Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Delight your friends with this cunning trick! Can you explain how
Find out what a "fault-free" rectangle is and try to make some of
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Explore the effect of reflecting in two intersecting mirror lines.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Got It game for an adult and child. How can you play so that you know you will always win?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
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?
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.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
This challenge asks you to imagine a snake coiling on itself.
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
A collection of games on the NIM theme
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.