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.

Can you explain the strategy for winning this game with any target?

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

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?

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

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

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?

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 work out how to win this game of Nim? Does it matter if you go first or second?

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

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

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

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

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

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

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

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.

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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.

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

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?

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

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 and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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

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?

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?

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.

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

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 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.

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

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

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.

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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

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

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 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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

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