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?”
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. . . .
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you use the diagram to prove the AM-GM inequality?
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
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
Generalise this inequality involving integrals.
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 ...?
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
First of all, pick the number of times a week that you would like
to eat chocolate. Multiply this number by 2...
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.
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?
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?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
An article which gives an account of some properties of magic squares.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
What is the value of the integers a and b where sqrt(8-4sqrt3) =
sqrt a - sqrt b?
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?
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
Can you describe this route to infinity? Where will the arrows take you next?
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.
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. . . .
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.
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
A game for 2 players
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.
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.
A collection of games on the NIM theme
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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?
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
To avoid losing think of another very well known game where the
patterns of play are similar.
Your data is a set of positive numbers. What is the maximum value
that the standard deviation can take?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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 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.
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.
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.