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?
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 DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
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 different triangles can you make on a circular pegboard that has nine pegs?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
Can you find all the different ways of lining up these Cuisenaire
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
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?
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....?
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
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.
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 extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
Can you recreate these designs? What are the basic units? What
movement is required between each unit? Some elegant use of
procedures will help - variables not essential.
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
Find out what a "fault-free" rectangle is and try to make some of
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
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?
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
A few extra challenges set by some young NRICH members.
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Find all the different shapes that can be made by joining five
equilateral triangles edge to edge.
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.
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?
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?
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Can you find all the different triangles on these peg boards, and
find their angles?
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?
Two sudokus in one. Challenge yourself to make the necessary
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 in them?
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
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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.
There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.
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?
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?
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?