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.
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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.
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?
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
Can you tangle yourself up and reach any fraction?
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
It would be nice to have a strategy for disentangling any tangled ropes...
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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.
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. . . .
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
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?
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?
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.
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?
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?”
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?
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.
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.
Generalise this inequality involving integrals.
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?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
An account of some magic squares and their properties and and how to construct them for yourself.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
A game for 2 players with similarities 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.
Charlie has moved between countries and the average income of both has increased. How can this be so?
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .
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.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Can you find the values at the vertices when you know the values on the edges?
Can you find the area of a parallelogram defined by two vectors?
What is the total number of squares that can be made on a 5 by 5 geoboard?
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.