I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Investigate the sequences obtained by starting with any positive 2
digit number (10a+b) and repeatedly using the rule 10a+b maps to
10b-a to get the next number in the sequence.
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 first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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 work through these direct proofs, using our interactive
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Can you rearrange the cards to make a series of correct mathematical statements?
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.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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?
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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.
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.
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.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
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...
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
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.
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
Take a number, add its digits then multiply the digits together,
then multiply these two results. If you get the same number it is
an SP number.
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.
Let a(n) be the number of ways of expressing the integer n as an
ordered sum of 1's and 2's. Let b(n) be the number of ways of
expressing n as an ordered sum of integers greater than 1. (i)
Calculate. . . .
Can you explain why a sequence of operations always gives you perfect squares?
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
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.
The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?
Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...
There are 12 identical looking coins, one of which is a fake. The
counterfeit coin is of a different weight to the rest. What is the
minimum number of weighings needed to locate the fake coin?
An equilateral triangle is sitting on top of a square.
What is the radius of the circle that circumscribes this shape?
Can you make sense of the three methods to work out the area of the kite in the square?
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
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
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
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.
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Explore what happens when you draw graphs of quadratic equations
with coefficients based on a geometric sequence.
Can you discover whether this is a fair game?
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.
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
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. . . .
Some diagrammatic 'proofs' of algebraic identities and
Tom writes about expressing numbers as the sums of three squares.
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.
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.
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
These proofs are wrong. Can you see why?
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.