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.
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 ...?
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.
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.
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.
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?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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. . . .
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?
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?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
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.
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. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
A game for 2 players
Can you use the diagram to prove the AM-GM inequality?
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.
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?”
A collection of games on the NIM theme
Can you tangle yourself up and reach any fraction?
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?
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
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?
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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.
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
To avoid losing think of another very well known game where the patterns of play are similar.
An article which gives an account of some properties of magic squares.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
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. . . .
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?