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

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

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?

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.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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.

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?

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?

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?

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?

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

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

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

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

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?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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

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?

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

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

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

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

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

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?

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?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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?

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

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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

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

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

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

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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

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?

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

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

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

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

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?

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?

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