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 relating to odd and even numbers always true, sometimes true or never true?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Here are two kinds of spirals for you to explore. What do you notice?
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?
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?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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?
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.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
Are these statements always true, sometimes true or never true?
This challenge asks you to imagine a snake coiling on itself.
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.
This activity involves rounding four-digit numbers to the nearest thousand.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Delight your friends with this cunning trick! Can you explain how
Got It game for an adult and child. How can you play so that you know you will always win?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
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?
This task follows on from Build it Up and takes the ideas into three dimensions!
An investigation that gives you the opportunity to make and justify
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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.
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
Can you explain how this card trick works?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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. . . .
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, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Find the sum of all three-digit numbers each of whose digits is
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
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?
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?
How many centimetres of rope will I need to make another mat just
like the one I have here?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?