Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Here are two kinds of spirals for you to explore. What do you notice?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
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?
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
This challenge asks you to imagine a snake coiling on itself.
Got It game for an adult and child. How can you play so that you know you will always win?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
This task follows on from Build it Up and takes the ideas into three dimensions!
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Find out what a "fault-free" rectangle is and try to make some of your own.
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?
A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
Find the sum of all three-digit numbers each of whose digits is odd.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?
This is a game for two players. Can you find out how to be the first to get to 12 o'clock?
How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
This challenge is about finding the difference between numbers which have the same tens digit.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
This activity involves rounding four-digit numbers to the nearest thousand.
Stop the Clock game for an adult and child. How can you make sure you always win this game?
What happens when you round these three-digit numbers to the nearest 100?
An investigation that gives you the opportunity to make and justify predictions.
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?
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.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?
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 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.