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?
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.
Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
Replace each letter with a digit to make this addition correct.
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. . . .
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
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. . . .
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Choose any three by three square of dates on a calendar page...
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?
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?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
What are the missing numbers in the pyramids?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square 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.
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.
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. . . .
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
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?
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?
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
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?
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.
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.
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. . . .
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?
How many noughts are at the end of these giant numbers?
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?
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?
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?
A huge wheel is rolling past your window. What do you see?
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?
Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches.
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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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.
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
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. . . .
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?