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.

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

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

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 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?

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

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

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.

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

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?

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

Can you find the values at the vertices when you know the values on the edges?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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

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

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?

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

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?

How many centimetres of rope will I need to make another mat just like the one I have here?

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?

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

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

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

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

It starts quite simple but great opportunities for number discoveries and patterns!

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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.

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?

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.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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?

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.

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

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

A package contains a set of resources designed to develop pupils’ mathematical thinking. This package places a particular emphasis on “generalising” and is designed to meet the. . . .

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Charlie has moved between countries and the average income of both has increased. How can this be so?