List

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.
If the World Were a Village
problem
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If the World Were a Village

Age
5 to 11
Challenge level
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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?

Magnifying glass
problem
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Number Detective

Age
5 to 11
Challenge level
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Follow the clues to find the mystery number.

Five Steps to 50
problem
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Five Steps to 50

Age
5 to 7
Challenge level
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Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?
Next-door Numbers
problem

Next-door Numbers

Age
5 to 7
Challenge level
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Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?
Eightness of Eight
problem
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Eightness of Eight

Age
5 to 7
Challenge level
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What do you see as you watch this video? Can you create a similar video for the number 12?
Digit Addition
problem
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Digit Addition

Age
5 to 11
Challenge level
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Try out this number trick. What happens with different starting numbers? What do you notice?

Shaping It
problem
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Shaping It

Age
5 to 11
Challenge level
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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.
Ring a Ring of Numbers
problem
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Ring a Ring of Numbers

Age
5 to 7
Challenge level
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Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.
Colouring Triangles
problem
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Colouring Triangles

Age
5 to 7
Challenge level
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Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
Chain of Changes
problem
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Chain of Changes

Age
5 to 7
Challenge level
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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?
Little Man
problem
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Little Man

Age
5 to 7
Challenge level
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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?
Light the Lights
problem
Favourite

Light the Lights

Age
5 to 7
Challenge level
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Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

More Numbers in the Ring
problem
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More Numbers in the Ring

Age
5 to 7
Challenge level
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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?
Consecutive Numbers
problem
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Consecutive Numbers

Age
7 to 14
Challenge level
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An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Number Differences
problem
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Number Differences

Age
7 to 11
Challenge level
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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?
Subtraction Surprise
problem
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Subtraction Surprise

Age
7 to 14
Challenge level
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Try out some calculations. Are you surprised by the results?
Brush Loads
problem
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Brush Loads

Age
7 to 11
Challenge level
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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.

Statement Snap
problem
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Statement Snap

Age
7 to 14
Challenge level
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You'll need to know your number properties to win a game of Statement Snap...
Tumbling Down
problem
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Tumbling Down

Age
7 to 11
Challenge level
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Watch this animation. What do you see? Can you explain why this happens?

Triangle transformation
problem

Triangle transformation

Age
7 to 14
Challenge level
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Start with a triangle. Can you cut it up to make a rectangle?
Fruity Totals
problem
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Fruity Totals

Age
7 to 16
Challenge level
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In this interactivity each fruit has a hidden value. Can you deduce what each one is worth?
The Number Jumbler
problem
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The Number Jumbler

Age
7 to 14
Challenge level
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The Number Jumbler can always work out your chosen symbol. Can you work out how?
Nice or Nasty
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Nice or Nasty

Age
7 to 14
Challenge level
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There are nasty versions of this dice game but we'll start with the nice ones...
Your number is...
problem
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Your number is...

Age
7 to 14
Challenge level
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Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
Three neighbours
problem
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Three neighbours

Age
7 to 14
Challenge level
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Take three consecutive numbers and add them together. What do you notice?
Two Clocks
problem
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Two Clocks

Age
7 to 11
Challenge level
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These clocks have only one hand, but can you work out what time they are showing from the information?

Pouring Problem
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Pouring Problem

Age
7 to 11
Challenge level
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What do you think is going to happen in this video clip? Are you surprised?

Curious number
problem
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Curious number

Age
7 to 11
Challenge level
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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?

Summing Consecutive Numbers
problem
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Summing Consecutive Numbers

Age
11 to 14
Challenge level
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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?

Perimeter Possibilities
problem
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Perimeter Possibilities

Age
11 to 14
Challenge level
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I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Semi-regular Tessellations
problem
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Semi-regular Tessellations

Age
11 to 16
Challenge level
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Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Blue and White
problem
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Blue and White

Age
11 to 14
Challenge level
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Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Satisfying Statements
problem

Satisfying Statements

Age
11 to 14
Challenge level
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Can you find any two-digit numbers that satisfy all of these statements?
What's it worth?
problem
Favourite

What's it worth?

Age
11 to 16
Challenge level
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There are lots of different methods to find out what the shapes are worth - how many can you find?
Largest product
problem
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Largest product

Age
11 to 14
Challenge level
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Which set of numbers that add to 100 have the largest product?
Elevenses
problem
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Elevenses

Age
11 to 14
Challenge level
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How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can they be equal?
problem
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Can they be equal?

Age
11 to 14
Challenge level
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Can you find rectangles where the value of the area is the same as the value of the perimeter?
Your number was...
problem
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Your number was...

Age
11 to 14
Challenge level
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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?
Special Numbers
problem
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Special Numbers

Age
11 to 14
Challenge level
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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?
Number Pyramids
problem
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Number Pyramids

Age
11 to 14
Challenge level
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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

How much can we spend?
problem
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How much can we spend?

Age
11 to 14
Challenge level
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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?
Shifting Times Tables
problem
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Shifting Times Tables

Age
11 to 14
Challenge level
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Can you find a way to identify times tables after they have been shifted up or down?

Tilted Squares
problem
Favourite

Tilted Squares

Age
11 to 14
Challenge level
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It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Charlie's delightful machine
problem
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Charlie's delightful machine

Age
11 to 16
Challenge level
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Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

What numbers can we make?
problem
Favourite

What numbers can we make?

Age
11 to 14
Challenge level
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Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Dicey Operations
game
Favourite

Dicey Operations

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

Arithmagons
problem
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Arithmagons

Age
11 to 16
Challenge level
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Can you find the values at the vertices when you know the values on the edges?

Non-Transitive Dice
problem
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Non-Transitive Dice

Age
11 to 14
Challenge level
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Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?
Right angles
problem
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Right angles

Age
11 to 14
Challenge level
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Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
What numbers can we make now?
problem
Favourite

What numbers can we make now?

Age
11 to 14
Challenge level
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Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Two's company
problem
Favourite

Two's company

Age
11 to 14
Challenge level
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Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?
Cosy corner
problem
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Cosy corner

Age
11 to 14
Challenge level
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Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?
Estimating time
problem
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Estimating time

Age
11 to 14
Challenge level
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How well can you estimate 10 seconds? Investigate with our timing tool.
Wipeout
problem
Favourite

Wipeout

Age
11 to 16
Challenge level
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Can you do a little mathematical detective work to figure out which number has been wiped out?
Reversals
problem
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Reversals

Age
11 to 14
Challenge level
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Where should you start, if you want to finish back where you started?
Unequal Averages
problem
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Unequal Averages

Age
11 to 14
Challenge level
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Play around with sets of five numbers and see what you can discover about different types of average...
Marbles in a box
problem
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Marbles in a box

Age
11 to 16
Challenge level
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How many winning lines can you make in a three-dimensional version of noughts and crosses?
Think of Two Numbers
problem
Favourite

Think of Two Numbers

Age
11 to 14
Challenge level
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Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Take Three From Five
problem
Favourite

Take Three From Five

Age
11 to 16
Challenge level
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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?

Searching for mean(ing)
problem
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Searching for mean(ing)

Age
11 to 16
Challenge level
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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?
Cuboid challenge
problem
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Cuboid Challenge

Age
11 to 16
Challenge level
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What's the largest volume of box you can make from a square of paper?

More Number Pyramids
problem
Favourite

More Number Pyramids

Age
11 to 14
Challenge level
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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

On the Edge
problem
Favourite

On the Edge

Age
11 to 14
Challenge level
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If you move the tiles around, can you make squares with different coloured edges?
Who's the best?
problem
Favourite

Who's the best?

Age
11 to 14
Challenge level
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Which countries have the most naturally athletic populations?
Sending a Parcel
problem
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Sending a Parcel

Age
11 to 14
Challenge level
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What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Opposite vertices
problem
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Opposite vertices

Age
11 to 14
Challenge level
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Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Square coordinates
problem
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Square coordinates

Age
11 to 14
Challenge level
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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?

Stars
problem
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Stars

Age
11 to 14
Challenge level
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Can you work out what step size to take to ensure you visit all the dots on the circle?
Litov's Mean Value Theorem
problem
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Litov's Mean Value Theorem

Age
11 to 14
Challenge level
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Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
A Chance to Win?
problem
Favourite

A Chance to Win?

Age
11 to 14
Challenge level
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Imagine you were given the chance to win some money... and imagine you had nothing to lose...
Cola Can
problem
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Cola Can

Age
11 to 14
Challenge level
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An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?
Which solids can we make?
problem
Favourite

Which solids can we make?

Age
11 to 14
Challenge level
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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?
How old am I?
problem
Favourite

How old am I?

Age
14 to 16
Challenge level
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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?
Beelines
problem
Favourite

Beelines

Age
14 to 16
Challenge level
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Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Data matching
problem

Data matching

Age
14 to 18
Challenge level
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Use your skill and judgement to match the sets of random data.
Which spinners?
problem
Favourite

Which spinners?

Age
14 to 18
Challenge level
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Can you work out which spinners were used to generate the frequency charts?
Curve Hunter
problem

Curve Hunter

Age
14 to 18
Challenge level
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This problem challenges you to sketch curves with different properties.

Curvy areas
problem
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Curvy areas

Age
14 to 16
Challenge level
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Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Pair Products
problem
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Pair Products

Age
14 to 16
Challenge level
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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
circles in quadrilaterals
problem
Favourite

Circles in quadrilaterals

Age
14 to 16
Challenge level
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Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Last one standing
problem
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Last one standing

Age
14 to 16
Challenge level
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Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?
A little light thinking
problem
Favourite

A little light thinking

Age
14 to 16
Challenge level
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Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Vector journeys
problem
Favourite

Vector journeys

Age
14 to 18
Challenge level
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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?
Triangle midpoints
problem
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Triangle midpoints

Age
14 to 16
Challenge level
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You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Odds and Evens made fair
problem
Favourite

Odds and Evens made fair

Age
14 to 16
Challenge level
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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.
What's Possible?
problem
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What's Possible?

Age
14 to 16
Challenge level
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Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Arclets
problem
Favourite

Arclets

Age
14 to 16
Challenge level
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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".
Trapezium Four
problem
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Trapezium Four

Age
14 to 16
Challenge level
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The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
Pick's Theorem
problem
Favourite

Pick's Theorem

Age
14 to 16
Challenge level
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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.
Triangles and petals
problem
Favourite

Triangles and petals

Age
14 to 16
Challenge level
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An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?
Same Number!
problem
Favourite

Same Number!

Age
14 to 16
Challenge level
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If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?
Partly Painted Cube
problem
Favourite

Partly Painted Cube

Age
14 to 16
Challenge level
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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?
Where to Land
problem
Favourite

Where to Land

Age
14 to 16
Challenge level
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Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?
Multiplication arithmagons
problem
Favourite

Multiplication arithmagons

Age
14 to 16
Challenge level
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Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Hexy-Metry
problem
Favourite

Hexy-Metry

Age
14 to 16
Challenge level
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A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
Mathsland National Lottery
problem
Favourite

Mathsland National Lottery

Age
14 to 16
Challenge level
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Can you work out the probability of winning the Mathsland National Lottery?
Fit for photocopying
problem
Favourite

Fit for photocopying

Age
14 to 16
Challenge level
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Explore the relationships between different paper sizes.
Three by One
problem
Favourite

Three by One

Age
16 to 18
Challenge level
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There are many different methods to solve this geometrical problem - how many can you find?
Rational request
problem

Rational request

Age
16 to 18
Challenge level
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Can you make a curve to match my friend's requirements?