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?
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.
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.
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?
This challenge asks you to imagine a snake coiling on itself.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Delight your friends with this cunning trick! Can you explain how
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?
Here are two kinds of spirals for you to explore. What do you notice?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
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 ...?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you explain how this card trick works?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
This activity involves rounding four-digit numbers to the nearest thousand.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
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?
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?
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.
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.
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. . . .
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?
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.
Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you tangle yourself up and reach any fraction?
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. . . .
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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
Find out what a "fault-free" rectangle is and try to make some of
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written