Are these statements always true, sometimes true or never true?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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?

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

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

Can you explain the strategy for winning this game with any target?

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

Got It game for an adult and child. How can you play so that you know you will always win?

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?

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

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

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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

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.

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.

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?

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 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.

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

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

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

Make some loops out of regular hexagons. What rules can you discover?

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

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

This challenge asks you to imagine a snake coiling on itself.

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

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

Are these statements always true, sometimes true or never true?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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.

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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

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?

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.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

A game for 2 players with similarities 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.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

What's the largest volume of box you can make from a square of paper?

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

Find the sum of all three-digit numbers each of whose digits is odd.