Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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 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?
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?
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?
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
Can you work out how to win this game of Nim? Does it matter if you
go first or second?
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?
Delight your friends with this cunning trick! Can you explain how
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Find out what a "fault-free" rectangle is and try to make some of
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Can you explain how this card trick works?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
This challenge asks you to imagine a snake coiling on itself.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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 many moves does it take to swap over some red and blue frogs? Do you have a method?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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.
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
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
Find the sum of all three-digit numbers each of whose digits is
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?
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?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?
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?
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?
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
Can you find the values at the vertices when you know the values on
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?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
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.
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. . . .
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.