Being Curious

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.
Challenge Level: Challenge Level:1

Being Curious - Number

Number problems for inquiring students.

Challenge Level: Challenge Level:1

Being Curious - Algebra

Algebra problems for inquiring students.

Challenge Level: Challenge Level:1

Being Curious - Geometry

Geometry problems for inquiring students.

Challenge Level: Challenge Level:1

Being Curious - Statistics

Statistics problems for inquiring students.

Statement Snap

KS 2 & 3 Challenge Level:

Challenge Level:1

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

Nice or Nasty

KS 2 & 3 Challenge Level:

Challenge Level:1

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

Your Number Was...

KS 3 Challenge Level:

Challenge Level:1

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?

Dicey Operations

KS 3 Challenge Level:

Challenge Level:1

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

Blue and White

KS 3 Challenge Level:

Challenge Level:1

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

Elevenses

KS 3 Challenge Level:

Challenge Level:1

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

Number Pyramids

KS 3 Challenge Level:

Challenge Level:1

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

Perimeter Possibilities

KS 3 Challenge Level:

Challenge Level:1

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

How Much Can We Spend?

KS 3 Challenge Level:

Challenge Level:1

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?

What Numbers Can We Make?

KS 3 Challenge Level:

Challenge Level:1

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

Can They Be Equal?

KS 3 Challenge Level:

Challenge Level:1

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

Shifting Times Tables

KS 3 Challenge Level:

Challenge Level:1

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

Summing Consecutive Numbers

KS 3 Challenge Level:

Challenge Level:1

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Satisfying Statements

KS 3 Challenge Level:

Challenge Level:1

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

Special Numbers

KS 3 Challenge Level:

Challenge Level:1

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?

Semi-regular Tessellations

KS 3 Challenge Level:

Challenge Level:1

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

Largest Product

KS 3 Challenge Level:

Challenge Level:1

Which set of numbers that add to 10 have the largest product?

Searching for Mean(ing)

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Stars

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

Pick's Theorem

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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.

More Number Pyramids

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Estimating Time

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Right Angles

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Marbles in a Box

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Cuboid Challenge

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

On the Edge

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Reversals

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Sending a Parcel

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Unequal Averages

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Cosy Corner

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

Non-transitive Dice

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Think of Two Numbers

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Who's the Best?

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Which countries have the most naturally athletic populations?

What Numbers Can We Make Now?

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Square Coordinates

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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?

Opposite Vertices

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Litov's Mean Value Theorem

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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

A Chance to Win?

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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

Cola Can

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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

Two's Company

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

Which Solids Can We Make?

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

What's it Worth?

KS 3 & 4 Challenge Level:

Challenge Level:1

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

Charlie's Delightful Machine

KS 3 & 4 Challenge Level:

Challenge Level:1

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

Pair Products

KS 4 Challenge Level:

Challenge Level:1

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

Last One Standing

KS 4 Challenge Level:

Challenge Level:1

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

Arithmagons

KS 4 Challenge Level:

Challenge Level:1

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

Same Number!

KS 4 Challenge Level:

Challenge Level:1

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

Triangle Midpoints

KS 4 Challenge Level:

Challenge Level:1

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

Multiplication Arithmagons

KS 4 Challenge Level:

Challenge Level:1

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

Mathsland National Lottery

KS 4 Challenge Level:

Challenge Level:1

Can you work out the probability of winning the Mathsland National Lottery? Try our simulator to test out your ideas.

Vector Journeys

KS 4 Challenge Level:

Challenge Level:1

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?

Curvy Areas

KS 4 Challenge Level:

Challenge Level:1

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

How Old Am I?

KS 4 Challenge Level:

Challenge Level:1

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?

A Little Light Thinking

KS 4 Challenge Level:

Challenge Level:1

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

Circles in Quadrilaterals

KS 4 Challenge Level:

Challenge Level:1

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

Beelines

KS 4 Challenge Level:

Challenge Level:1

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

What's Possible?

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Odds and Evens Made Fair

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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.

Take Three from Five

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Trapezium Four

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Arclets

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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".

Partly Painted Cube

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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?

Triangles and Petals

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

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

Where to Land

KS 4 Challenge Level:

Challenge Level:2 Challenge Level:2

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Fit for Photocopying

KS 4 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Explore the relationships between different paper sizes.

Hexy-metry

KS 4 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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

Which Spinners?

KS 4 & 5 Challenge Level:

Challenge Level:1

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

Three by One

KS 5 Challenge Level:

Challenge Level:1

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

Data Matchinglive

KS 5 Challenge Level:

Challenge Level:1

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