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?
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.
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
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.
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?
Special clue numbers related to the difference between numbers in two adjacent cells and values of the stars in the "constellation" make this a doubly interesting problem.
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?
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.
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. . . .
Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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?
I added together some of my neighbours house numbers. Can you explain the patterns I noticed?
This Sudoku combines all four arithmetic operations.
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?
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
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.
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?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
A Sudoku with clues as ratios.
A Sudoku with a twist.
Four small numbers give the clue to the contents of the four surrounding cells.
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?
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
A Sudoku that uses transformations as supporting clues.
This sudoku requires you to have "double vision" - two Sudoku's for the price of one
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.
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.
How many different symmetrical shapes can you make by shading triangles or squares?
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?
Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?
Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!
This Sudoku requires you to do some working backwards before working forwards.
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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.
A Sudoku with a twist.
This Sudoku, based on differences. Using the one clue number can you find the solution?
Solve the equations to identify the clue numbers in this Sudoku problem.
The challenge is to find the values of the variables if you are to solve this Sudoku.
Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.
A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"