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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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?

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.

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.

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.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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?

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.

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?

Delight your friends with this cunning trick! Can you explain how it works?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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?

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

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

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

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.

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

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

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?

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.

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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?

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Can you find all the ways to get 15 at the top of this triangle of numbers?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

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

It would be nice to have a strategy for disentangling any tangled ropes...

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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