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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
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.
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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?
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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Delight your friends with this cunning trick! Can you explain how
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
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.
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.
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
A game for 2 players
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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.
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
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 NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Can you explain how this card trick works?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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?
Great Granddad is very proud of his telegram from the Queen
congratulating him on his hundredth birthday and he has friends who
are even older than he is... When was he born?
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Can you describe this route to infinity? Where will the arrows take you next?