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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
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?
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. . . .
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.
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?
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?
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. . . .
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
A huge wheel is rolling past your window. What do you see?
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. . . .
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.
In the following sum the letters A, B, C, D, E and F stand for six
distinct digits. Find all the ways of replacing the letters with
digits so that the arithmetic is correct.
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. . . .
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Which set of numbers that add to 10 have the largest product?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
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.
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?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
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?
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. . . .
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
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.
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.
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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.
Replace each letter with a digit to make this addition correct.
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
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.
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. . . .
We have exactly 100 coins. There are five different values of
coins. We have decided to buy a piece of computer software for
39.75. We have the correct money, not a penny more, not a penny
less! Can. . . .
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?
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?
Here are some examples of 'cons', and see if you can figure out where the trick is.
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Can you find all the 4-ball shuffles?
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. . . .
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.
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.
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
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.
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. . . .
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?