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?”
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 use the diagram to prove the AM-GM inequality?
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
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 terms.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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 ...?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
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.
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 variables.
An article which gives an account of some properties of magic squares.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
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?
Can you find the area of a parallelogram defined by two vectors?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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?
Can you find the values at the vertices when you know the values on the edges?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Generalise this inequality involving integrals.
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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. . . .
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?
A game for 2 players
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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.
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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?
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 collection of games on the NIM theme
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 moves.
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.
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.
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
To avoid losing think of another very well known game where the patterns of play are similar.
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.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
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.
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.