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Investigate circuits and record your findings in this simple introduction to truth tables and logic.
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!
Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
Can you discover whether this is a fair game?
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. . . .
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.
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
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. . . .
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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. . . .
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees?
If you think that mathematical proof is really clearcut and universal then you should read this article.
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
A huge wheel is rolling past your window. What do you see?
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .
An article which gives an account of some properties of magic squares.
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
Can you make sense of these three proofs of Pythagoras' Theorem?
This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
What fractions can you divide the diagonal of a square into by simple folding?
Can you rearrange the cards to make a series of correct mathematical statements?
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?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?