2) Do you want to do some extra Maths this summer?
If your answer to both questions is ``Yes!'', then I have the following suggestion.
We can make this a thread a place where interesting Maths problems of pre-Undergraduate/Undergraduate level are discussed. The questions' difficulty can vary considerably, and they can be suggested by current undergraduate students, but the main focus must be on educational problems.
** I don't want people showing off with extremely difficult, not interesting and practically unsolvable with pre-Undergraduate/Undergraduate knowledge, questions. **
If you want to play this game, I have a few rules for a start.
[1] Post a question only if you believe it is of the right caliber and you have the complete solution.
[2] Indicate any required Undergraduate knowledge that people need to know in order to produce a solution to the given problem.
[3]Hints and solutions go in spoilers. In case they are required, the person who posted the question must provide them; with a reference to their origin.
List of problems offered by universities for practice during the summer.
Spoiler
List of suggested books that people have decided to spend time on over the summer:
Spoiler
List of additional resources that people suggested:
We use induction to prove that n horses are the same colour: Base case, n=1, true.
assume true for n=k \geq 1 If we have k+1 horses. Consider the first k horses. By assumption, they are all the same colour. Take one of these k horses, and the (k+1)th horse - again, by assumption, they are the same colour. Hence all k+1 are the same colour (here we use 'transitivity of colour' - to be formalised below)
Later that evening, after an unsatisfactory meal produced by Mrs Hilbert, Mr Hilbert is hungry again. Mr Hilbert eats another key from the typewriter, leaving just one key!
ii)It is possible for the receptionist to keep track of the room numbers, but how?
At the Nottingham University Open Day there was a "Maths Trail" with several interesting questions on it. The questions ranged from requiring a working knowledge of arithmetic progressions and combinations to lowest common multiples and counting squares; and more emphasis was put on thinking about them than slogging through endless manipulation. I thought I'd share one with you. The question is of the kind that could be set in C1, but is an interesting one:
A rectangle is inscribed inside a circle of radius 6 units such that each of the vertices of the rectangle lie on the circumference of the circle. Given that the perimeter of the rectangle is 28 units, what is the area?
What's the difference between this thread and the TSR maths society thread?
TSR Maths Society seems to be a bit more general thread than what we need. However, I am here for the Maths, so it makes no difference to me where I will post it.
Do people prefer to post in the TSR Maths Society thread?