MATH 480 Senior Seminar (Spring 2015)

Most of the course information is now at the Spring 2015 Laulima “Wiki”.

Word Distance Project

Word distance metrics in one of the following databases:

Make a definition of distance between words. For instance, you could use the number of occurrences of $A$ and $B$ exactly 9 words apart, divided by the number of occurrences of $A$ and $B$ at most 9 words apart.

Select 10 words (or more) and compute all distances between them.

Check whether your distance $d(A,B)$ satisfies the following axioms for a metric (if they don’t, your definition may still be okay if it has other nice features):

$d(A,A)=0$

$d(A,B)=d(B,A)$

$d(A,B)\le d(A,C)+d(C,A)$

Make a graphic representation of the 10 words so that words $A$ and $B$ are close to eachother in the graphic if $d(A,B)$ is small.

Pocket cube project

There are many articles available about Rubik’s cube but not as many about the simpler pocket cube (2x2x2).

Here are some draft instructions:

First solve the white side.
Get two of same color in the layer
Face the opposite two and have white face up.
Do:
$$R^{-1} D R L D^{-1} L^{-1} R^{-1} D R$$
Now we have the layer.

Now to swap two corners (recall that yellow should be opposite white),
face them towards you and do
$$L^{-1} U^{-1} L F U F^{-1} L^{-1} U L U^2$$

Then finally do (with problem corners facing you and white at bottom)
$$R U R^{-1} U R U^2 R^{-1} U^2$$
(possibly repeat)

But if two already solved put solved corners furthest away from you
do $R U …$

If not done yet,
put the newly solved bottom *right* in bottom *left* and repeat $R U…$ maybe twice (this may intermittently mess up the two solved ones!).

To complete this project, your write-up will have to show exactly how the algorithm works on a couple of random cubes that I provide.