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?

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.

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

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

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

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

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

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?”

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.

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

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.

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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?

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. . . .

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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.

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

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

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

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?

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

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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

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

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?

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

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

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?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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

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

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

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

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

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

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.

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. . . .

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

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

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. . . .

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.

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.

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.