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

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?

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

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

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.

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

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?

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

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.

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

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.

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

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

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?

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 ...?

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

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

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?

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?

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

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

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

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

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?

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?

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?

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

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?

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?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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. . . .

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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.

What happens when you round these numbers to the nearest whole number?

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

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?

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

Delight your friends with this cunning trick! Can you explain how it works?