What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
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.
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
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.
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. . . .
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Can you find the area of a parallelogram defined by two vectors?
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?
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
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?
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.
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
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?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Can you tangle yourself up and reach any fraction?
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.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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?
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 use the diagram to prove the AM-GM inequality?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
An article which gives an account of some properties of magic squares.
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?
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
What is the total number of squares that can be made on a 5 by 5 geoboard?
Generalise this inequality involving integrals.
Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
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?”
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?
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
It would be nice to have a strategy for disentangling any tangled ropes...
Can you find the values at the vertices when you know the values on the edges?
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?
Charlie has moved between countries and the average income of both has increased. How can this be so?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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.
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.
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?