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 NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A game for 2 players
Can you explain the strategy for winning this game with any 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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
To avoid losing think of another very well known game where the patterns of play are similar.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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.
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.
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.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Delight your friends with this cunning trick! Can you explain how it works?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Can you explain how this card trick works?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
It would be nice to have a strategy for disentangling any tangled ropes...
Can you find sets of sloping lines that enclose a square?
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?
Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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 Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
Can all unit fractions be written as the sum of two unit fractions?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Can you find the values at the vertices when you know the values on the edges?
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?
A collection of games on the NIM theme
Got It game for an adult and child. How can you play so that you know you will always win?
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.
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.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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 find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
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?”
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
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.
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.