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.
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Can you discover whether this is a fair game?
Explain why, when moving heavy objects on rollers, the object moves
twice as fast as the rollers. Try a similar experiment yourself.
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF.
Similarly the largest. . . .
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Prove Pythagoras' Theorem using enlargements and scale factors.
Take any rectangle ABCD such that AB > BC. The point P is on AB
and Q is on CD. Show that there is exactly one position of P and Q
such that APCQ is a rhombus.
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
Can you use the diagram to prove the AM-GM inequality?
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Investigate the number of points with integer coordinates on
circles with centres at the origin for which the square of the
radius is a power of 5.
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
A point moves around inside a rectangle. What are the least and the
greatest values of the sum of the squares of the distances from the
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Find all positive integers a and b for which the two equations:
x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .
Four jewellers share their stock. Can you work out the relative values of their gems?
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
Some diagrammatic 'proofs' of algebraic identities and
The sum of any two of the numbers 2, 34 and 47 is a perfect square.
Choose three square numbers and find sets of three integers with
this property. Generalise to four integers.
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
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 make sense of these three proofs of Pythagoras' Theorem?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?
Can you make sense of the three methods to work out the area of the kite in the square?
You have twelve weights, one of which is different from the rest.
Using just 3 weighings, can you identify which weight is the odd
one out, and whether it is heavier or lighter than the rest?
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Peter Zimmerman from Mill Hill County High School in Barnet, London
gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is
divisible by 33 for every non negative integer n.
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Tom writes about expressing numbers as the sums of three squares.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Suppose A always beats B and B always beats C, then would you
expect A to beat C? Not always! What seems obvious is not always
true. Results always need to be proved in mathematics.
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.