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.

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?

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

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

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

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

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. . . .

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

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.

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

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?

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

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. . . .

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

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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

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

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.

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

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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?

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.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

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

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

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

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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

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?

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

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

What happens when you round these three-digit numbers to the nearest 100?

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

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?

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?

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

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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.

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?