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. . . .
Here are three 'tricks' to amaze your friends. But the really
clever trick is explaining to them why these 'tricks' are maths not
magic. Like all good magicians, you should practice by trying. . . .
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 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. . . .
What are the missing numbers in the pyramids?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
A introduction to how patterns can be deceiving, and what is and is not a proof.
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Replace each letter with a digit to make this addition correct.
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. . . .
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .
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.
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
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.
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?
A huge wheel is rolling past your window. What do you see?
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. . . .
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. . . .
Make an eight by eight square, the layout is the same as a
chessboard. You can print out and use the square below. What is the
area of the square? Divide the square in the way shown by the red
dashed. . . .
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some. . . .
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.
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?
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. . . .
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Four jewellers possessing respectively eight rubies, ten saphires,
a hundred pearls and five diamonds, presented, each from his own
stock, one apiece to the rest in token of regard; and they. . . .
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.
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
Can you find all the 4-ball shuffles?
Here are some examples of 'cons', and see if you can figure out where the trick is.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Can you rearrange the cards to make a series of correct
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you fit Ls together to make larger versions of themselves?
What can you say about the angles on opposite vertices of any
cyclic quadrilateral? Working on the building blocks will give you
insights that may help you to explain what is special about them.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Can you make sense of these three proofs of Pythagoras' Theorem?
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.