Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Find out what a "fault-free" rectangle is and try to make some of your own.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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 moves does it take to swap over some red and blue frogs? Do you have a method?
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?
Can you explain how this card trick works?
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 challenge asks you to imagine a snake coiling on itself.
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.
Delight your friends with this cunning trick! Can you explain how it works?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
Find the sum of all three-digit numbers each of whose digits is odd.
Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Can you find the values at the vertices when you know the values on the edges?
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
A package contains a set of resources designed to develop pupils’ mathematical thinking. This package places a particular emphasis on “generalising” and is designed to meet the. . . .
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Can you find sets of sloping lines that enclose a square?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.
A collection of games on the NIM theme
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
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?