An article which gives an account of some properties of magic 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?
Can you tangle yourself up and reach any fraction?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Generalise this inequality involving integrals.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
It would be nice to have a strategy for disentangling any tangled ropes...
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A game for 2 players
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.
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 collection of games on the NIM theme
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?”
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. . . .
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?
Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.
Can you use the diagram to prove the AM-GM inequality?
Can you find the values at the vertices when you know the values on the edges?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Charlie has moved between countries and the average income of both has increased. How can this be so?
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
What's the largest volume of box you can make from a square of paper?
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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 find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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?
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. . . .
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.
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.
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?
Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.
An account of some magic squares and their properties and and how to construct them for yourself.
What is the total number of squares that can be made on a 5 by 5 geoboard?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.