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?
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
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.
What are the missing numbers in the pyramids?
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Can you use the diagram to prove the AM-GM inequality?
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.
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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. . . .
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. . . .
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
Replace each letter with a digit to make this addition correct.
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. . . .
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. . . .
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?
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?
Choose any three by three square of dates on a calendar page...
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
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. . . .
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
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. . . .
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?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
A huge wheel is rolling past your window. What do you see?
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
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?
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. . . .
Can you make sense of these three proofs of Pythagoras' Theorem?
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. . . .
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.
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.
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. . . .
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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.
After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?
I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
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.
Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?
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?
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .