Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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. . . .
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.
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. . . .
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?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
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.
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?
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?
You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
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.
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?
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. . . .
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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.
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
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?
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. . . .
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.
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.
Replace each letter with a digit to make this addition correct.
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
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. . . .
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?
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?
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. . . .
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
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.
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 you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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?
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. . . .
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.
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?
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.
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. . . .