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.
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.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Can you find the area of a parallelogram defined by two vectors?
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.
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
An account of some magic squares and their properties and and how to construct them for yourself.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
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?
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.
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.
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
A game for 2 players
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.
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.
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.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A collection of games on the NIM theme
Can you find the values at the vertices when you know the values on the edges?
Can you use the diagram to prove the AM-GM inequality?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?”
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
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.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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. . . .
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 value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
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 ...?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
An article which gives an account of some properties of magic squares.
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
Generalise this inequality involving integrals.
To avoid losing think of another very well known game where the patterns of play are similar.
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?
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?
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.
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 work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?