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?
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?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
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?
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.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E
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.
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.
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?
Find the values of the nine letters in the sum: FOOT + BALL = GAME
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
Find out about Magic Squares in this article written for students. Why are they magic?!
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
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?
Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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 second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
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.
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?
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
The clues for this Sudoku are the product of the numbers in adjacent squares.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
You need to find the values of the stars before you can apply normal Sudoku rules.
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.
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.
An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.
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 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?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
This sudoku requires you to have "double vision" - two Sudoku's for the price of one
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.
A pair of Sudokus with lots in common. In fact they are the same problem but rearranged. Can you find how they relate to solve them both?
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?
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?
Have a go at this game which has been inspired by the Big Internet Math-Off 2019. Can you gain more columns of lily pads than your opponent?
A Sudoku with a twist.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?