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 ...?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
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.
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
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.
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?
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.
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.
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?
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.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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. . . .
An account of some magic squares and their properties and and how to construct them for yourself.
This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.
You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.
Can you find the area of a parallelogram defined by two vectors?
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.
Can you use the diagram to prove the AM-GM inequality?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
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?
Can you tangle yourself up and reach any fraction?
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 find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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?
What is the total number of squares that can be made on a 5 by 5 geoboard?
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?”
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Charlie has moved between countries and the average income of both has increased. How can this be so?
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Can you see how to build a harmonic triangle? Can you work out the next two rows?
A game for 2 players
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.
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?
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?