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?
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....?
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?
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?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
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?
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?
How many triangles can you make on the 3 by 3 pegboard?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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?
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?
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?
Can you find all the different ways of lining up these Cuisenaire
Can you put the 25 coloured tiles into the 5 x 5 square so that no
column, no row and no diagonal line have tiles of the same colour
Find out what a "fault-free" rectangle is and try to make some of
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
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?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
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.
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
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 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?
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?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
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.
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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?
Arrange the four number cards on the grid, according to the rules,
to make a diagonal, vertical or horizontal line.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Try out the lottery that is played in a far-away land. What is the
chance of winning?
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.
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.
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other 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?