Collatz Conjecture Ramblings.

So playing with the Collatz conjecture, and I feel like this is something where if we can show for any n > 1, there exists a m such that f_m(n) < n , and then we simply prove this by induction, and we’ll be done.

I can show this is true for a list of items mod 3, 9, etc. and there is a clear pattern to demonstrate this by building on this:

n = k (mod 4) has 3n+1 = 3k+1 (mod 4), thus if k=1, 3n+1 | 4, thus Collatz will go 3n+1 / 4 before it could possibly become odd again, and thus we simply show 3n/4+1/4 <= n , which is relatively easy  and we are good to go with this one.

Similarly if n is even, we have immediately that n -> n/2 which is less than n.

So we have if n=k (mod 4) k=0,1,2, then n will eventually drop below itself using the Collatz procedure. What about if k=3?

Well, we have

3k+1 = 10 = 2 (mod 4). Now, obviously we will need to divide by 2, and we will get another odd number, because 2 / 2 (mod 4) = 1 (mod 2) (since 3*2, and 1*2 = 2 (mod 4)). This means though, that we have (3n+1)/2 = 1 or 3 (mod 4), and it isn’t hard to see how this becomes exponentially complex. Yet, to me, I feel like if I can show that f_m(n) < n for some m, then we have proven the conjecture. It just would take some chewing of this irritant to get there. It isn’t hard to see that you can expand this to cover n = k (mod 8) and get most n’s covered in this circumstance, but then you miss a few and have to go to mod 16, etc.

The question is, is there an obvious enough pattern to be able to prove that for n = k (mod 2^l) and use that as leverage…

We can wash rinse and repeat somewhat, but the math becomes somewhat complicated (and ugly) as the different pathways diverge. Yet, I think if there is a way to chew up the modular arithmetic, you’d be able to approach this using an inductive method.

Hrm… Times like this makes me wonder how hard it would be to get paid a decent salary doing math research again.

Happy Hockey Season

Blog - Panda Rose Consulting Studios, Inc.

Looks like we’re not the only one’s who had fun with Rogers Hometown Hockey’s Be a Broadcaster! Evanka Osmak and Ken Reid stopped by Rogers Hometown Hockey, where they interacted with the Be a Broadcaster activity, programmed by Panda Rose. First up Evanka Osmak, interacting with…herself?

Next up, her co-anchor Ken Reid

It’s great to watch these two cut loose and have a great time. To see where Rogers Hometown Hockey will be next, check out their website!

Happy Hockey Season!

View original post

Gameday MVP

This was an awesome project to work on.

Blog - Panda Rose Consulting Studios, Inc.

Pepsi Gameday 00006

We love our sports here at Panda Rose and we had a fantastic time working on the Roger’s Hometown Heroes project. When we were approached to develop another fun green screen program for Pepsi and Ruffles Grey Cup Game Day MVP we got really excited and go to work with a really fun interactive green screen program.

The program puts you in the Stadium, running through the entry way as the MVP of your favorite team! You can run though as your own great self, or in your favorite team’s uniform.Pepsi Gameday 00002

That was one of our favorite moments from last year. This year, we’re looking forward to creating even more fun and exciting programs to help people connect to their favorite activities.

Pepsi Gameday 00013

View original post

Forgetting Results

This is a wonderful discussion on some pretty straightforward math.

Gödel's Lost Letter and P=NP


Henry Smith was a mathematician of the 19th century who worked mainly in number theory. He especially did important work on the representation of numbers by various quadratic forms. We have discussed how even in something seemingly settled, like Joseph Lagrange’s theorem that every natural number is representable as a sum of four squares, new questions are always around—especially when one considers complexity.

Today Ken and I want to discuss a private slip of forgetfulness, and how often others may have done the same.

For a variety of reasons we recently were thinking about one of the most basic questions in linear algebra: the solvability of

$latex displaystyle Ax = b, &fg=000000$

where $latex {A}&fg=000000$ and $latex {b}&fg=000000$ are fixed and $latex {x}&fg=000000$ is to be determined. Over a field there is a polynomial time algorithm that determines whether there is a solution and finds one if there is. The…

View original post 2,062 more words

Roger’s Hometown Hockey and Panda Rose

Panda Rose has been doing lots of fun stuff.

Blog - Panda Rose Consulting Studios, Inc.

Hometown Hockey

It’s hockey season! And if you’re like us you’re excited to gather round with friends and family and watch all the big games. That’s why we were so excited when SDI marketing contracted us to help created a feature for Roger’s Hometown Hockey Tour’s Kid’s Zone!

The tour is rolling across Canada this winter, giving people all across the nation to come together and celebrate our national past-time with an exciting roster of events for each city. It’s an awesome day out for the whole family.

For our part, we were asked to help program the Roger’s Sportsnet’s Be A Broadcaster, a really fun way for Sportsnet fans to step onto the shoes of their favorite broadcasters and deliver their own highlight reel for their favorite Canadian hockey team.


Panda Rose took on the challenge, programming the portable green screen studio with a quick and easy way to place…

View original post 109 more words

Four ways we get drawn in by magical literature

At least 20% of my current to-read list...

At least 20% of my current to-read list…

Shadowed figures stare at lit boxes
Power of knowledge from all ages at their touch
Feline pleasures instead feel their eyes
And the story will continue without them.

I’ve been reading a lot lately, working my way through the Story of Fire and Ice series (aka. Game of Thrones books). I realized a trend to this series which seems to be overwhelmingly common amongst most science fiction and fantasy literature. A “trope,” if you may – A character or some all knowing creature who presents cryptic clues to the protagonists or antagonists which hints to future events to come.

This acts as a driver of interest for the reader, but leaves them guessing about what is going to happen next. I know that I enjoy this style of literature, but I am left wondering, why do I find this so fun to read? Why does it work so well as a story structure and keep me coming back for more?

Most importantly, how can a potential writer use this to their advantage to make interesting books?

Why does partially spoiling a book make that book so appealing?

It allows a small bit of the author’s real person to enter the story

When an author adds mystical clues to their story, they are subtly spoiling it for you. They want you to know what is coming, but they cannot just outright tell you. If they did, then they could skip the entire series that leads up to that release. However, by using an omniscient character in the novel, the author enters into the story, speaking directly with their own characters and the reader.

The writer has been dropping these clues throughout the book  like hidden treasure, giving all of the readers something to talk about around the water cooler. I have read tons of theories about Game of Thrones specifically because there are tons of cryptic clues hidden throughout the book and TV show. Each one built up from these cryptic clues and prophecies.

Note though, the prophecies must be cryptic. If they are not, then the book would be spoiled and boring. So remember that…

Most wrapped boxes are fun and exciting, until they are opened.

This is simple way to explain why LOST was so awesome until it wasn’t. The big problem with LOST was it felt like a Christmas tree, every episode being  loaded with more and more awesomely shaped and sized gifts, each of them hinting at some very cool and deep underlying universe where all of the bat shit craziness made sense. Each one a prophecy of something mind blowing to come. Yet, around season 4, viewers started getting a bit pissed, the piles of boxes kept piling up, but no one was opening them, there was no release.

Then, the worst came to past and it was clear why they weren’t being opened. They contained garbage and disappointment. Most of the prophecies were answered in a way tangential to the overlying story, and when they were found out, it felt more like seeing the wizard behind the curtain, than learning something foundational that made all of the other pieces fit together in a satisfying way.

If the wrapped box is a puzzling prophecy, that almost implies that the answers should not be lame or silly. Why? Because they are the structural answers to the story. Regardless, a puzzling prophecy is a mystery and…

Mysteries are cool and fun to figure out

Everyone loves mysteries. You can talk about it with friends. Come up with theories, each one progressively more complex. Yet, no one loves mysteries when you have to bang your head against the wall every page of a book.

I think cryptic prophecies are a fantastic halfway point for this. They give you just enough pieces to come up with intricate theories, but then reel it in every once and a while so you are brought back to the main story thread, and generally you learn the solution to the prophecy in pieces, not with one grand reveal (a la Sherlock Holmes.)

If you spread this out over multiple books, I think you pretty much guarantee yourself a fan club. If the book is good, people will find others online and create communities solely dedicated to trying to figure it all out. This works particularly well because they know the writer already has an answer to it and is slowly revealing it.

I’ve seen this happen with reality TV shows even. People love to try to suss out the intentions of the writer. We, in a way, enjoy being spoiled. As long as we are doing the spoiling to ourselves.

It gives the appearance of magic and purpose to our lives

This is deeper and more metaphysical. When we read stories, we find it easier to write our own story of our lives. The integration of prophecy and magic in a storybook seems to open prophecy and magic in our own lives. I don’t mean that people who read Harry Potter will suddenly discover how to cast spells and fly, nor do I mean that people who read Game of Thrones are going to see dragons flying around London.

Instead, it gives the seed of the child in our psyche. The possibility of a grander narrative to our lives. We read stories, sometimes of grand characters and sometimes of peons, but in the process we take on a bit of those characters, if only for an instant. We become the warrior, the maiden, the smith, the crone, the father, the mother or even the stranger, and while that moment is fleeting, it gives us a bit of magic ourselves. Those prophecies only apply to those characters in the book, but possibly there is some prophecy we don’t know that applies to us.

Some sort of magic that might give us purpose.

It draws us in to solve the cryptic prophecy, but in the end that didn’t even matter. It was only a dance between the reader and the writer.

The transcendence of being

It takes a couple of reads to understand, but it’s a good analysis of some pretty major philosophical structures underlying the idea of God.

Just Thomism

Hypothesis: Physics and mathematics are those discourses in which Nominalism involves no immediate contradiction. 

By “Nominalism” I mean the claim that predicated natures are beings of reason.* When we say “John is a man” then the predicate can be viewed without immediate contradiction as an ens rationis, abstracted from the concrete reality of “John” or “this man”.

But there is an immediate contradiction in assuming the same thing about “John is real” or “a being”. The real is divided by contradiction from the unreal or the being of reason. One is free to see “being” or “real” as the limit of abstraction, but the limit is not homogeneous with what leads up to it – at the most general level one hits something that does engender a contradiction if it is seen as merely logical, abstract, or ens rationis. 

But to leave it at this would give us the universe…

View original post 249 more words