# Rubik’s Cube Cracked

I am no expert when it comes to the Rubik’s cube. As a kid, my strategy was to start with a completely solved cube and to rearrange in such a way that I could always reverse my steps. I’d go a little further each time until I couldn’t get it back, at which point I would try a series of random rearrangements until I eventually gave up and waited for someone else to “heal” my cube. Occasionally I would even resort to the guaranteed method of disassembling the cube by popping out the pieces and sticking them back in their appropriate location. It didn’t take long for this to destroy the cube so that it was practically impossible to play with any longer. Only once did I resort to removing the stickers and putting them back on. After only once they start to fall off on their own. That wasn’t too bad since a Rubik’s cube with missing stickers is certainly easier to solve.

I know you’d probably expect a mathematician to be able to have mastered the art of the cube but I just never took the time to think it through and apply a logical approach. Before my mind goes, I plan on getting one and conquering it once and for all. I was quite impressed when one of my students from Wayland was able to quickly solve one from any arrangement. By quickly I mean a couple minutes, maybe even less than a minute. That was impressive, but not as impressive as the kid who can study a Rubik’s cube then put on a blindfold and completely solve the cube. If you don’t believe me: Check this out.

I was reading MathTrek, and learned that Daniel Kunkle, a computer scientist at Northeastern University has proved that 26 moves are enough to solve ANY Rubik’s cube (3x3x3), no matter how scrambled. There are approximately 43 quintillion possible configurations for a 3x3x3 Rubik’s cube (that’s 43,000,000,000,000,000,000). When you eliminated equivalent configurations such as rotations and color symmetry, you still end up with just over a quintillion. It was interesting to see how you can show that 26 is enough considering that is beyond the scope of having a computer simulate every possible configuration.

# Meteor Shower Weekend

Be sure not to miss out on the meteor showers this weekend (Sunday night – Monday morning). It is a new moon coinciding with the Perseid shower. Mars will also be visible to the naked eye.
Just in case you are curious:

The Perseids (PURR-see-idz) are a prolific meteor shower[1] associated with the comet Swift-Tuttle. The Perseids are so called because the point they appear to come from, called the radiant, lies in the constellation Perseus. However, they can be seen all across the sky. Because of the path of Swift-Tuttle’s orbit, Perseids are mostly visible on the northern hemisphere. The shower is visible from mid-July each year, with the greatest activity between August 8 and 14, peaking about August 12. During the peak, the rate of meteors reaches a hundred or more per hour.(from Wikipedia)

Because this shower is visible from any direction, it is best to pick out a darker patch of sky and simply wait for “shooting star”. At its peak, you can expect to see 60 per hour.

# Consecutive Composite Integers (or Prime Gaps)

I was browsing through some unread blogs in my RSS feeder and saw something that I thought was interesting, so I thought I’d pass it along.

From think again!

– I read somewhere that there are an infinite number of them, but at the same time I have found areas where there are none!
– What do you mean?
– I have found one hundred consecutive integers none of which is a prime!
– What do you mean by consecutive?
– They follow one after the other. For example 32, 33, 34, 35, 36 are five consecutive integers none of which are prime.
– I see. But you have found one hundred consecutive integers and none of them is a prime?
– Did I say one hundred? I meant one hundred thousand!
– You are pulling my leg right?
– I thought I was pulling the left.

It turns out that the longest stretch of non-prime, or composite, integers in a row that has been found so far is 1,442. This is based on a search of numbers up to 10^18. (http://hjem.get2net.dk/jka/math/primegaps/maximal.htm). But someone posted an even better answer in the comments of the think again! website. It’s fairly easy to show that for any natural number, n, there exists a prime gap of that size, that is, n consecutive composite integers:

Theorem: For any natural number, n, there exists a set of n consecutive integers such that none are prime.

Proof: Let n be a natural number. Clearly (n+1)! + 2 is divisible by 2, since both (n+1)! and 2 are divisible by 2. By the same reasoning:
(n+1)! + 3 is divisible by 3
(n+1)! + 4 is divisible by 4

(n+1)! + n is divisible by n
(n+1)! + (n+1) is divisible by n+1

Thus the numbers from (n+1)! +2 up to (n+1)! + (n+1) represent a list of n consecutive integers, none of which are prime. Q.E.D.

I thought that was pretty slick.

# Personality.info

ISTJ Career Matches

ISTJs are often happy with the following jobs which tend to match well with the Examiner/Protector personality.

 Accountant Administrator Auditor Computer Programmer Computer Specialist Dentist Detective Doctor Electrician Executive Financial Officer Judge Lawyer/Attorney Librarian Manager Marketer Math Teacher Mechanical Engineer Military Leader Police Scientist Steelworker Systems Analyst Technical Specialist Technician

What does all this have to do with the title of this entry, “Academia versus Industry”? Well, during lunch, my colleague and I began discussing with our trainer exactly how he came to be in this line of work. He told of his prior employment in academia, at which point my colleague inquired as to why he made the transition from academia to industry. This started a very long discussion on the pitfalls of academia and the many advantages of entering into industry. You see, my colleague is seriously considering moving into industry, away from academia, and he has a number of strong opinions on why the current system of obtaining funding and “publish or perish” is flawed.

Among the advantages the listed for industry were the following:

1. Cooperation/Teamwork:
In academia, you are an island. If you are the principal investigator, you gain very little by helping your collaborators since you make your name by the papers you produce and funding you obtain. Cooperation only goes so far as if it benefits your lab and your research. In the end, you are the business all by yourself, hiring assistants, post-docs, technicians; obtaining necessary equipment; account management; obtaining grant funding; etc. In industry, you are part of a much larger team, working together with many resources at hand.
2. It is not publish or perish:
There is still competition, but in a completely different environment. Many people have argued that the peer review system is broken so the publishing of papers is not as significant as it once seemed. Your success as an individual does not depend solely on your ability to publish your research, it depends on the success of your team in accomplishing its goals, largely in the application of the science you are researching.
3. Avoids the flawed funding system:
The case they made is that currently only about 10% of all grant proposals get funded and that is still declining. In fact, 7 years ago the NIH funding rate was closer to 20%. Now, before you can get a position as an assistant professor you have to prove that you can obtain extramural funding. But the current state of affairs is that only those people who have made a name for themselves are getting funding. So you can’t make a name for yourself until you get funded, but you can’t get funded until you have advanced in the field. The problem is just as bad for tenure-track professors trying to break into the funding cycle. Shouldn’t there be more money set aside and available for new researchers?

All of these arguments were made by the trainer and my colleague and, in fact, there were several more that I eventually tuned out. I occasionally tried to make the case for academia, but I was outgunned considering my lack of experience in this particular field. I, personally, have made no decisions about what I am going to do at the end of my couple of years of post-doc research as a bioinformaticist. In my mind, the field is wide open. I may even end up back at a teaching university like Wayland or DBU. I still am turned off by the closed, proprietary nature of industry. Clearly, the competition between corporations stands starkly opposed to openness in scientific innovation. Nevertheless, the current system in academia doesn’t promote this openness either.

Where will I be in 3-5 years? This question is more open than ever.

# Friday Random 10

The Friday Random ten is finally back, after months on sabbatical:

“God” by Rebecca St. James
“Do You Dream Of Me” by Michael W. Smith
“Mambo Number 5” by Lou Bega
“The Long Run” by Eagles
“I Am (Album Version – Guitars Up Version)” by Mark Schultz
“Crown Him With Many Crowns” by 2nd Chapter Of Acts
“Solid Men To The Front” by John Philip Sousa
“Stagger Lee [Bandstand Version]/Bandstand Version” by Lloyd Price
“Bring Me Life” by Evanescence
“Polkarama! [Main Version]/Main Version” by Weird Al Yankovic