Rubik's cube talk at CNU in Newport News, VA
I gave a 30 minute talk tonight to the CNU Pi Mu Epsilon Math Honors Society at Christopher Newport University (CNU)! There were about 60 people attending, with 10 inductees, their family members, the math department faculty, and other math enthusiasts at the school. The talk was more math based than other talks I've done before, but I was told to make the talk entertaining to a lay audience as well. I did a demo speedsolve and demo blindfolded solve at the start too, and the audience loved those!
The room we were in had a huge projector screen, so after the demos I showed the calculation for the number of positions to a 3x3x3 cube (8!*3^8*12!*2^12) / (3*2*2) and I talked about how the calculation is an overcount which we adjust by dividing by the (3*2*2). I did a very brief overview of cube laws, enough to explain where each factor in (3*2*2) comes from. I showed them the expanded form for the number of calculations (43,252,003,274,489,856,000) and I told them that the number of unsolved configurations to the cube is a prime number
After that I talked about God's number being 20 and it's discovery and announcement in 2010. I gave a brief overview of the history of the lower bounds and upper bounds being narrowed down, as well as explaining that it was a computer aided proof and that they split up all the positions of a cube into sub-cases (cosets) in order to prove the result. I didn't do very much detail on the math of God's number, I gave more of a historical overview of how it was proven.
I then brought up Randelshofer's 3x3x3 applet and showed them how you can use the order of a permutation as the base concept to twist three corners in place. Basically (RUR'U')*6 is an identity, and (RUR'U')*2 flips one corner in the bottom layer, I showed them how the alg:
(RUR'U')*2 D' (RUR'U')*2 D' (RUR'U')*2 D2
works to flip corners and explained how it is using the identity property of (RUR'U')*6 mixed with the fact that (RUR'U')*2 only flips the one corner on the bottom layer to flip all 3 corners. I didn't ever show them notation, I just did that alg on the main screen using the applet while explaining as I did it.
Lastly I did show them commutators (but I did not name them at first). I told them that to cycle a small number of pieces they can follow the process:
Identify a slice of the cube then:
1) Find a sequence of moves that destroys (i.e. changes) one piece on that slice
2) Turn the slice
3) Perform the destroy maneuver backwards to restore the piece from step 1
4) undo the slice turn
Again using Randelshofer's applet I showed them (R U R') (D') (R U' R') (D) and how it moves only 3 corners. I left this part very general and just said that these steps were a process to move a small, targeted number of pieces around on the cube. At the end of this segment I did tell them that this process had a name, and that these were commutators, which were a composition of permutations.
And that was the end of the talk! The audience seemd to enjoy it very much. There was enough math to keep the Pi Mu Epsilon inductees interested. A few of them came up and talked to me afterward and said they really enjoyed the talk, and one of them was a cuber! The parents and non-math relatives and friends really enjoyed the demos, and they liked the more interesting trivia info from the talk (# unsolved configurations is prime, God's Number, hearing the explanation that there was a structure for algs like the 3 corner flipper and 3 corner mover even if they didn't grasp the concepts fully).
I had a great time, and the faculty was really glad to have me as a presenter! Yay! I hope I get to go back again for another presentation at some point! I am also doing a cubing demo at a middle school next week for a math camp being held by the CNU faculty. That one will pretty much just be demo solves as I only have about 20 minutes. I may give them a very basic overview of the cube (center pieces determine the color of that face when solved, there are corners and edges not "sides" to solve, etc.). I will let everyone know how that goes afterward too!
Have fun all, and happy cubing!