Movie Game: Fred Astaire to Dane Cook, John Lennon to Gene Kelly

Some movie game fun to fill in for lack of maths. Now, these are far out there enough that even I had to resort to the IMDB, but there are still some interesting connections. First, John Lennon to Gene Kelly.

John Lennon to Michael Crawford, in How I Won the War (1967).
Michael Crawford to Phil Silvers, in A Funny Thing Happened On The Way to the Forum (1966)
Phil Silvers to Stockard Channing, in The Cheap Detective (1978 )
Stockard Channing to Oliva Newton-John, in Grease (1978 )
Olivia Newton-John to Gene Kelly, in Xanadu (1980)

Look out also for a brief appearance by the Third Doctor in A Funny Thing Happenend.

Next, Fred Astaire to Dane Cook, two different ways of doing this that I found.

Fred Astaire to Paul Newman, in The Towering Inferno (1974)
Paul Newman to Bruce Willis, in The Verdict (1982) [One of Willis' first parts!]
Bruce Willis to Justin Long, in Live Free or Die Hard (2007)
Justin Long to Dane Cook, in Waiting (2005)

I really like that one for getting to use that Bruce Willis – Paul Newman jump, but this next one also has its points.

Fred Astaire to Audrey Hepburn, in Funny Face (1957)
Audrey Hepburn to Sean Connery, in Robin and Marian (1976)
Sean Connery to Kevin Costner, in Robin Hood: Prince of Thieves (1991)
Kevin Costner to Dane Cook, in Mr. Brooks (2007)

This one is nice because not only does it use that little used Hepburn – Connery connection, but it also uses two Robin Hood movies in a row. And I do enjoy Kevin Costner. I really do.

These aren’t guaranteed to be the best solutions, but they’re certainly interesting ones.

BBC on the Brain

I love the BBC. Love ‘em. But that’s a personal thing.

More importantly, today the BBC has three interesting articles relating to brains, maths, and culture.

Firstly, dolphins. It appears as though a dolphin who spent some time in captivity 20 years ago (in a dolphinarium! Neat!) before being released has taught other members of her group ‘tail-walking’, a traditional trick among trained dolphins. If she did do so, and did so without instruction or prompting, it would suggest all kinds of interesting, like cultural transmission. And it serves as further proof of how neat dolphins are.

Secondly, it seems as though magpies have some degree of self-recognition. And this would be the first time such a thing has been documented in a non-mammal species. In short, stickers were placed on the magpie, and when confronted with its own reflection, the bird would attempt to scratch the sticker off itself. Once it did so, it calmed down. Certainly not conclusive, but interesting.

Thirdly, it seems as though human brains are, to some degree, hard wired for maths, and at least counting. In an experiment that involved counting without language, children whose native languages have no counting words were found to be just as numerically proficient as children who spoke english. It certainly seems to me that the near universality of mathematical thought and logic speaks to some inherent property of the human brain. Or, if I am to be honest, at least some universal property of human culture.

Interesting.

Sorry for linkage – getting ready to go back to campus in two days. Much to do, and miles to go before I sleep.

Nesting for Fun and Profit

Something for you to consider.

Interestingly, the above identity implicitly contains the solution the to puzzler posted two entries ago.

A Sequence of interest

For your consideration, I offer

1, 1, 1, 2, 1, 2, 1, 20, 1, 10, 1, 8, 5, 2, 5, 4, 1, 130, 1, 4000, 1, 2, 5, 52, 5, 494, 1, 40, 1, 10, 13, 4, 25, 38, 5, 16, 13, 230, 13, 20, 1, 46, 5, 104, 475, 62, 1, 20, 1, 130, 3, …

Make of it what you will. And I’ll make of it something else : )

Puzzler #Eleventy: Non-linear Fun

Consider the set of values that satisfy the following equations.

…

In general, for any n

Solve for, in general, . Bonus if you can figure out why the sequence of a’s is of interest.

Back…

Back from New Orleans … sort of. I had a great time : ) Walked through most of the French Quarter and the Garden District. Went down to the old Mint, but it was closed : ( Most things seem to be closed on Mondays, no doubt because people are still recovering from the weekend. I was shown around a cemetery by Shawn, the research director there. Have you ever wondered how they can fit 10, 15 people in those above ground tombs? I met a nice australian/german lady who was on a walkabout, and a couple from Wichita who were on a street car, just to see where it would take them. Picked up some interesting things there, including an 1853 3-cent silver piece, and a dollar coin from the 1962 Seattle World’s Fair.

I find them interesting, anyway.

So I’m back … in theory, at least. Coming up dry on maths to talk about. Maybe I’ll talk about vectors tomorrow … Or who knows. Hmmm.

Break!

Going to New Orleans for a few days : ) No maths until I get back. Consider this an open invitation to post, discuss, solve, etc, your own problems in the comments until then.

The Necessity of Quantum Mechanics

I’d really love to tell you all about why quantum mechanics, with all it’s bizarre results and logic-breaking conclusions that defy convention, is an absolute necessity, but Michael Nielsen has done a much better job than I ever could. The punch line is that whatever else you may say about quantum mechanics, to the best of our ability to tell, -that’s the way the world works-. Convention and intution is all well and good, just so long as they conform to and describe what is actually happening in the world. And what is actually happening … is truly bizarre.

Happy Thoughts

Consider a number. Any whole number will do. Consider that number, and then take the sum of the squares of the digits of that number. For example, 17 becomes 1² + 7² = 1 + 49 = 50. Repeat as necessary. 50 becomes 5² + 02 = 25 + 0 = 25. Repeating this process yields the sequence

After that 89, the sequence begins to repeat in periods of 89, 145, 42, 20, 4, 16, 37, 58.

Picking any number to start with, one of three possible behaviors occurs in the resulting sequence. A) the sequence starts with 0, and stays there, B) the sequence eventually hits a 1, and then that 1 repeats, or C) the sequence ultimately repeats in the loop given above.

7 is a good example of a number that produces a sequence terminating in 1’s. The sequence is 7, 49, 97, 130, 10, 1, 1, … etc.

Only these behaviors will occur.

It’s easy enough to show that this algorithm must produce periodic results. Consider an n digit number. It’s actual value is, at most, 10ⁿ⁺¹-1. Like the maximum value of a 5 digit number is 99999. But the maximum value of the next term in the sequence is 9²*n, giving 405 in the case of a 5 digit number. So most large numbers are drastically reduced in size. Smaller numbers, around 3 digits, are always kept between 1, 2, and 3 digits in the next step of the sequence. This means that the terms of the sequence are ultimately chosen from a finite number of possibilities. And, assuming worst case, where the sequence visited every single one of the possible 1, 2, or 3 digit values, eventually it runs out, and has to repeat, using numbers it’s already visited.

Very handwavy, but the sequence has to repeat. And it does, repeating 0’s, 1’s, or {89, 145, 42, 20, 4, 16, 37, 58}.

Which makes me think, basically what we’re doing is taking a number n, and then taking the sum over the digits of n of f(x), where f(x) = x². What about other polynomials of x? What if we cubed instead of squared? It’s easy enough to show, using a similar argument as above, that any polynomial f(x) must yield periodic behavior for any starting number.

Jumping into Mathematica quickly, for f(x) = x² + 1, starting with 15 yields a sequence that ultimately repeats with groups of {28, 70, 51}. Starting with 16 repeats in groups of {52, 31, 12, 7, 50, 27, 55}. And there are likely other periods as well.

Which makes me wonder, for general f(x), how many different loops will there be? How large will they be?

I don’t really know… Something to think about later, when I’m not working v.v

Oh. And I completely forgot to say. For using the f(x) = x² we started with, if a number produces a sequence of 1’s, that number is said to be happy. If it produces that longer loop, it is said to be unhappy. 7 is the first prime number that is happy. Happy primes : )

Coin Tosses and Probabilities

Flipping a coin 200 times in a row, what is the probability of getting, at any point, 6 heads in a row?

Disclaimer: I hate probability calculations. I’ve never felt myself to be any good at it. So there is certainly a chance I’m wrong. But even if I’m wrong, what follows below the fold certainly seems interesting.