# What is the Sparsest Word?

Side note: If you know a better way to describe an anagram that doesn't use all the letters except for "words within the word" or some other lengthy phrase, please let me know. I considered asking over at English Language & Usage but decided against it.

ORIGINAL IDEA: I had an idea for a puzzle that would be a challenge to find the word with the most other words within it. You didn't have to use all the letters in the original word and you couldn't use any letter more than the number of times it appeared in the original word. For instance, HELICOPTER contains all the letters for CLOTH but not HELLO because that would require using the same L twice. Imagine that someone gave you refrigerator magnets with each magnet having a single letter on it. They spell a word with the magnets and you can then pick any number of the magnets they used and rearrange them to make a new word. To normalize the scoring, the points for each word would be $\frac{\text{Number of words made from the word}}{\text{Number of letters in the word}}$. The highest scoring word would have lots of words from few letters and, therefore, would be very dense.

THE PROBLEM WITH THE ORIGINAL IDEA: If you start with the largest words you can find, there are so many words you can make from them that their scores are probably the best you're going to get.

NEW IDEA: Try to find the sparsest word possible. The ideally sparse word would be very long but there would be very few words you could make from those same letters.

Guidelines / Rules:

1. The score for each word will be $\frac{\text{Number of letters in the word}}{\text{Number of words made from the word}}$
2. The word with the highest score wins.
3. All words must appear in Litscape's Mammoth Uncensored word list.
4. All words made from your word must be found by the Words Containing Only tool on Litscape.
5. The Litscape tool can only handle words up to a length of 30 characters but I suspect this won't be an issue since the highest scoring word is likely to be shorter. I checked all the words in Litscape that were 30 characters or more and the highest sparseness score was 0.00807537. For a comparison, bob has a score of 3.0.
6. You are allowed to disparage my choice of dictionary tools. (Yes, SCOWL is very neat but there isn't a ready online tool I could find to find words in the word based on SCOWL's )

Bonus Round: Can you find a word that is even more dense than hippopotomonstrosesquipedaliophobic? Its Density Score is 1,514.46. I had to create my own tool for this check since Litscape couldn't process it so you are allowed to create your own as well. You still have to use Litscape's Mammoth Uncensored word list, though.

• Answer to your side note: You can use the term subanagram. This term is not as commonly used as the word anagram nonetheless it conveys the idea. It is used by the Zyzzva word search application published by Collins. – CodeNewbie Sep 28 '16 at 9:02

I found a word with a score 6:

'syzygy'

Also two more, scoring 5 (different from Sconibulus answer):

'crwth' and 'cwtch'

I found a 5 letter word that contains no shorter words, scoring 5, it might be hard to beat.

'Queue'

• This depends on the dictionary you use. The official wordlist used for international Scrabble contains EE, giving this word a score of 2.5. – paolo Oct 3 '16 at 10:16

Another score of 6:

glycyl

Interestingly it is in the html page for six letter words but the "words made from" tool gives zero results rather than one.

• Are you using the same link? It gives 1 result when I try, so it appears normal. – Maria Ivanova Sep 28 '16 at 19:48
• The "words out of these letters" tool does, but not the "words made from" tool (which does give the base result, for others) – Jonathan Allan Sep 28 '16 at 20:02
• The "words made from" uses a shorter word list. – Maria Ivanova Sep 28 '16 at 20:05
• I hardly think it should be a censored word – Jonathan Allan Sep 28 '16 at 20:15
• @JonathanAllan Depends how you use it. – Dan Russell Sep 29 '16 at 3:37