I added together some of my neighbours house numbers. Can you explain the patterns I noticed?
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.
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
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.
A huge wheel is rolling past your window. What do you see?
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?
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 how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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. . . .
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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.
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. . . .
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
What is the largest number of intersection points that a triangle and a quadrilateral can have?
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
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.
Which hexagons tessellate?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
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.
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 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. . . .
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
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.
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.
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. . . .
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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.
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. . . .
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
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. . . .
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Can you rearrange the cards to make a series of correct mathematical statements?
Can you discover whether this is a fair game?
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. . . .
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!
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
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. . . .
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.
Here are some examples of 'cons', and see if you can figure out where the trick is.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
What are the missing numbers in the pyramids?
Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?