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?”
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 game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
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.
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.
A collection of games on the NIM theme
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
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.
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.
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
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?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
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?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Here are two kinds of spirals for you to explore. What do you notice?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
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. . . .
Are these statements always true, sometimes true or never true?
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .
Can you describe this route to infinity? Where will the arrows take you next?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
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.
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
This challenge asks you to imagine a snake coiling on itself.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
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?
This activity involves rounding four-digit numbers to the nearest thousand.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
An investigation that gives you the opportunity to make and justify
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?