It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
Can all unit fractions be written as the sum of two unit fractions?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Can you see how to build a harmonic triangle? Can you work out the next two rows?
This challenge asks you to imagine a snake coiling on itself.
Are these statements always true, sometimes true or never true?
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.
Here are two kinds of spirals for you to explore. What do you notice?
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?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Find out what a "fault-free" rectangle is and try to make some of
How many centimetres of rope will I need to make another mat just
like the one I have here?
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 you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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
An investigation that gives you the opportunity to make and justify
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?
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
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.
A collection of games on the NIM theme
Can you find the values at the vertices when you know the values on
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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
What happens when you round these three-digit numbers to the nearest 100?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
What happens when you round these numbers to the nearest whole number?
This activity involves rounding four-digit numbers to the nearest thousand.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.