Christopher Mowla Math 3900 November 18, 2011

1. Christopher Mowla Math 3900 November 18, 2011

2. The Rubik’s Cube

3. Rubik’s Cubes of Order n • A Rubik’s Cube of order n is referred to as the nxnxn cube, where • A cube with n = an even integer value is referred to as an even cube. • A cube with n = an odd integer value is referred to as an odd cube. • A cube of size n = 4 and larger are referred to as big cubes.

4. Other commonly solved cube sizes are:

5. n = 2

6. n = 4

7. n = 5

8. n = 6

9. n = 7

10. n = 9

11. n = 11

12. Piece Types of the 3x3x3

13. Corners: They each have 3 stickers.

14. Edges: They each have 2 stickers.

15. (Fixed) centers: They each have 1 sticker.

16. They are fixed on a xyz axis.

17. Piece Types of the nxnxn • Unlike the common 3x3x3 Rubik’s Cube size, there are 4 additional types of pieces on the nxnxn cube, in general. • Therefore, there is a total of 7 different pieces types which can be seen on larger cube sizes. • Side Note: We will exclude the 1x1x1 cube size for consistency (it’s an exception in more than one respect).

18. Types of Pieces (7) • Corners • Edges • Middle Edges • Wing Edges • Centers • Fixed Centers • Non-Fixed Centers • X-center pieces, oblique center pieces, and + center pieces.

19. For illustration, we will construct cube sizes up to the 7x7x7 to see all 7 of the possible piece types. • The standard 3x3x3 Rubik’s Cube does not have all of the possible piece types. • The 7x7x7 Rubik’s Cube has all of the possible piece types.

20. We can construct all cube sizes from the 2x2x2 cube because all cube sizes fromnù 2 have 8 corners.

21. Constructing the nxnxn

22. Odd Cube Size Construction

23. 2x2x2  3x3x3

27. 3x3x3  5x5x5

28. New piece types on the 5x5x5 • There are three new piece types on the 5x5x5: wing edges, X-center pieces, and + center pieces.

29. Wing Edges

30. (They are commonly called wing edges by the cubing community due to their symmetry.)

31. X-Center Pieces

32. They are called X-center pieces by the cubing community because they form an X about the composite center.

33. + Center Pieces

34. These are called + center pieces (many refer to them as T-center pieces as well) because they form a plus sign about the big cube center.

35. 5x5x5  7x7x7

36. Wing Edges

37. For the 7x7x7, we have a second set of 24 wing edges. • The term orbit is used to differentiate different sets of wing edges. It means where the pieces are able to move. • On the 7x7x7 cube, we have 2 orbits of wing edges.

38. X Center Pieces

39. + Center Pieces

40. New piece type on the 7x7x7

41. Oblique Center Pieces

42. The term oblique is used to describe these center pieces because they are neither X-center pieces nor + center pieces.

43. Permutations • A permutation is an arrangement of objects of the same type in some order. • Permutations can be decomposed into cycles.

44. Definition of a Cycle • An n-cycle is moving n pieces (2 or more) of the same type (i.e. edges, corners, or centers) at the same time so that, when the algorithm generating the cycle is repeated exactlyn times (and not until then), the cube will be restored to the original state it was in.

46. Examples of N-Cycles

47. 3-Cycles(Permutations of Middle Edges and Corners on the 3x3x3)

48. 2-Cycles(Permutations of Wing Edges on the 5x5x5)

49. 4-Cycles(Permutations of Wing Edges on the 5x5x5)

50. Combinations of Disjoint Cycles • Definition:If an algorithm affects 4 or more pieces of the same type on a cube (corners, edges, or centers), all of the pieces affected need not be part of an n-cycle. • In other words, n pieces of the same type affected by an algorithm  an n-cycle of pieces.