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. . . .

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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

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.

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

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.

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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. . . .

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

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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

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.

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?

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 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.

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

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

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

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.

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

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

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

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

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.

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.

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. . . .

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

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.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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.

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?

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

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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?

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.

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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.

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

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

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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.

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?

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