Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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?

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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 loser is the player who takes the last counter.

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.

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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?

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

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

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. . . .

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.

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.

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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.

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

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

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.

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

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

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

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

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

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.

To avoid losing think of another very well known game where the patterns of play are similar.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

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

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. . . .

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

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.

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. . . .

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

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

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. . . .

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?

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?