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. . . .
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
Can you use the diagram to prove the AM-GM inequality?
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
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
An article which gives an account of some properties of magic squares.
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 ...?
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?”
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
Generalise this inequality involving integrals.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
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.
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
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.
A game for 2 players
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
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.
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
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 winner is the player to take the last counter.
Can you find the values at the vertices when you know the values on the edges?
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.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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.
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?
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.
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
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.
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
To avoid losing think of another very well known game where the
patterns of play are similar.
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
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.
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
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.
Can you find the area of a parallelogram defined by two vectors?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
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 find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?