All 3 entries tagged Miscellany

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October 30, 2011

Balls in boxes and other miscellaneous thoughts

It is a reasonably common statement in probability that problems can be simplified down to spinners or balls in boxes. So:
Let there be k balls that we wish to place in n boxes. If we can distinguish between the balls, then each ball can go in n different places, so the total is \textstyle n^k. If we cannot distinguish between the balls, this is answered by the occupancy theorem: the multiset formula for n and k.

What, then, if it is the boxes we cannot distinguish between? If the balls also cannot be distinguished between, this is equivalent to partitioning k into at most n parts. The OEIS implies that this has a closed-form solution: \textstyle \frac{1}{n!} \frac{d^n\left( \frac{1}{\prod_{i=1}^k(1-x^i)} \right)}{dx^n}(0), which is nice to know, even if it suggests there’s no ‘nice’ combinatorial solution.
If we can distinguish between the balls but not the boxes, I believe this would come to a sum of Stirling numbers of the second kind: \sum_{i=0}^n \left\{\begin{matrix} k \\ i \end{matrix}\right\}.

Other miscellany: Mario Micallef told us a few integration tricks this week: being able to cancel entire functions while integrating from 0 to 2pi (due to them containing sine), \textstyle\int_0^{2\pi}sin^2(x)dx=\pi by comparison with \cos^2(x) across the same interval. Massively time-saving tricks I’d never considered before. Great!

From combinatorics: wondered if there was a way to check transitivity of a relation expressed in matrix form quickly. Found nothing more than checking each row one step at a time, but I least I can do that fairly quickly now.


July 23, 2011

Golden Boomerang

In 2004, there existed a competition known as the Golden Boomerang – solve riddles at associated website (now nonexistant) www.goldenboomerang.co.uk. The riddles were by a company named Boomerang which fortunately still exists. I find this interesting mostly because I can find no record of it ever existing, although admittedly “golden boomerang” is not a terribly unique name.

Four postcards of four different designs were distributed, each with a riddle and two puzzle pieces. I was in Year 7 at the time and my general knowledge was awful so I couldn’t answer the riddles at all, and mostly collected the postcards as bookmarks.

You were invited to submit answers to the riddles on the website, so I’m presuming something more happened there. I vaguely remember them sending emails with poor HTML coding that displayed as code instead of visuals, but that might have been something else – it wasn’t exactly an uncommon mistake.


June 02, 2011

Post #1

And so I decide, at the end of the year, to get a blog here. A blog seems rather an awkward callback to the halcyon days of livejournal or such. I suppose these things were rather more popular approximately five years ago.

I was pretty late on the social scene in the first place – I am an incredibly introverted person – I got a Facebook account last year, I believe.

The general aim of this is to make some random posts on whatever strikes my fancy. I’d also like to use it to reflect on how exams went – which is the main point I’ve set it up now, in all honesty.

But for now, miscellany:
I thought I knew pi to 30 digits (why did I memorise them? I wanted to, I guess); but while attempting to speak the digits of pi in Japanese, I discovered that I actually only knew it to 4. I knew how to say the 30 digits in English, but just thinking of the numbers directly – I couldn’t connect them. I realised this on the first go, but did much better on the second and third try by means of just remembering what I’d done before – still, the first time was a surprise.

Costcutters’ is currently having a sale on chocolate fingers, so I indulge in my terrible vice. I feel they’re far too expensive at £1.79 each, but upon cutting down to £1 I suddenly feel the urge to stock up. I can finish off a pack a day, and sometimes more. Pringles, too.

The current Mathstuff logo is the first time I’ve seen GIF compression in a loong while.


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