The last time I wrote about this, I’d found a formula that you can solve algorithmically to find the minimum number of shuffles it takes to resurrect an even numbered deck of cards. I’d also found two solution series – for decks of size 2^{N} , and 2^{N} -2. Since then, i’ve made some exciting progress!

Continue reading “Card Shuffling – oh, how the turn tables”

I’ve written about this before (here).

When we left off, we had a formula to solve that would give the length of a single loop:

P*2^{N} mod (C+1) = P

P is the position of a card in the loop, N is the number of shuffles, and C is the number of cards. Essentially, you take the starting position, and keep doubling it – if it goes above the number of cards in the deck, take the remainder after dividing it by (C+1). Once it hits it’s starting position again, the loop has ended.

Continue reading “Some more card shuffling”

Yes, that’s right – i’m jumping on the Vsauce bandwagon. Well, not entirely – this is actually a different method for answering this question, which I was given in an assignment, and I thought I’d share.

If you haven’t seen the Vsauce video, the problem goes something like this. You take a sphere, and push a cylinder through it; the cylinder carves out a chunk of the sphere.

Continue reading “Napkin ring problem”

NOTE: I did not make this explanation. I based it on the explanation from the book “Journey through Genius” by William Dunham, and changed a few parts – I liked how much clearer this book’s explanation was than the other, more formal proofs I have seen.

## What is the Basel problem?

The Basel problem had stumped mathematicians for years before Euler came along. It asked a question about this infinite sum:

Continue reading “An explanation of how Euler solved the Basel problem”

My old high school has a program where they invite primary school kids to the school to do science and maths activities – and this year, they asked me to help run one of these, the Mini Mathematicians class. This problem actually came out of a class we did there, on card shuffling.

Continue reading “Card shuffling – it’s actually really mathematical”

I want to have a look at a very famous (and bizarre) result, the infinite sum

At first, it doesn’t seem to make any sense. If you add all these numbers up, the result keeps getting larger, so how does it approach a finite number – and a negative one for that matter?

Continue reading “1+2+3+4+5…=-1/12”

In this post, I want to have a look at a really neat part of calculus, Taylor series. It’s essentially a process you can use to approximate some graph, e.g. e^{x}, using a polynomial – the idea is, the higher order the polynomial, the more accurate the approximation gets.

## Let’s dive in! (This gets a bit tricky)

How does it work? First, some terminology – the first order approximation is a linear approximation (so ax+b), second order is quadratic (ax^{2} + bx + c), and so on. So let’s start with a first order approximation of, say, e^{x}.

Continue reading “Taylor Series”

(The above photo is from Philippe Petit’s famous tightrope walk across the WTC)

This is a really nice question about limits from a uni assignment I was given. Say you are at the start of a stretchy tightrope, a kilometre long. You walk along it at 1 metre per second – but at the end of every second, the rope stretches another kilometre longer. Do you ever reach the end?

Continue reading “Infinitely stretchy ropes and the Harmonic Series”

Here’s a paper I promised some time ago, about creating Pythagorean triples.I’m not really too sure about how maths papers should be constructed, so it’s probably a bit rubbish, but oh well.