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.
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?
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?
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?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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?
Find out what a "fault-free" rectangle is and try to make some of
How many different symmetrical shapes can you make by shading triangles or squares?
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.
How many models can you find which obey these rules?
A Sudoku with a twist.
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?
A Sudoku with clues as ratios.
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you find all the different ways of lining up these Cuisenaire
Solve this Sudoku puzzle whose clues are in the form of sums of the
numbers which should appear in diagonal opposite cells.
Use the interactivity to find all the different right-angled
triangles you can make by just moving one corner of the starting
Two sudokus in one. Challenge yourself to make the necessary
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?
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....?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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
My two digit number is special because adding the sum of its digits
to the product of its digits gives me my original number. What
could my number be?
Here are four cubes joined together. How many other arrangements of
four cubes can you find? Can you draw them on dotty paper?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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?
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?
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
A Sudoku with clues as ratios or fractions.
A Sudoku that uses transformations as supporting clues.
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
Given the products of diagonally opposite cells - can you complete
Arrange the four number cards on the grid, according to the rules,
to make a diagonal, vertical or horizontal line.
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Can you find all the different triangles on these peg boards, and
find their angles?
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?