Where Grey Matter meets Dark Matter |

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Episode 13 - 10 June 2009

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Special Relativity is possibly the most widely misunderstood theory in physics. It deals with relatively simple things like spaceships moving in a straight line (albeit really, really fast) and light beams, has a straightforward mathematical description, yet it has some of the strangest and most counter-intuitive outcomes.

Tie Interceptors. Credit: Jay

For example, in quantum mechanics someone might say, "Well, the electron isn't really orbiting the nucleus of an atom. And it can only have these particular energies, but none of these other ones. And what's more...blah blah blah." And that kinda makes sense: electrons are little buggers that you can't see or touch, so who knows? They probably do all sorts of crazy things. But Special Relativity lets us say things like, "If your twin brother goes up in that spaceship and orbits the earth for a few months, he's going to travel through time into the future. Then he'll be older than you." You can touch your brother (although he might not like it), and you can get some idea of how weird time travel is. Like woah!

The man, without his tongue poking out. Credit: Ferdinand Schmutzer (1870-1928).

A certain Albert Einstein put all this together back in 1905, by realising what certain fundamental assumptions about the universe resulted in. These assumptions are: That science is the same in all Inertial Rest Frames; and that the speed of light (in a vacuum, ie. not interacting with any stuff) is constant for everybody, regardless of their velocity.

An Inertial Rest Frame means when you don't feel any acceleration. The second assumption about the speed of light being constant is the thing that makes everything go crazy, but it has been shown to be absolutely true.

These two small assumptions lead to all the crazy predictions of Special Relativity:

- Lack of Simultaneity - Things can never be said to be simultaneous anymore. Something that happens at exactly the same time to one person happen at different times to someone else who moving at a different speed.
This video gives an excellent explanation of the effect on simultaneity:

- Time Dilation - Forward time travel. A clock going fast ticks slower than one that is stationary (for you).
This vid goes on about light speed and time, including the infamous light clock:

- Length Contraction - Things that are going fast are shorter than stationary ones.
- Universal speed limit - Because you have mass, you can never go as fast as light can go - regardless of how much Lean Cuisine you eat.
- E = mc
^{2}- This says that matter and energy are essentially the same thing, and if you can convert some matter directly into energy then you can get a BIG explosion (ie. nuclear explosions).

The really hard part for people to grasp is that all this is related to the fact that there is no 'preferred' frame. That is, there is no 'correct' measurement of time or length. Anybody's measurement is just as good as anybody else's. You can't just say that time 'appears' to be slower, although it 'really' isn't - who is to say that their measurement isn't the correct one?

This one has a more geometric explanation:

It's really hard to explain the whole thing without heaps of diagrams, lots of time and over-the-top hand gestures, but the above videos may help.

BETA:

Card counting is not the province of the human calculator, despite what the film 21 would have us believe. Nor is it only for the idiot-savant, so forget Rainman. Card counting in the game of blackjack requires only a few things: the ability to count up to about 20, divide by numbers 1 through 8, a moderate attention span, and deep pockets.

Marty Ross (Courtesy Marty Ross)

For this section, we called on the wisdom of Melbourne's beloved mathematical bum, Marty Ross.

There's two distinct sides to strategy in blackjack. First assume you know nothing about the cards, the best you can guess is that every card has a 1 in 52 chance of being dealt. Well then you should apply what is known as Basic Strategy. This amounts to looking at what you have in your hand, and what the dealer has in theirs - then deciding what the odds favour. It was first worked out by four US army guys in the 50s on their calculators. Depending on the exact rules and number of decks being used in a particular casino, you can figure out a table that tells you exactly what to do at each point. Or you can just find one that somebody else has worked out:

Your hand | Dealer's face-up card | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A | |

Hard totals (excluding pairs) | ||||||||||

17-20 | S | S | S | S | S | S | S | S | S | S |

16 | S | S | S | S | S | H | H | SU | SU | SU |

15 | S | S | S | S | S | H | H | H | SU | H |

13-14 | S | S | S | S | S | H | H | H | H | H |

12 | H | H | S | S | S | H | H | H | H | H |

11 | Dh | Dh | Dh | Dh | Dh | Dh | Dh | Dh | Dh | H |

10 | Dh | Dh | Dh | Dh | Dh | Dh | Dh | Dh | H | H |

9 | H | Dh | Dh | Dh | Dh | H | H | H | H | H |

5-8 | H | H | H | H | H | H | H | H | H | H |

Soft totals | ||||||||||

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A | |

A,8 A,9 | S | S | S | S | S | S | S | S | S | S |

A,7 | S | Ds | Ds | Ds | Ds | S | S | H | H | H |

A,6 | H | Dh | Dh | Dh | Dh | H | H | H | H | H |

A,4 A,5 | H | H | Dh | Dh | Dh | H | H | H | H | H |

A,2 A,3 | H | H | H | Dh | Dh | H | H | H | H | H |

Pairs | ||||||||||

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A | |

A,A | SP | SP | SP | SP | SP | SP | SP | SP | SP | SP |

10,10 | S | S | S | S | S | S | S | S | S | S |

9,9 | SP | SP | SP | SP | SP | S | SP | SP | S | S |

8,8 | SP | SP | SP | SP | SP | SP | SP | SP | SP | SP |

7,7 | SP | SP | SP | SP | SP | SP | H | H | H | H |

6,6 | SP | SP | SP | SP | SP | H | H | H | H | H |

5,5 | Dh | Dh | Dh | Dh | Dh | Dh | Dh | Dh | H | H |

4,4 | H | H | H | SP | SP | H | H | H | H | H |

2,2 3,3 | SP | SP | SP | SP | SP | SP | H | H | H | H |

**S ** = Stand

**H ** = Hit

**Dh ** = Double (if not allowed, then hit)

**Ds ** = Double (if not allowed, then stand)

**SP ** = Split

**SU ** = Surrender (if not allowed, then hit)

This table was taken from Wikipedia, and it applies to using 3 or more decks. For instance, if you've got 12 and the dealer has an 8, then you should 'hit' (ie. take another card). But if the dealer only has a 5 compared to your 12 then you should stand. There are fancy things like splitting and doubling which you can find out about if you're really keen. Have a look at the Wizard of Odds for heaps more info and exact details on returns using different strategies.

But even the the best use of this strategy will still result in the casino taking your money. What you need is more information, and that's where card counting comes in.

Card counting all starts with the observation that if you get dealt a card, say a 7, then there is now one fewer 7 in the deck remaining to be dealt. So if you keep track of which cards are dealt, then you know what's left to come. The other piece of the puzzle is knowing that basic strategy tells you that if there are more high cards than low cards, the player is more likely to win. So you just need to play when there's more high cards.

Let's use a thought experiment:

- Let's say that when we play the game we keep in mind a number called the Running Count.
- It starts at 0 at the start of the game.
- For every low card (2,3,4,5,6) we add 1 to the Running Count
- For every high card (say 10, Jack, Queen, King and Ace) we take 1 away from the Running Count.
- We just ignore the other cards.

Then, we have come up with a clever way of keeping track of how many high cards and low cards are left in the game, while having to only remember one number at a time. If the Running Count is 15, then that means there are 15 more high cards than low cards left. Which is good for you. If it's negative 15, maybe you should go get a drink at the bar.

Let's add one small extra piece of the puzzle. Now if the Running Count is +15 and there are 200 cards left to be dealt, then you know that somewhere in those 200 cards are buried more high cards than low cards. But if instead there was only 15 cards left to be dealt, then you know EVERY card left is a high card - and it's time to bet the farm. So take the Running Count and divide by the number of decks remaining. This gives you the True Count. The True Count is what really tells you how likely you are to win.

What we have just invented is the High-Low system of card counting (many years after it was really invented), and it was the secret of success of the MIT card counters in the film, and a whole host of others.

Of course, as Marty points out, the edge gained by this technique is very slim and, even if you stick by it to the letter, you can still wind up losing everything you walked into the casino with. So if you're going to gamble then maybe you should keep these rules in mind.

Note that although the hero in the film was able to do complicated mental accountancy in his head, card counting essentially involves adding and subtracting one number, then dividing by another number - hardly the stuff that requires off-the-charts mental powers. In fact, your average small business owner would make a better card counter than the best mathematician (who probably can't calculate a bus fare).

Child's Play.

Of course Rainman could do it.

Refs:

- Here's the book that the film was based on: Mezrich, Ben (2002),
*Bringing Down the House*, Arrow Books, London. - But this is the book that started it all (you can even see a copy of it in one scene of the film): Thorp, Edward O. (1966),
*Beat the Dealer*, Random House, New York. - The website Wizard of Odds has heaps of info on all sorts of gambling strategies.
- You can find out more info about Marty Ross at Maths Masters. He and another cool dude do public lectures, school demonstrations, articles for the Age, plus other stuff.

Quiz

- Here's the WMAP (Wilkinson Microwave Anisotropy Probe) image. The colours represent the small variations in temperature in the universe left over from the Big Bang. The average temperature is -270.41 C (that's 2.73 degrees above absolute zero, only slightly cooler than a Melbourne winter).

Source: WMAP Science Team and NASA - Here's the published data from the WMAP survey: Hinshaw et al. (2008), "Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Data processing, sky maps and basic results", arXiv:0803.0732.
- The furthest galaxy was reported here: http://www.universetoday.com/2004/03/01/record-for-furthest-galaxy-is-broken-again/. The info was published in:
Bradley L. D., Bouwens R. J. , Ford H. C., Illingworth G. D., Jee M. J., Benítez N., Broadhurst T. J., Franx M., Frye B. L., Infante L., Motta V., Rosati P., White R. L. & Zheng W., 2008, Discovery of a Very Bright Strongly Lensed Galaxy Candidate at z ~ 7.6. The Astrophysical Journal
**678**, 647-654.