An article which gives an account of some properties of magic squares.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you use the diagram to prove the AM-GM inequality?
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. . . .
An account of some magic squares and their properties and and how to construct them for yourself.
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
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. . . .
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Can you find the area of a parallelogram defined by two vectors?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
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?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?”
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nĀ² Use the diagram to show that any odd number is the difference of two squares.
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?
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?
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?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Can you describe this route to infinity? Where will the arrows take you next?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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
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.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?