Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
This challenge asks you to imagine a snake coiling on itself.
Here are two kinds of spirals for you to explore. What do you notice?
Delight your friends with this cunning trick! Can you explain how it works?
Can you explain how this card trick works?
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?
Can you explain the strategy for winning this game with any target?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
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.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
This activity involves rounding four-digit numbers to the nearest thousand.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
This task follows on from Build it Up and takes the ideas into three dimensions!
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
Can you find all the ways to get 15 at the top of this triangle of numbers?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
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.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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.
An investigation that gives you the opportunity to make and justify predictions.
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?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
It would be nice to have a strategy for disentangling any tangled ropes...