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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. . . .
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
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?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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.
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 number...
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.
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.
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 vertices?
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?
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.
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?
Can you find the value of this function involving algebraic fractions for x=2000?
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.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
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.
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.
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
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.
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
Can you make sense of these three proofs of Pythagoras' Theorem?
Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.
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.
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 article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
Tom writes about expressing numbers as the sums of three squares.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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.
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.
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. . . .
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.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Can you discover whether this is a fair game?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
These proofs are wrong. Can you see why?
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.
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.
Can you make sense of the three methods to work out the area of the kite in the square?
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?