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.
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.
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.
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 moves.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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 collection of games on the NIM theme
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. . . .
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?”
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.
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
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.
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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. . . .
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.
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.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
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 NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Are these statements relating to odd and even numbers 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. . . .
Here are two kinds of spirals for you to explore. What do you notice?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
Can all unit fractions be written as the sum of two unit fractions?
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
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?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Can you describe this route to infinity? Where will the arrows take you next?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
This challenge asks you to imagine a snake coiling on itself.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
An investigation that gives you the opportunity to make and justify predictions.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?