Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

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?

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 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.

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

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

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

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

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?

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

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

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Can you find a way to identify times tables after they have been shifted up?

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

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

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

Given the products of adjacent cells, can you complete this Sudoku?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

Can you find any two-digit numbers that satisfy all of these statements?

Given the products of diagonally opposite cells - can you complete this Sudoku?

Is there an efficient way to work out how many factors a large number has?

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

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.

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Find the highest power of 11 that will divide into 1000! exactly.

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Benâ€™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?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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.

The clues for this Sudoku are the product of the numbers in adjacent squares.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?