# My (Not-So-Successful) Quest to Conquer the NYT Connections Game with Word2Vec

The New York Times’ Connections game: a fairly simple puzzle that has been rising in popularity. The objective? Find four groups for four within a larger sample of sixteen total words such that each subgroup has an overarching theme.

I thought this would be fairly easy to solve with some simple usage of word embedding and K-means clustering. After all, if it can figure out king – man + woman = queen, then surely it can figure out that these are all sandwich ingredients. There are enough models out there for topic modelling that it was easy to install a model in under 1 minute, and I just used a simple K-means.

However, I quickly ran into problems. The most major is the fact that K-means doesn’t always give four groups of four. Seeing as this was the case, I switched to a constrained K-means algorithm. Another thing I noticed is that the word embedding probably doesn’t account for the fact that repetition might be used (e.g. ‘tom tom’ rather than ‘tom’).

It’s curious to wonder what a better approach would be, as spending some 2 hours on this little question has proved to be not as fruitful, even for some relatively simple puzzles. Maybe a contextual embedding is needed, rather than just a GLOVE word2vec model.

I also thought a more curated, greedy algorithm might work rather than K-means. Take the two most similar words, and assume they must be a group. Average the two word vectors then find the next word from the now reduced list. I gave this a whack, but also didn’t turn out too well…

… maybe this is a more difficult puzzle than I originally thought.

Nevertheless, below is some sample code:

```import gensim.downloader
from sklearn.metrics.pairwise import cosine_similarity
from k_means_constrained import KMeansConstrained

words = [
'plastic', 'foil', 'cellophane', 'lumber',
'organism', 'harpoon', 'trudge', 'limber',
'stomp', 'elastic', 'glove', 'bassinet',
]

# Generate similarity matrix
word_vectors = [
model[word] for word in words # We assume all words exist in corpus
]
sim_matrix = cosine_similarity(word_vectors)

clf = KMeansConstrained(n_clusters=4, size_min=4, size_max=4, random_state=0)
clf.fit_predict(sim_matrix)

print([x for _, x in sorted(zip(clf.labels_, words))])
print(sorted(clf.labels_))```

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