Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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 continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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 investigation that gives you the opportunity to make and justify predictions.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
How many centimetres of rope will I need to make another mat just like the one I have here?
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 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.
Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
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 ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
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?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
What happens when you round these numbers to the nearest whole number?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
What happens when you round these three-digit numbers to the nearest 100?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Delight your friends with this cunning trick! Can you explain how it works?
This task follows on from Build it Up and takes the ideas into three dimensions!
Find the sum of all three-digit numbers each of whose digits is odd.
Find out what a "fault-free" rectangle is and try to make some of your own.
Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?
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?
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?
Are these statements always true, sometimes true or never true?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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?
This activity involves rounding four-digit numbers to the nearest thousand.
This challenge asks you to imagine a snake coiling on itself.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
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?
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?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?
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?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?