Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Find out what a "fault-free" rectangle is and try to make some of
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Delight your friends with this cunning trick! Can you explain how
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you explain how this card trick works?
Can you find the values at the vertices when you know the values on
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
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?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
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. . . .
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
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. . . .
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?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Can you tangle yourself up and reach any fraction?
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?
It would be nice to have a strategy for disentangling any tangled
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. . . .