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