# Mathematics for the English major

An entry in Fortnightly Topic Challenge #48: Unusual tag mix

I was looking at the unusual tag mixes post, and one of the ones listed is and . I thought "who's going to be silly enough to blend those?". +15 points for correct answer: this guy!

Here's the question: how many different grammatically correct simple sentence diagrams can be made starting with the base sentence "bludgeons humanise", and using all four of the words "blonde", "nigh", "midst", and "abaft"?

Note that I use "sentence diagram", instead of sentence, because what we're really counting is the number of ways these six words can relate to each other as parts of speech, ignoring the complications of making an actual sentence out of them. This would generally involve the addition of connecting words and articles, but we do not wish to consider those here.

### Rules

1. The sentence diagram should be for a simple sentence, having one independent clause with subject "bludgeons" and verb "humanise".
2. For the purpose of this puzzle, the words given can be used as the following parts of speech, independent of context:
• bludgeons: noun
• humanise: verb
• midst: noun, preposition
3. Word ordering is irrelevant here, so "quick brown fox" and "brown quick fox" are considered the same. All that matters is the "what modifies what" relationships, much as is usually displayed in a sentence diagram. This includes ordering within the diagram.
6. Prepositions must have a noun object, which may itself be modified by an adjective, which may itself be further modified.
7. A prepositional phrase may be used as either an adjective or an adverb, and may as a whole be modified according to the rules of the part of speech it is taking on.
8. The verb may be used either transitively or intransitively.

Examples

Two similar, but slightly different examples:

"abaft" is an adverb, which is modifying the verb "humanise". The adverb "abaft" is modified by the prepositional phrase "midst blonde", acting as an adverb. The noun "blonde", as an object of a preposition, is being modified by the adjective "nigh". To make this into a "real" sentence, it would read something like "Bludgeons become more human behind the middle of a nearby blonde."

Again "abaft" is an adverb, which is modifying the verb "humanise". The adverb "abaft" is modified by the prepositional phrase "midst blonde", acting as an adverb. But now the prepositional phrase "midst blonde", acting as an adverb, is being modified the adverb "nigh". This might be realized via "Bludgeons become more human behind the nearby middle of a blonde."

The meaning difference is subtle, but potentially important for the blonde, since the former implies the integrity of the blonde (since the blonde is nearby), but the latter only insists that the middle of the blonde be nearby.

As with any "real" combinatorics problem, there is some case bashing involved, but I hope the context keeps it interesting. I hope you enjoy!

• All four words exactly once each and no others? Feb 1 '21 at 0:19
• Rather than referring solvers to another reference work to identify what parts of speech your six words should be considered as, wouldn't it be better to say so directly in the question? That would make it more self-contained and save redundant effort on the part of solvers. Feb 1 '21 at 0:21
• @GarethMcCaughan Yes, all four words exactly once each and no others. I understand what you're saying about the reference, but there is method behind the madness. There is an interesting aspect here that I think moves this from math problem to math puzzle, and I'm trying very hard not to give it away. Feb 1 '21 at 0:28
• +1 just for this quote: "As with any "real" combinatorics problem, there is some case bashing involved" Feb 1 '21 at 3:03
• If you were just hoping people would miss the possibility that the verb might become transitive, I'll say that it seems to me that puzzles are seldom improved merely by having opportunities for error. And when you're considering "sentences" like "abaft nigh midst bludgeons humanize blonde", I think you have to make the grammar you're using formal and explicit, because it sure as hell isn't anything like "whatever is actual valid English" :-). Feb 1 '21 at 4:38

Disclaimers and checking-we're-on-the-same-page:

• The ritual of "diagramming sentences", which as I understand it is popular in the United States, has thankfully never really been a thing in the UK. It is therefore possible that I may not be understanding the question quite right.
• I think that one thing we are supposed to keep intact the intransitive use of "humanise" here; that is, there is no question of blondes or midsts being humanised.
• The Moby list has various derivatives of "humanise" but for the verb itself has only "humanize". I take it this is what we are supposed to use.

Moby says:

• "blonde" is noun and adjective
• "abaft" is adverb and preposition
• "midst" is noun and preposition
• "nigh" is adjective, verb, and preposition. (We have no way to use it as a verb; I assume we aren't supposed to be allowed to inflect it into "nighing" or anything like that.)

Suppose first of all that we don't use any of those words as prepositions. Then "midst" is a noun, and in fact we have no way to use it without changing the fundamental sentence structure (e.g., by making "midst" the object of "humanise"). So we need at least one preposition.

OK, back to the possibility that I expect to yield a lot more cases: we have exactly one preposition. If so, then "midst" must be either that preposition or its object. (If not, "midst" is not a preposition since something else is our only preposition, and it is not a noun because the only role for a noun is as preposition-object, and something else is playing the only such role; but it can't be anything else.) If "midst" is preposition, the only noun to attach it to is "blonde". So, our options for the prepositional phrase are "midst blonde", "abaft midst", and "nigh midst" -- except that the noun might be modified by an adjective (blonde, nigh), so add "midst nigh blonde", "abaft nigh midst", "abaft blonde midst", "nigh blonde midst". If we have a spare adverb ("abaft" is the only one) we can modify the adjective with it, adding "midst abaft nigh blonde" and "nigh abaft blonde midst". (Again, this is not real English, it's artificially-formalized English.) In all these cases, any remaining words must function as adjectives or adverbs; we have no further use for nouns, verbs, or prepositions.

Let's take those possibilities one by one. For each, our prepositional phrase might function as adjective or adverb; we'll split those out too. First, the 2-word prepositional phrases.

• "midst blonde" (adj). We now have "midst blonde" (adj), "abaft" (adv), "nigh" (adj). The only thing to do with our adjectives is to modify "bludgeons" with them. Then we can attach "abaft" to either adjective or to "humanise". 3 possibilities.
• "midst blonde" (adv). We now have "midst blonde" (adv), "abaft" (adv), "nigh" (adj). The adjective has to modify "bludgeons". Now the two adverbs can modify one another and then either "nigh" or "humanise" (2 orders for them times 2 things to attach to: 4 possibilities); or they can act separately (2 choices for which word each one modifies: 4 possibilities). So 8 possibilities overall.

Now the three-word prepositional phrases.

• "midst nigh blonde" (adj) gives us one adjective, one adverb. The adjective must modify "bludgeons" and then as usual the adverb can go in two places. 2 possibilities.
• "midst nigh blonde" (adv) gives us two adverbs, so 6 possibilities as we had before when we made two prepositional phrases and used them both as adverbs.
• "abaft nigh midst" (adj) gives us two adjectives. They must both modify "bludgeons": 1 possibility.
• "abaft blonde midst": this can be adj or adv, "nigh" is only an adjective, so like "abaft nigh midst" we have 1+2 = 3 possibilities.
• "nigh blonde midst": this can be adj or adv, "abaft" is only an adverb, so like "midst nigh blonde" we have 2+6 = 8 possibilities.

Finally, we had two possible four-word prepositional phrases. Either can be an adjective (modifying "bludgeons") or an adverb (modifying "humanise"), so 4 possibilities here.

So our 1-preposition options ended up numbering 3+8+1+3+3+8 + 2+6+1+2+3+8 + 4 = 23+22+4 = 49 in all.

And therefore our total number of possibilities is (if I haven't miscounted something, which most likely I have) 16+49 = 65.

• Definitely a good start! Regarding one of your second assumption, I was trying to be cagey, but this assumption is not what I meant. Also, when you use a prepositional phrase, you have a noun that can be modified with another adjective, which may itself be a prepositional phrase. Also, "nigh" can be an adverb, not a verb...I thought the legend for the Moby database was in the file, but it is in a separate README file (now linked). Feb 1 '21 at 3:02
• @gareth, I'm not sure where you got that "humanize" needs to be kept to its intransitive use. Feb 1 '21 at 3:15
• @justhalf We're told: a simple sentence, having one independent clause with subject "bludgeons" and predicate "humanise". If the verb has a direct object, the predicate isn't "humanise" but something like "humanize blondes". At least, that's my understanding of what Jeremy wrote. Maybe it's not what he actually meant. Feb 1 '21 at 4:30
• Ah, I think in layman terms the predicate in that sentence is still "humanize" (as in the main verb, not the main verb phrase), but indeed the verb phrase in syntax analysis does include the object. Feb 1 '21 at 4:32
• @justhalf I think that as soon as you start talking about predicates and drawing diagrams you're no longer talking in layman's terms. Though I think the idea that sentences should be thought of as subject + predicate is rather out of fashion now. Feb 1 '21 at 4:35

I see no [no-computer], so I'll post an answer found using computer (just complete search). For a manual approach, see Gareth's answer.

The question is asking for how many possible distinct sentence structures are there with the given rules. I take the part-of-speech as summarized by Gareth in his answer. I will represent the grammar using context-free grammar rules, as follows (explanation below):

S -> NP1 VP1


S represents the main sentence, which should contain a subject and a verb. Here, NP1 and VP1 represents the verb phrase that should contain "bludgeons" and "humanize", respectively, since we want "bludgeons" to be the subject and "humanize" to be the main verb. The rule S -> NP1 VP1 means a sentence should contain a NP1 followed by a VP1. Any | you see means alternation, and symbols in brackets means optional, while the plus sign means repetition (one or more). So NP1 -> (AdjP+) 'bludgeons' means an NP1 should contain the noun "bludgeons" modified by any number of adjective phrase where order doesn't matter. And so on. The rest of the rules are my encoding of the written rules in the question. Note that for adverbs I consider the order to matter, so "abaft_Adv nigh_Adv humanize" and "nigh_Adv abaft_Adv humanize" are different.

What we want here is all possible trees that produce the 6 words that we want. Since the realization of each of the tree should correspond to some permutation of the 6 words, we can simply run through all permutations and parse the permutation using the grammar above to get the trees, then remove duplicates (trees with different realization but the same relations)

This gave me 100 distinct trees.

One example where the realization has the most interpretations is this:

bludgeons abaft nigh midst blonde humanize


Which has these trees (note that the adjective phrases are sorted since order doesn't matter):

Parse 1:

Parse 2:

Parse 3:

The actual grammar I used in my code (since it doesn't allow repetition or optional) is as follows:

S  -> NP1 VP1
NP1 -> AdjP N1 | N1
NP -> AdjP N | N
VP1 -> AdvP V1 | AdvP V1 NP | V1 | V1 NP
N1 -> 'bludgeons'
N -> 'blonde' | 'midst'
Prep -> 'abaft' | 'nigh' | 'midst'
V1 -> 'humanize'


As per request, the trees with intransitive humanize and at most one word as Prep (total 40).

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

• I have updated the question to be more clear regarding the rules. Apologies for the earlier confusion, and thank you for your feedback to date. Regarding your solution, the grammar parse into trees looks correct, but doubtless there is more than one assignment of words to each tree. For example, in your "Parse 1" above, I see three different possible assignments of words (which does not include the one show, which uses midst as an adjective); from left to right in the top AdvP: AMNB, ANBM, NABM. Perhaps this explains the discrepancy? Feb 3 '21 at 15:35
• Thanks for response! The different assignments should be covered in other permutation of the words, I believe. Note that in your puzzle we don't care about realization at all. My solution uses realization as part of the solution, but is not central to the problem. The discrepancy seems to be that I used "midst" as "adjective" or "preposition". I've fixed my answer according to the correct parts of speech, and I have 100 trees now. Feb 4 '21 at 4:34
• I came to a larger number of diagrams, but we've approached the problem in such radically different ways (my approach is much closer to GarethMcCaughan's) that I'm having trouble reconciling the discrepancy. Can you easily determine, say, the number of your trees that are intransitive and use just one prepositional phrase? Feb 4 '21 at 15:22
• Sure! I just need to disable some of the rules. Will do that in a few hours. Feb 5 '21 at 4:48
• So, to be clear, your intended answer is much higher than 100? Feb 5 '21 at 4:48