Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Find out what a "fault-free" rectangle is and try to make some of your own.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
What happens when you round these three-digit numbers to the nearest 100?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Delight your friends with this cunning trick! Can you explain how it works?
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?
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.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
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 could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
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.
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.
It starts quite simple but great opportunities for number discoveries and patterns!
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?
Can you explain the strategy for winning this game with any target?
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you find all the ways to get 15 at the top of this triangle of numbers?
Find the sum of all three-digit numbers each of whose digits is odd.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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?
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?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
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.
Here are two kinds of spirals for you to explore. What do you notice?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.