Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Find out what a "fault-free" rectangle is and try to make some of
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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?
Delight your friends with this cunning trick! Can you explain how
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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
Got It game for an adult and child. How can you play so that you know you will always win?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
Find the sum of all three-digit numbers each of whose digits is
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
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?
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 many centimetres of rope will I need to make another mat just
like the one I have here?
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.
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.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?