Are these statements relating to odd and even numbers always true, sometimes true or never true?

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?

Here are two kinds of spirals for you to explore. What do you notice?

Are these statements always true, sometimes true or never true?

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

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

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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?

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?

Got It game for an adult and child. How can you play so that you know you will always win?

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.

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?

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

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.

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.

Nim-7 game for an adult and child. Who will be the one to take the last counter?

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

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.

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?

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.

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

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

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

This activity involves rounding four-digit numbers to the nearest thousand.

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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.

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

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

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?

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

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.

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

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?

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?

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

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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?

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

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?

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?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?