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. . . .
Can you use the diagram to prove the AM-GM inequality?
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?”
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
An account of some magic squares and their properties and and how to construct them for yourself.
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?
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. . . .
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?
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 explain the surprising results Jo found when she calculated the difference between square 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.
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
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. . . .
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
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.
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?
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
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.
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.
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.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Charlie has moved between countries and the average income of both has increased. How can this be so?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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
A collection of games on the NIM theme
Can you find the values at the vertices when you know the values on the edges?
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
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
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. . . .
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?
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?