Which set of numbers that add to 10 have the largest product?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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?
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
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Keep constructing triangles in the incircle of the previous triangle. What happens?
How many noughts are at the end of these giant numbers?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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?
An article which gives an account of some properties of magic squares.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
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.
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 looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
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. . . .
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you
notice when successive terms are taken? What happens to the terms
if the fraction goes on indefinitely?
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
Three points A, B and C lie in this order on a line, and P is any
point in the plane. Use the Cosine Rule to prove the following
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. . . .
Four jewellers share their stock. Can you work out the relative values of their gems?
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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. . . .
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.
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF.
Similarly the largest. . . .
The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. . . .
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!
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
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Can you rearrange the cards to make a series of correct mathematical statements?
What fractions can you divide the diagonal of a square into by simple folding?
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?
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
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.
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?