

Age 11 to 14

Challenge Level

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Age 11 to 14

Challenge Level

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Age 11 to 16

Challenge Level

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

Age 11 to 16

Challenge Level

There are lots of different methods to find out what the shapes are worth - how many can you find?

Age 11 to 16

Challenge Level

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Age 11 to 16

Challenge Level

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

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

Age 11 to 16

Challenge Level

What's the largest volume of box you can make from a square of paper?

Age 11 to 16

Challenge Level

Can you do a little mathematical detective work to figure out which number has been wiped out?

Age 11 to 16

Challenge Level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

Age 11 to 16

Challenge Level

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Age 14 to 16

Challenge Level

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Age 14 to 16

Challenge Level

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

Age 14 to 16

Challenge Level

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Age 14 to 16

Challenge Level

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Age 14 to 16

Challenge Level

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Age 14 to 16

Challenge Level

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

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Age 14 to 16

Challenge Level

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Age 14 to 16

Challenge Level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Age 14 to 16

Challenge Level

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Age 14 to 16

Challenge Level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Age 14 to 16

Challenge Level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Age 14 to 16

Challenge Level

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

Age 14 to 16

Challenge Level

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

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

Age 14 to 16

Challenge Level

Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?

Age 14 to 16

Challenge Level

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Age 14 to 16

Challenge Level

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

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Age 14 to 16

Challenge Level

Explore the relationships between different paper sizes.

Age 14 to 16

Challenge Level

Can you work out the probability of winning the Mathsland National Lottery?

Age 14 to 18

Challenge Level

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

Use your skill and judgement to match the sets of random data.

Age 14 to 18

Challenge Level

Can you work out which spinners were used to generate the frequency charts?

Age 16 to 18

Challenge Level

There are many different methods to solve this geometrical problem - how many can you find?