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
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?
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
An article which gives an account of some properties of magic squares.
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.
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. . . .
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.
A collection of games on the NIM theme
An account of some magic squares and their properties and and how to construct them for yourself.
Can you find the values at the vertices when you know the values on
the edges of these multiplication arithmagons?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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?
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
A game for 2 players
These gnomons appear to have more than a passing connection with
the Fibonacci sequence. This problem ask you to investigate some of
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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.
Generalise this inequality involving integrals.
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?”
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.
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?
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?
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.
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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
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.
To avoid losing think of another very well known game where the
patterns of play are similar.
For which values of n is the Fibonacci number fn even? Which
Fibonnaci numbers are divisible by 3?
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 a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
Can you find the area of a parallelogram defined by two vectors?
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
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.
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.