Challenge Level

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.

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.

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.

An account of some magic squares and their properties and and how to construct them for yourself.

Challenge Level

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.

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Challenge Level

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.

Challenge Level

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Challenge Level

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Challenge Level

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

Challenge Level

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?

Challenge Level

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Challenge Level

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.

Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Challenge Level

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Challenge Level

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?

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.

Challenge Level

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Challenge Level

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. . . .

Challenge Level

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?”

Challenge Level

Can you find the values at the vertices when you know the values on the edges?

Challenge Level

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.

Challenge Level

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?

Challenge Level

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Challenge Level

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Challenge Level

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?

Challenge Level

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Challenge Level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Challenge Level

What's the largest volume of box you can make from a square of paper?

Challenge Level

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.

Challenge Level

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Challenge Level

Charlie has moved between countries and the average income of both has increased. How can this be so?

Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Challenge Level

It would be nice to have a strategy for disentangling any tangled ropes...

Challenge Level

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. . . .

Challenge Level

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.

Challenge Level

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Challenge Level

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.

Challenge Level

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?

Challenge Level

What is the total number of squares that can be made on a 5 by 5 geoboard?

Challenge Level

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Challenge Level

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

Challenge Level

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.