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.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
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.
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
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
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.
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.
A game for 2 players
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 opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
An account of some magic squares and their properties and and how to construct them for yourself.
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
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?
A counter is placed in the bottom right hand corner of a grid. You
toss a coin and move the star according to the following rules: ...
What is the probability that you end up in the top left-hand. . . .
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?
To avoid losing think of another very well known game where the
patterns of play are similar.
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
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. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
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.
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?
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.
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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 describe this route to infinity? Where will the arrows take you next?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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?
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?”
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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.
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
An article which gives an account of some properties of magic squares.
Generalise this inequality involving integrals.
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?
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of