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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
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?
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?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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.
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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
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
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
This challenge asks you to imagine a snake coiling on itself.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
An investigation that gives you the opportunity to make and justify
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.
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?
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. . . .
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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. . . .
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 challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Got It game for an adult and child. How can you play so that you know you will always win?
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?
What happens when you round these three-digit numbers to the nearest 100?
Find the sum of all three-digit numbers each of whose digits is
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 activity involves rounding four-digit numbers to the nearest thousand.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
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.
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?