Are these statements always true, sometimes true or never true?
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?
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.
Got It game for an adult and child. How can you play so that you know you will always win?
Ben’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?
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.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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 explain the strategy for winning this game with any target?
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
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?
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?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
An investigation that gives you the opportunity to make and justify predictions.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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?
This activity involves rounding four-digit numbers to the nearest thousand.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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?
This task follows on from Build it Up and takes the ideas into three dimensions!
This challenge asks you to imagine a snake coiling on itself.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
Can you find all the ways to get 15 at the top of this triangle of numbers?
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. . . .
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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.
Here are two kinds of spirals for you to explore. What do you notice?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?