path in word graph 
The Word graph $W_n$ is the graph whose vertex set consists of all $n$-letter English words and whose edge set consists of all vertices that differ by exactly one letter in one position. For instance, in $W_4, \{hard, card\}$ is an edge. Find a path from wrong to right in $W_5.$

There needs to be a way to shift the r in wrong one place to the left, though I'm not sure how to do this. A word adjacent to wrong that might be useful include wring.  A path from wring to price would be wring, bring, brine, bride, pride, price, prise, prose, prone, drone. However, this doesn't seem very useful.
 A: I'm not really sure you're in the right tags, or even necessarily that this should be under math, but I've got an answer for you. The words in the answer are pretty uncommon, but googling showed me that they are indeed words. Later I will attempt the same using only more common words, and see if I can find a more "real" answer. In any case, here you go:
wrong
wrung
drung
drunt
daunt
dasnt
dasht
hasht
hacht
hicht
richt
right

Edit: Using only words in the Scrabble Dictionary, we have:
wrong
prong
prone
phone
phons
pions
lions
linns
lines
sines
sinhs
sighs
sight
right

I accomplished this with the following python program that uses Dijkstra's algorithm:
from dijkstra import Graph, DijkstraSPF
from tqdm import tqdm

def load_words():
    with open('words_alpha.txt') as word_file:
        valid_words = set(word_file.read().split())

    return valid_words

def d(word1,word2):
    t = 0
    for i in range(len(word1)):
        if word1[i] != word2[i]:
            t += 1
    return t

def near(word):
    s = set()
    for w in w5:
        if d(w,word) == 1:
            s.add(w)
    return s

w = load_words()
w5 = set([word if len(word) == 5 else '' for word in w])
w5.remove('')
w5 = list(w5)

w5g = Graph()

for i in tqdm(range(len(w5)), "Building Graph"):
    for n in near(w5[i]):
        w5g.add_edge(w5[i],n,1)

d = DijkstraSPF(w5g, "wrong")
print("\n".join(d.get_path("right")))

If you want to use the above to solve similar problems, you will need to install the packages. The commands to do so vary based on your OS and python installation, but in general they look like this:
$ pip3 install tqdm
$ pip3 install dijkstra

I pieced this together from a couple places, so it's probably not as graceful as it could be - it's definitely not as fast as it could be.
The dictionaries of words I used can be found here and here.
