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?
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find 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.
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?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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?
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?
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.
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
How many different symmetrical shapes can you make by shading triangles or squares?
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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?
Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E
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?
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 it up?
A Sudoku with a twist.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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.
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?
Solve the equations to identify the clue numbers in this Sudoku problem.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
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?
You need to find the values of the stars before you can apply normal Sudoku rules.
A Sudoku with a twist.
Find out about Magic Squares in this article written for students. Why are they magic?!
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.
If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Investigate the different ways that fifteen schools could have given money in a charity fundraiser.
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
This sudoku requires you to have "double vision" - two Sudoku's for the price of one
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?
A Sudoku with clues given as sums of entries.
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?