Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Delight your friends with this cunning trick! Can you explain how it works?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A game for 2 players
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.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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 you explain the strategy for winning this game with any target?
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 for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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 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?
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.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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 explain how this card trick works?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
To avoid losing think of another very well known game where the patterns of play are similar.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
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.
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?
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.
Can you find sets of sloping lines that enclose a square?
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 likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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?
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.
Can you find the values at the vertices when you know the values on the edges?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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?
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?
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.
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?
A collection of games on the NIM theme
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
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