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.





Next-door numbers

Eightness of eight

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


Shaping it

If the world were a village
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?

Five steps to 50

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

Ring a ring of numbers

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

Chain of changes
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

More numbers in the ring


Dicey operations
In these addition and subtraction games, you'll need to think strategically to get closest to the target.


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

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

Number differences

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

More Dicey operations
In these multiplication and division games, you'll need to think strategically to get closest to the target.

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.

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


Your number is...

Statement snap


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


Curious number
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?

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?


Your number was...

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?

Semi-regular tessellations
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

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?

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


Elevenses

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

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

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

How much can we spend?

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

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

Satisfying statements

Square coordinates
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

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

Two's company

Cosy corner


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

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

Think of two numbers

Non-transitive dice

Take three from five
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)

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



Wipeout

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


Unequal averages

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

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

Which solids can we make?
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?

Litov's mean value theorem

A chance to win?

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


Last one standing

How old am I?
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

Vector journeys



Curvy areas

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

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

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

Where to land

Same number!

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

What's possible?

Arclets

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

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

Pick's theorem

Odds and evens made fair

Triangles and petals

Partly painted cube

Mathsland national lottery


Hexy-metry

Three by one
