How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
A challenging activity focusing on finding all possible ways of stacking rods.
How many different symmetrical shapes can you make by shading triangles or squares?
You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Systematically explore the range of symmetric designs that can be
created by shading parts of the motif below. Use normal square
lattice paper to record your results.
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
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?
Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?
Four small numbers give the clue to the contents of the four
Can you use your powers of logic and deduction to work out the missing information in these sporty situations?
Use the clues about the symmetrical properties of these letters to
place them on the grid.
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?
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?
This Sudoku, based on differences. Using the one clue number can you find the solution?
Use the differences to find the solution to this Sudoku.
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
A few extra challenges set by some young NRICH members.
Each clue number in this sudoku is the product of the two numbers in adjacent cells.
This Sudoku combines all four arithmetic operations.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Pentagram Pylons - can you elegantly recreate them? Or, the
European flag in LOGO - what poses the greater problem?
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.
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Remember that you want someone following behind you to see where
you went. Can yo work out how these patterns were created and
In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.
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.
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.
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 task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
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?
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
Can you find which shapes you need to put into the grid to make the
totals at the end of each row and the bottom of each column?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Each of the main diagonals of this sudoku must contain the numbers
1 to 9 and each rectangle width the numbers 1 to 4.
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.
Can you find all the different triangles on these peg boards, and
find their angles?
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?
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
Two sudokus in one. Challenge yourself to make the necessary
Can you draw a square in which the perimeter is numerically equal
to the area?
A Sudoku with clues as ratios.
A Sudoku with a twist.
Rather than using the numbers 1-9, this sudoku uses the nine
different letters used to make the words "Advent Calendar".