• Best of luck to the class of 2019 for their HSC exams. You got this!

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  1. seanieg89

    Interesting problem

    Here is an interesting problem. If you are not comfortable with higher dimensional spaces, replace "affine hyperplane" with "line" (not necessarily through the origin), and fix n=2 in the below. Suppose we have finitely many affine hyperplanes \mathcal{H}_1,\mathcal{H}_2,\ldots,\mathcal{H}_m...
  2. seanieg89

    Linear Algebra Marathon & Questions

    Linear Algebra Marathon & Questions This is a marathon thread for linear algebra. Please aim to pitch your questions for first-year/second-year university level maths. Excelling & gifted/talented secondary school students are also invited to contribute. (mod edit 7/6/17 by dan964)...
  3. seanieg89

    Numerical Integration

    The last question in this years BoS trials involved a closer look at the trapezoidal rule than we usually take. The below article of mine further explores this topic, and has a few exercises that are good practice for the interested HSC student. Nothing outside MX2 is used (except the...
  4. seanieg89

    Why is the fundamental theorem of algebra true?

    The fundamental theorem of algebra asserts that any non-constant polynomial with complex coefficients has a complex root. It is notorious as being quite hard to prove for what it is, typically requiring some machinery at least as advanced as complex analysis / vector calculus / algebraic...
  5. seanieg89

    Logic Puzzle!

    (Not created by me.) Cheryl Welcome, Albert and Bernard, to my birthday party, and I thank you for your gifts. To return the favor, as you entered my party, I privately made known to each of you a rational number of the form n-1/2^k-1/2^{k+r}, where n and k are positive integers and r is a...
  6. seanieg89

    Convergence lemma.

    Here is a cute little convergence lemma that gets used a lot in dynamics (eg in the study of entropy). A good exercise for students learning real analysis for the first time. Given a sequence (x_n) of positive real numbers such that x_{m+n} \leq x_n+x_m for every pair of positive integers m...
  7. seanieg89

    A taste of higher mathematics!

    Hi all. Sometime over the next couple of months I intend to start a professional blog to help me organise my mathematical writing and practice my exposition. (It will also give me a relatively easy way to answer the dreaded questions from friends on the contents of my research.) Most of this...
  8. seanieg89

    Smash 64 / Melee

    Anyone up for some online play of ssb64 (via the project64k emulator on your pc) or ssbm (via dolphin)? Ideally, you should have some kind of usb controller, but some people get by with keyboards.
  9. seanieg89

    Gambling Systems Thread (Roulette).

    A surprising number of people I have met seem to believe in the existence of profitable "betting systems" for casino games like Roulette. Casinos are very happy about this misconception. Here is a thread for high school students to practice their combinatorics skills by suggesting and/or...
  10. seanieg89

    Fun linear algebra exercise.

    Motivated by Sy's last question, here is a fun problem for those with some linear algebra knowledge: $1. If $T:X\rightarrow Y$ is a linear map between vector spaces, and for some $q\in Y$, $p\in X$ is a solution to the equation $Tx=q$, prove that the general solution is given by $x=p+r$ where...
  11. seanieg89


    A well known but nice practical question: You have a boiling cup of coffee (so 100 degrees) and you wish to drink it in exactly 5 minutes time. You like to dilute your coffee with milk (at say 5 degrees) in the ratio 1 part milk to 19 parts black coffee. You like your coffee to be as cool...
  12. seanieg89

    Nice probability question.

    A man tosses a fair coin repeatedly until he throw n heads in a row. (n a positive integer). What is the average number of tosses required? If you cannot solve the general problem, feel free to post solutions for small values of n.
  13. seanieg89

    Why proof is important!

    A lot of friends have asked me why the standard of rigour in mathematical proof is so high. Why it is necessary to fiddle with things like epsilons and deltas? Why do people waste their time looking for proofs of theorems that we have verified up until massive numbers using computers? Eg...
  14. seanieg89

    Isoperimetric inequality 1.

    $Prove that the area of a polygon with perimeter $1$ is alway smaller than $\frac{1}{4\pi}$, and that we can find polygons with perimeter $1$ that have area arbitrarily close to $\frac{1}{4\pi}$. $ NB: This is a polygonal version of the isoperimetric inequality. The general isoperimetric...
  15. seanieg89

    Mathematics: Is it discovered or created?

    I have my own views on the matter which would have come up elsewhere on bos before, but I would be interested in hearing the thoughts of the mixture of (mostly) HS students, HS teachers, and undergraduates who frequent this site (without any bias from reading what I have to say first). Some...
  16. seanieg89

    Coloured Hats.

    One hundred mathematicians are in a line facing the same direction, each of them wearing a hat that is coloured either red or blue. Each mathematician can only see the colour of the hats of the mathematicians in front of him. Starting from the back of the line, each mathematician announces a...
  17. seanieg89

    Worms Armageddon

    Anyone here remember Worms Armageddon (not W2A, the original released around the year 2000)? It holds up incredibly well and I have been playing a ton of multiplayer with housemates and friends from interstate recently. BoS should get some online games going!
  18. seanieg89


    A man is in the middle of a circular lake and a monster is at the edge of the shore. The monster runs around the lake at a speed X times the swimming speed of the man (but much slower than the man's running speed), and chooses his running direction at all times optimally in order to catch the...
  19. seanieg89

    Monty Hall Variant

    Many of you will have heard of the Monty Hall problem, presented below: 1. There are three briefcases, two are empty and one contains one million dollars. Monty knows which one contains the prize. 2. The contestant chooses a case but does not open it. 3. Monty opens an empty case out of the two...
  20. seanieg89

    Cranks and Crackpots

    A crank/crackpot is basically someone who makes claims of immense academic achievements (they exist in every discipline, seemingly most of all in physics and mathematics) such as the the discovery of an elementary proof of Fermat's Last Theorem. An old form of procrastination for me was...