How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
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.
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?
Explore the effect of reflecting in two intersecting mirror lines.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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. . . .
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Delight your friends with this cunning trick! Can you explain how
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
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.
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?
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
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Can you explain how this card trick works?
Explore the effect of reflecting in two parallel mirror lines.
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Can you tangle yourself up and reach any fraction?
This challenge focuses on finding the sum and difference of pairs of two-digit 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?
Can you find the values at the vertices when you know the values on
A collection of games on the NIM theme
It would be nice to have a strategy for disentangling any tangled
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.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
This task follows on from Build it Up and takes the ideas into three dimensions!
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Can you find all the ways to get 15 at the top of this triangle of numbers?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Find out what a "fault-free" rectangle is and try to make some of
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?