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.
How many different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted. . . .
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. . . .
Start with any triangle T1 and its inscribed circle. Draw the
triangle T2 which has its vertices at the points of contact between
the triangle T1 and its incircle. Now keep repeating this. . . .
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. . . .
This addition sum uses all ten digits 0, 1, 2...9 exactly once.
Find the sum and show that the one you give is the only
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. . . .
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.
A 'doodle' is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be 'double points'
not. . . .
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
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!
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
This jar used to hold perfumed oil. It contained enough oil to fill
granid silver bottles. Each bottle held enough to fill ozvik golden
goblets and each goblet held enough to fill vaswik crystal. . . .
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
Can you find all the 4-ball shuffles?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Write down a three-digit number Change the order of the digits to
get a different number Find the difference between the two three
digit numbers Follow the rest of the instructions then try. . . .
Carry out cyclic permutations of nine digit numbers containing the
digits from 1 to 9 (until you get back to the first number). Prove
that whatever number you choose, they will add to the same total.
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 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.
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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?
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.
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.
An article which gives an account of some properties of magic squares.
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.
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?
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. . . .
If you think that mathematical proof is really clearcut and
universal then you should read this article.
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are
natural numbers and 0 < a < b < c. Prove that there is
only one set of values which satisfy this equation.
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.
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. . . .
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
Here are some examples of 'cons', and see if you can figure out where the trick is.
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
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?
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
From a group of any 4 students in a class of 30, each has exchanged
Christmas cards with the other three. Show that some students have
exchanged cards with all the other students in the class. How. . . .
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
I start with a red, a blue, a green and a yellow marble. I can
trade any of my marbles for three others, one of each colour. Can I
end up with exactly two marbles of each colour?