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. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
An article which gives an account of some properties of magic squares.
Can you use the diagram to prove the AM-GM inequality?
Can you tangle yourself up and reach any fraction?
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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 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.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
To avoid losing think of another very well known game where the patterns of play are similar.
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
I added together some of my neighbours house numbers. Can you explain the patterns I noticed?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?”
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.
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A game for 2 players
It would be nice to have a strategy for disentangling any tangled ropes...
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
What's the largest volume of box you can make from a square of paper?
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?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
A collection of games on the NIM theme
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?
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.
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Can you find the values at the vertices when you know the values on the edges?
Charlie has moved between countries and the average income of both has increased. How can this be so?
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.
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Make some loops out of regular hexagons. What rules can you discover?
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?
Can you describe this route to infinity? Where will the arrows take you next?
What is the total number of squares that can be made on a 5 by 5 geoboard?