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.
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
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. . . .
How many tours visit each vertex of a cube once and only once? How
many return to the starting point?
Have a go at being mathematically negative, by negating these
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 any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
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?
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 has. . . .
Sort these mathematical propositions into a series of 8 correct
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. . . .
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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 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.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
How many noughts are at the end of these giant numbers?
An article which gives an account of some properties of magic squares.
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.
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
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
Here the diagram says it all. Can you find the diagram?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Can you find the value of this function involving algebraic
fractions for x=2000?
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
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. . . .
A polite number can be written as the sum of two or more
consecutive positive integers. Find the consecutive sums giving the
polite numbers 544 and 424. What characterizes impolite numbers?
With n people anywhere in a field each shoots a water pistol at the
nearest person. In general who gets wet? What difference does it
make if n is odd or even?
Keep constructing triangles in the incircle of the previous triangle. What happens?
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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.
Follow the hints and prove Pick's Theorem.
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!
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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.
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
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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.
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. . . .
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?
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.