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?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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

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

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

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.

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

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

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?

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

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.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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.

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

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

Can all unit fractions be written as the sum of two unit fractions?

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.

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

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?

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 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?

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

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

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

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Can you find sets of sloping lines that enclose a square?

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

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

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

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?

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.

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

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.

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?

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

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

Can you describe this route to infinity? Where will the arrows take you next?