Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Delight your friends with this cunning trick! Can you explain how it works?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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?
What happens when you round these numbers to the nearest whole number?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
What happens when you round these three-digit numbers to the nearest 100?
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?
Can you explain how this card trick works?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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.
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
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.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
How many centimetres of rope will I need to make another mat just like the one I have here?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
Find the sum of all three-digit numbers each of whose digits is odd.
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?
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?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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?
Find out what a "fault-free" rectangle is and try to make some of your own.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
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?
Can you explain the strategy for winning this game with any target?
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.
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
This task follows on from Build it Up and takes the ideas into three dimensions!