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

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?

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

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?

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

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.

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

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

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.

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?

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

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?

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

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

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

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

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?

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

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

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

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

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

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?

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

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

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?

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

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. . . .

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 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?

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

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?

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

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six 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?

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

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

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

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

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

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

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.

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.

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.