Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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.
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.
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.
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.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
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.
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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.
A collection of games on the NIM theme
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
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.
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.
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.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Fractional calculus is a generalisation of ordinary calculus where you can 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.
Generalise this inequality involving integrals.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?
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.
To avoid losing think of another very well known game where the patterns of play are similar.
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 ...?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
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?
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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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 work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
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?”
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?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
A game for 2 players
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
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.
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.
Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?
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?
Find the vertices of a pentagon given the midpoints of its sides.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?