Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
A huge wheel is rolling past your window. What do you see?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Which hexagons tessellate?
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
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.
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
An equilateral triangle is constructed on BC. A line QD is drawn,
where Q is the midpoint of AC. Prove that AB // QD.
In the following sum the letters A, B, C, D, E and F stand for six
distinct digits. Find all the ways of replacing the letters with
digits so that the arithmetic is correct.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Baker, Cooper, Jones and Smith are four people whose occupations
are teacher, welder, mechanic and programmer, but not necessarily
in that order. What is each person’s occupation?
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
Replace each letter with a digit to make this addition correct.
Can you find all the 4-ball shuffles?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Choose any three by three square of dates on a calendar page...
Three teams have each played two matches. The table gives the total
number points and goals scored for and against each team. Fill in
the table and find the scores in the three matches.
This addition sum uses all ten digits 0, 1, 2...9 exactly once.
Find the sum and show that the one you give is the only
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
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. . . .
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
What are the missing numbers in the pyramids?
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Can you discover whether this is a fair game?
After some matches were played, most of the information in the
table containing the results of the games was accidentally deleted.
What was the score in each match played?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Carry out cyclic permutations of nine digit numbers containing the
digits from 1 to 9 (until you get back to the first number). Prove
that whatever number you choose, they will add to the same total.
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
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. . . .
We are given a regular icosahedron having three red vertices. Show
that it has a vertex that has at least two red neighbours.
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. . . .
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are
natural numbers and 0 < a < b < c. Prove that there is
only one set of values which satisfy this equation.
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. . . .