assignments
due Thu Sep 8
getting started
- Play around with the online TopSpin puzzle. What sorts of things are you trying? How are you thinking about it?
- Play around with a Rubik's Cube. Again, pay attention to what sorts of things you're trying. What do you think a "solution" will look like?
- Browse the links I've set up at the left of this page.
- Start looking the notion of a mathematical group, as described at wikipedia:group (mathematics) and intro group theory at dogschool. (Both on the resources page in the links at the left.)
- Using the "submit work" link, tell me:
- What is your math background?
- How much experience do you have with Rubik-ish puzzles?
- How did this assignment go?
due Thu Sep 15
commutators and the dihedral group
- In class we defined (R, L, F) as the (right, left, flip) operations on the TopSpin. We examined the sequence (R F L F), found how many times that can be repeated before it loops around back to the original, and discussed why that is the "commutator" of R and F, written [R, F], and what that means.
- First, review what we did from your notes (or a friends), and look at the "commutator" definition on the definitions page (see the links in the left margin).
- Second, do a similar analysis for R2 and F. In other words,
- Write down the [RR, F] commutator in terms of R, F, and L. (In words its "rotate right twice, flip, undo the rotations, undo the flips").
- Do that on the TopSpin puzzle, and write down where the numbers go.
- Figure out how many repetitions of that it takes to return the puzzle to its original state. Does anything interesting happen along the way?
- Also in class we started to work out the group operation table (sometimes called the "Cayley table") of D4, rotating and flipping the numbers (1,2,3,4) in a square grid.
- Do a similar analysis for D3, the symmetries of solid equilateral triangle. There are 6 group elements: the identity, two rotations, and three ways to flip it upside down. Name these six elements, and work out the 6 x 6 group operation table describing what happens when you do two operations in a row. Try this on your own first. If you need help, you can look online at wikipedia:Cayley Table and wikipedia:dihedral group of order 6.
- Start thinking about the rotations of a solid cube: how many there are, and how they combine.
due Thu Sep 22
permutations and cycle notation
- Browse through wikipedia's article on permutations.
- S4 (the symmetric or permutation group of 4 symbols) has 4! = 24
elements. The rotational symmetries of a solid cube also has 24
elements, as discussed in class. Are these the same groups? Why or
why not? (Think about the cycles of the different group elements and
which ones are similar to each other.)
- We've now discussed in class several times one way to solve TopSpin,
namely to do (R F L F) twice, which shifts once piece relative to the
others. (I'm using left-to-right notation here.) Then by lining that
piece up where it was and doing this many times you can shift one
piece all the way around, eventually shifting it by just one place.
Use this trick (and any others you like) to solve
the TopSpin puzzle (if you haven't already), and discuss how that goes.
- Consider a 2x2x2 Rubik's cube, which is mostly like just
looking at the corners of a 3x3x3 and ignoring everything else.
If I unfold it and put different number for each it would
look like this :
+----------+
| |
| 1 2 |
| |
| 4 3 |
| |
+----------+----------+----------+----------+
| | | | |
| 6 5 | 15 16 | 17 20 | 21 24 |
| | | | |
| 7 8 | 14 13 | 18 19 | 22 23 |
| | | | |
+----------+----------+----------+----------+
| |
| 9 12 |
| |
| 10 11 |
| |
+----------+
Now if I rotate the top face that has the numbers one
through four, the numbers below also cycle. In the
permutation notation from class, this move is then
rotate_top = (1,2,3,4)(15,6,21,17)(16,5,24,20)
What are some of the other moves on the 2x2x2 cube
in this notation?
- Optional: what is the smallest non-Abelian group?
due Thu Sep 29
similarity; odd & even; 2x2x2 Rubik
- Last class I defined what it meant for two group operations to be similar, namely that A and B are similar if there is X such that A = X B inverse(X) , where "inverse(X)" means the operator that undoes X.
- For the 24 element solid cube group, find X in the relation above to show that two different 1/4 turn face-center axis rotations (A and B) are similar.
- Do the same for two opposite corner axis 1/3 turn rotations.
- If a 1/4 rotation is one move, which of the 24 positions are an odd or even number of moves from the solved position?
- Start working through a group theory examination of the 2x2x2 rubik's cube (i.e. a regular cube ignoring everything but the corners).
- What are the generators?
- How many group elements (i.e. positions) are there?
- What is the commutator of two of the 1/4 turn generators? (We discussed this in class a bit.)
- What is the order (i.e. length of the cycle) of that commutator?
- What is commutator of a 1/4 turn move with a 1/2 turn on an adjacent side.
- What is its order?
- Can you find any other pattern of moves that only
- Is it possible for a commutator to move only two of the corners? Why or why not?
due Thu Oct 6
2 x 2 x 2
- Your main job this week is try to find a way to solve the 2x2x2, either by finding your own tricks based on our class discussion, or to learn and explore one that's described online. (Google "Rubik 2x2x2 solutions" or similar phrases).
- Besides using a real cube (the corners on a 3x3x3 works if you ignore the edges and centers), you should also try typing move sequences in; use either the CubeTwister or Jaap's cubie webpage :
- On the submissions page, describe your experience:
- Were you able to sove the 2x2x2?
- What move sequences did you find helpful, and what do they do?
- Are any of those sequences similar to the commutators and conjugates discussed in class?
- Come to class ready to discuss.
- Also, browse through the puzzles at http://www.jaapsch.net/puzzles/ and think about what you might want to do for the term project.
due Thu Oct 13
3 x 3 x 3
Now that we've talked all the way around the ways that people think about the Rubik's Cube, the
group theory that goes into, the notions behind the move sequences and all the rest:
- Using the ideas presented in class or solutions you can find online (google "rubik cube solutions"), find a way to solve the 3 x 3 x 3 Rubik's cube.
- Describe which method you adopted, and how easy or hard it was.
- If you already can solve the 3x3x3, look into another different method.
due Thu Oct 20
propose term project
- Pick a puzzle for your term project. Start thinking about it, and describe what you know so far and your plan of attack.
due Thu Oct 27
change ringing
- Browse through the links on my Oct 20 lecture notes. Pay particular attention to the plain hunt on 6 bells, and make sure you know how that works.
- Come to class on Thursday ready to try the plain hunt.
- Come up with your own version of the exercise that we did in class, namely, invent and/or explain how to change ring all 24 permutations of 4 bells. Only neighbor swaps are allowed. Explain what each person 1 through 4 does by giving the pattern that they ring (i.e. twenty four numbers 1 though 4, for where they ring their bell) and how you think they might remember that pattern. (You might want to start with the plain hunt, and find something to add.)
due Thu Nov 3
wallpaper 1
- Read the articles listed at the top of the Nov 3 lecture notes, and come to class ready to discuss 'em.
- Describe the symmetry structure of at least two of the Escher patterns.
due Thu Nov 10
NxNxN and 4D cubes
- Browse through the material and links on my Nov 10 lecture page.
- Come to class ready to discuss solutions and number of positions for the 4x4x4, 5x5x5 and larger cubes.
- And take a look at the 3x3x3x3 4D cube and see if you can make sense out of how it's pieces move around.
- Write down the answers to these questions:
- 1. On a 6x6x6 cube, how many different sorts of cubies (i.e. disjoint sets that can move into each other's positions) are there? The corners, for example, are one sort.
- 2. Also on the 6x6x6, what parity and turniness/twistiness constraints are there on what can move where?
- 3. On the 3x3x3x3 four dimensional cube, how many cubies are there?
- 4. If we ignore orientations of the 4D cubies, and give each cubie a single number, what permutations of those numbers are the quarter turn moves (i.e. the group generators) ?
due Thu Dec 1
project presentations
- Come to class ready to show and discuss your puzzle.
due Fri Dec 9
final projects
- Submit your final project paper, in which you
discuss a puzzle we haven't done in class, including
- how to solve it and how to think about and/or understand it
- and/or how other people solve it (cite sources; explain)
- how to write it down (notations)
- how the group theory concepts we've discussed fit: generators, number of positions, commutators, orbits, ...
- anything else you think is interesting or cool about the puzzle
- Pictures are good. Citations and sources are good. Clarity counts.
term grade
- Just a place for Jim to record a semester grade.
