It would be nice to have a strategy for disentangling any tangled ropes...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
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
Charlie has moved between countries and the average income of both has increased. How can this be so?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
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 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.
An article which gives an account of some properties of magic squares.
To avoid losing think of another very well known game where the patterns of play are similar.
A game for 2 players
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?
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.
This task encourages you to investigate the number of edging pieces and panes in different sized windows.
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.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you describe this route to infinity? Where will the arrows take you next?
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
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?
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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 loser is the player who takes the last counter.
A collection of games on the NIM theme
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 a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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?”
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.
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?
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Investigate the different ways that fifteen schools could have given money in a charity fundraiser.
Is there an efficient way to work out how many factors a large number has?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
It starts quite simple but great opportunities for number discoveries and patterns!
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?