In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
Are these statements always true, sometimes true or never true?
Got It game for an adult and child. How can you play so that you know you will always win?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Find the sum of all three-digit numbers each of whose digits is
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
This activity involves rounding four-digit numbers to the nearest thousand.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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.
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
What happens when you round these three-digit numbers to the nearest 100?
Can you find all the ways to get 15 at the top of this triangle of numbers?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
This challenge asks you to imagine a snake coiling on itself.
This task follows on from Build it Up and takes the ideas into three dimensions!
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
An investigation that gives you the opportunity to make and justify
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
Here are two kinds of spirals for you to explore. What do you notice?
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
A collection of games on the NIM theme
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?
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.