Rubik's Cube Conversation over Breakfast

If you read my previous post then you know that I'm at HCSSiM for the second half of the second half, teaching a mini on the Fundamental Theorem of Algebra. I'm very impressed with everything, from the teaching faculty to the students, and am really enjoying soaking in interesting mathematics every day. There is really high level mathematics going on here, and the students are truly immersed in mathematical knowledge and culture.

Conversation at breakfast with Lucas and Gabe.

I walked into a conversation about Rubik's Cube records this morning at breakfast. Talking about new cube records, Lucas complained about kids who ask him if he solves the cube by just doing the same sequence of moves over and over again. Of course not! This would only work in a cyclic group!

However, I argued that it is possible, if you look at it another way. Suppose that g is a long sequence of moves that traverses through all possible cube positions. Then you only have to do the sequence g once, and somewhere along the way, you'll have solved the cube. Notice that the end result of performing the moves in sequence g is the identity permutation on the cube.

We can improve this by finding a cube permutation g that generates a large cyclic subgroup of the cube group. Let G be the cyclic subgroup generated by g. If we can express g as a long sequence of cube moves that traverses through a complete set of coset representatives of G, then we have the cube neophyte's dream: a sequence of cube moves, that it you do over and over again, will eventually solve the cube (of course, in the worst case scenario you'll move through all possible configurations of the cube, but I'm not making any claim about the efficiency of this method!)

We finished our breakfast conversation by posing a more reasonable problem: try this for a small group.

1. Show that g = (1, 2, 3)(4, 5) is an element of S_5 of maximal order.
2. Find a sequence of 120/6 = 20 permutations s_1, s_2, ..., s_20 whose product is g, and whose partial products (s_1), (s_1 s_2), (s_1 s_2 s_3), ..., (s_1 s_2 s_3 ... s_20) is a set of coset representatives of the cyclic subgroup .
3. Solve problem 2, where each s_i is a transposition.
4. Solve problem 2, where each s_i is a transposition of adjacent elements.

This seems like a good start to investigating the breakfast conversation problem. Let me know what you think!

HCSSiM 2010: Alive and Well!

I visited the Hampshire College Summer Studies in Mathematics (HCSSIM) program for the annual YP Day, and am happy to report the program is as vibrant as ever this summer.

Yellow Pigs Day

For those not familiar with HCSSiM lore and what YP day is, the initials YP stand for Yellow Pig, one of the two mascots of the program. David Kelly, the director of the program, has perhaps the largest collection of yellow pigs in the world.

The other mascot of the program is 17, the favorite number of scads of summer studies students and alumni. YP Day (Yellow Pig's Day) falls each year on July 17th, and is the traditional time for alumni to revisit the campus and reconnect with HCSSiM.

This Year's Faculty

Arriving at HCSSiM on Friday in time for lunch, I met the faculty. In addition to Kelly, we have Rob Hochberg and Josh Greene also leading the workshops. Rob, a computer scientist, and I go way back to the 1980's when we were both summer studies students, but this was my first time meeting Josh, a Columbia University mathematician, who attended HCSSiM in the 1990's. To round out the crew we have the amazing:

• Gabe, recent University of Chicago graduate
• Misha, math graduate student at CUNY
• Nadine, math graduate student at the University of Pittsburgh
• Achyut, Hampshire College physics major
• Peter, Harvey Mudd math major
• Nate, U Mass math major
• Julia, recent Hampshire College graduate
• Emil, Harvey Mudd math major

and of course the indefatigable Susan Goff, who has been the program assistant for over a decade.

Prime Time Theorem

I got my chance to meet the students later that afternoon, when I explained How to Solve a Cubic to them as Friday's Prime Time Theorem speaker. The students are as sharp as ever, and an enthusiastic audience. I ended my talk with some open-ended questions.

• Is there a quartic formula?
• Is there a quintic formula?
• Does every polynomial have a zero?

Although most of the summer studies students know the general answer, I hope that these questions and my talk will pique their interest for my Mini later this summer on the Fundamental Theorem of Algebra. Apparently another Mini will focus on Galois Theory, so HCSSiM definitely has this topic covered!

After dinner I had a chance to talk with the students during their problem session. They're enjoying some Algebraic Number Theory, Cardinality Proofs, and Group Theory.

Saturday, July 17th

Yellow Pig's Day fell on a Saturday this year, just perfect timing for the traditional Ultimate Frisbee Game, Kelly's talk on The Mathematical and Social History of 17, the Yellow Pig Cake and the Singing of Traditional Yellow Pigs Day Carols.

Saturdays start off with the morning workshops, though. I was able to wander between the three workshops and observe a wonderful proof of Wilson's Theorem, a listing of Group Theory axioms (after two weeks of examples), and some interesting Cardinality Proofs.

Mathcamp

Surprisingly close this summer is Mathcamp, another wonderfully rich math environment for high school students. Math camp is just a 15 minute drive from Hampshire College at Mount Holyoke College, I visited Dan Zaharopol's class on Real Analysis, and Alison Miller's class on p-adic numbers.

Among the more than 100 Mathcamp students, there are several who previously studied at HCSSiM, and I was intrigued discussing their positive experiences at both programs.

Ultimate Frisbee Game, HCSSiM 2010 vs. HCSSiM <>

My visit to HCSSiM 2010 ended with a grueling game of ultimate in excruciating heat. The alums were outnumbered three-to-one, but did well until the 2010 students developed a strategy that took advantage of their strength in numbers: they lined up and passed the frisbee down the field. Luckily, the aging alums still managed to come out ahead by dividing each team's score by the number of players, and then multiplying by the average age.I drove back home soon after dinner, and enjoyed a beautiful double rainbow all the way through the Berkshires. I'm looking forward to returning in a few weeks to lead a Mini on The Fundamental Theorem of Algebra.