Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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 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?
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.
Delight your friends with this cunning trick! Can you explain how it works?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
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?
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...
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
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?
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.
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.
Can you explain how this card trick works?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A game for 2 players
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
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.
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?
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?
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?
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
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?
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.
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
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.
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?
An account of some magic squares and their properties and and how to construct them for yourself.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.