Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
A game for 2 players
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?
To avoid losing think of another very well known game where the patterns of play are similar.
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.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
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.
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 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 collection of games on the NIM theme
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.
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.
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.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you find the area of a parallelogram defined by two vectors?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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.
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?
Can you use the diagram to prove the AM-GM inequality?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
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.
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?”
Can you find the values at the vertices when you know the values on the edges?
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.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
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?
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?
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
An account of some magic squares and their properties and and how to construct them for yourself.
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Can you explain the strategy for winning this game with any target?
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .