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?
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
Here are some examples of 'cons', and see if you can figure out where the trick is.
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. . . .
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. . . .
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
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
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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.
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 discover whether this is a fair game?
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
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.
The diagram shows a regular pentagon with sides of unit length.
Find all the angles in the diagram. Prove that the quadrilateral
shown in red is a rhombus.
Prove Pythagoras Theorem using enlargements and scale factors.
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Take any rectangle ABCD such that AB > BC. The point P is on AB
and Q is on CD. Show that there is exactly one position of P and Q
such that APCQ is a rhombus.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
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.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
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.
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) =
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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.
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!
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. . . .
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
Show that if three prime numbers, all greater than 3, form an
arithmetic progression then the common difference is divisible by
6. What if one of the terms is 3?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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. . . .
This is the second article on right-angled triangles whose edge lengths are whole numbers.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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. . . .
Points A, B and C are the centres of three circles, each one of
which touches the other two. Prove that the perimeter of the
triangle ABC is equal to the diameter of the largest circle.
An article which gives an account of some properties of magic squares.
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 looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
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.
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?