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.
Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?
An account of some magic squares and their properties and and how to construct them for yourself.
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.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Can you find the values at the vertices when you know the values on the edges?
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?
Charlie has moved between countries and the average income of both has increased. How can this be so?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
A collection of games on the NIM theme
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.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
A game for 2 players
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.
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
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.
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.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Generalise this inequality involving integrals.
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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.
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.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. 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?
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.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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.
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 see how to build a harmonic triangle? Can you work out the next two rows?
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?”
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.
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
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 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 for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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.
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?