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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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?
Show that if three prime numbers, all greater than 3, form an
arithmetic progression then the common difference is divisible by
6. What if one of the terms is 3?
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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. . . .
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) =
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
The Tower of Hanoi is an ancient mathematical challenge. Working on
the building blocks may help you to explain the patterns you
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.
Which set of numbers that add to 10 have the largest product?
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
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
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.
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?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
A introduction to how patterns can be deceiving, and what is and is not a proof.
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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?
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Can you fit Ls together to make larger versions of themselves?
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 quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4.
Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .
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. . . .
An article which gives an account of some properties of magic squares.
Is the mean of the squares of two numbers greater than, or less
than, the square of their means?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you
notice when successive terms are taken? What happens to the terms
if the fraction goes on indefinitely?
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
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. . . .
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
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.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Can you make sense of the three methods to work out the area of the kite in the square?
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...