Can you find all the different triangles on these peg boards, and
find their angles?
How many different triangles can you make on a circular pegboard that has nine pegs?
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
Is it possible to place 2 counters on the 3 by 3 grid so that there
is an even number of counters in every row and every column? How
about if you have 3 counters or 4 counters or....?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
A tetromino is made up of four squares joined edge to edge. Can
this tetromino, together with 15 copies of itself, be used to cover
an eight by eight chessboard?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
In how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
Using different numbers of sticks, how many different triangles are
you able to make? Can you make any rules about the numbers of
sticks that make the most triangles?
Find out what a "fault-free" rectangle is and try to make some of
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
How many triangles can you make on the 3 by 3 pegboard?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Can you find all the different ways of lining up these Cuisenaire
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
What is the best way to shunt these carriages so that each train
can continue its journey?
Can you shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
10 space travellers are waiting to board their spaceships. There
are two rows of seats in the waiting room. Using the rules, where
are they all sitting? Can you find all the possible ways?
Mr McGregor has a magic potting shed. Overnight, the number of
plants in it doubles. He'd like to put the same number of plants in
each of three gardens, planting one garden each day. Can he do it?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
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?
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.
This problem is based on a code using two different prime numbers
less than 10. You'll need to multiply them together and shift the
alphabet forwards by the result. Can you decipher the code?
An activity making various patterns with 2 x 1 rectangular tiles.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
In this matching game, you have to decide how long different events take.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
These practical challenges are all about making a 'tray' and covering it with paper.
This challenge extends the Plants investigation so now four or more children are involved.
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?