To avoid losing think of another very well known game where the patterns of play are similar.
A game for 2 players
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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
A game for 2 players with similarities 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 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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
Can all unit fractions be written as the sum of two unit fractions?
Can you find sets of sloping lines that enclose a square?
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Make some loops out of regular hexagons. What rules can you discover?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
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?
Can you use the diagram to prove the AM-GM inequality?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you find the values at the vertices when you know the values on the edges?
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.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
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.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .
A collection of games on the NIM theme
Can you explain the strategy for winning this game with any target?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
It starts quite simple but great opportunities for number discoveries and patterns!
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
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.
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Got It game for an adult and child. How can you play so that you know you will always win?
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
Can you describe this route to infinity? Where will the arrows take you next?