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?

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

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?

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

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

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?

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

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

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

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.

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

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 numbers from a sequence. What numbers can we make now?

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?

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

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

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?

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

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.

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

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. . . .

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

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?

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?

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

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?

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, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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

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.

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

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?

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.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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

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

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?

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

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

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?

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?

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

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