Today I was trying out a “Math Jeopardy” game that a colleague had created, and one of the categories was “7 Letter Words”. An example of the sort of answer/question pair for that category:
ANSWER: This often is seen when the sun shines following a rain storm.
Question: What is a “rainbow”?
As I was reading through the questions in this category, my brain started anticipating “A word meaning `seven letter word'”.
Offhand, I didn’t know of a word meaning “seven letter word”. For that matter, I couldn’t immediately think of any words that meant “a word with n letters”, for any particular value of n.
But if such words existed… clearly, a word meaning “one letter word” would have more than one letter in it, since we can easily enumerate all the one letter words in English, and check their meanings. And it seemed pretty likely that a word meaning “one hundred letter word” would have fewer than 100 letters.
AH HA! I thought, for a fleeting moment… if such names start off too long, and eventually are too short, then somewhere in between they must be just right…, until I realized that there was no expectation of continuity, that any putative function for which f(n) = “the number of letters in a word meaning ‘word with n letters'” would map the natural numbers into the natural numbers, and so the intermediate value theorem need not hold.
A bit of thought, a trip to a latin dictionary, and then a forehead slap later, we had a few such words in mind:
- monoliteral (having one letter)
- centiliteral (having 100 letters)
Now the root “literal” has seven letters, so we cannot slap a prefix in front of it and get a 7 letter word, much less a 7 letter word meaning “has seven letters”. But if we can find number prefixes whose length is 7 less than the number they signify, we’d at least be able to create words whose length matched the length they aimed to describe. And happily, I did manage to create a few examples:
- duodeliteral (having 12 letters)
- undeliteral (having 11 letters)
- decliteral (having 10 letters)
Playing around with this suggests some other fun avenues for exploration:
- In English, the word “four” has 4 letters. A bit of thought is perhaps enough to convince you that no other english word could use the same number of letters as the word it represents. What would a proof of that look like?
- What happens in other languages? Are there languages where more than one word uses “its” number of letters? Are there languages where there are no such coincidences?
A far more general linguistic/logic topic: adjectives that apply to themselves. “Short”, or “polysyllabic”, or “English”. Perhaps “ostentatious”, or “unabbreviated”. Does “mispelled” count?
But then what of “Nonselfapplicable”? Does it apply to itself? Is “nonselfapplicable’ a nonselfapplicable word?
(I see this last paradox is just over 100 years old. That’s me, always late to the party.)
From now on, I will always associate Goldilocks and the Three Bears with the intermediate value theorem.