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.
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.
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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.
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
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
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.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A collection of games on the NIM theme
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you find the area of a parallelogram defined by two vectors?
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?
A game for 2 players
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.
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?
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?
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
Can you use the diagram to prove the AM-GM inequality?
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. . . .
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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
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.
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.
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?”
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
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?
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?
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
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
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.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
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 ...?
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?
Find the vertices of a pentagon given the midpoints of its sides.
Can you describe this route to infinity? Where will the arrows take you next?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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.