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. . . .
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.
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.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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 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.
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
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
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
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.
Replace each letter with a digit to make this addition correct.
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?
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?
A huge wheel is rolling past your window. What do you see?
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?
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.
How many noughts are at the end of these giant numbers?
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?
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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. . . .
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
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?
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
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?
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. . . .
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Which hexagons tessellate?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Can you fit Ls together to make larger versions of themselves?
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?
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.
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. . . .
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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. . . .
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?
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.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.