Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
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. . . .
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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.
Take a complicated fraction with the product of five quartics top
and bottom and reduce this to a whole number. This is a numerical
example involving some clever algebra.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
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.
Relate these algebraic expressions to geometrical diagrams.
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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.
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
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. . . .
How many tours visit each vertex of a cube once and only once? How
many return to the starting point?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
This is the second of two articles and discusses problems relating
to the curvature of space, shortest distances on surfaces,
triangulations of surfaces and representation by graphs.
Prove that you cannot form a Magic W with a total of 12 or less or
with a with a total of 18 or more.
Can you find the value of this function involving algebraic
fractions for x=2000?
Find the missing angle between the two secants to the circle when
the two angles at the centre subtended by the arcs created by the
intersections of the secants and the circle are 50 and 120 degrees.
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
Can you make sense of these three proofs of Pythagoras' Theorem?
A connected graph is a graph in which we can get from any vertex to
any other by travelling along the edges. A tree is a connected
graph with no closed circuits (or loops. Prove that every tree. . . .
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
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
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
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. . . .
Given any two polynomials in a single variable it is always
possible to eliminate the variable and obtain a formula showing the
relationship between the two polynomials. Try this one.
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .
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.
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.
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.
Can you discover whether this is a fair game?
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
By proving these particular identities, prove the existence of general cases.
Some diagrammatic 'proofs' of algebraic identities and
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Solve this famous unsolved problem and win a prize. Take a positive
integer N. If even, divide by 2; if odd, multiply by 3 and add 1.
Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
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. . . .
Can you make sense of the three methods to work out the area of the kite in the square?