Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
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?
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 ...?
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
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?
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?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
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?
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. . . .
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
Can you describe this route to infinity? Where will the arrows take you next?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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 game for 2 players
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.
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.
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.
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?”
A collection of games on the NIM theme
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
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.
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 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.
An account of some magic squares and their properties and and how to construct them for yourself.
An article which gives an account of some properties of magic squares.
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
Generalise this inequality involving integrals.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
To avoid losing think of another very well known game where the patterns of play are similar.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
Find the vertices of a pentagon given the midpoints of its sides.
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.