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?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?
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.
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 maths trails.
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?
Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
A Sudoku with clues as ratios.
This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.
in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?
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 Sudoku with clues as ratios.
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
In this game you are challenged to gain more columns of lily pads than your opponent.
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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.
A Sudoku with a twist.
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
A few extra challenges set by some young NRICH members.
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.
You need to find the values of the stars before you can apply normal Sudoku rules.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A Sudoku with clues as ratios or fractions.
Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".
Two sudokus in one. Challenge yourself to make the necessary connections.
Find out about Magic Squares in this article written for students. Why are they magic?!
A Sudoku that uses transformations as supporting clues.
This Sudoku combines all four arithmetic operations.
The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .
It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
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?
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.
You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
A Sudoku with clues given as sums of entries.
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?
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.
Solve the equations to identify the clue numbers in this Sudoku problem.
This Sudoku requires you to do some working backwards before working forwards.
Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
How many different symmetrical shapes can you make by shading triangles or squares?