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?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

This challenge is about finding the difference between numbers which have the same tens digit.

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 find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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?

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?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

An investigation that gives you the opportunity to make and justify predictions.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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?

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

This task follows on from Build it Up and takes the ideas into three dimensions!

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

What happens when you round these three-digit numbers to the nearest 100?

Find out what a "fault-free" rectangle is and try to make some of your own.

Find the sum of all three-digit numbers each of whose digits is odd.

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?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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?

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?

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 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 problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

This challenge asks you to imagine a snake coiling on itself.

What happens when you round these numbers to the nearest whole number?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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?

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?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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.

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.

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?