Table of contents

1    Introduction

You can study English while having fun with puzzle graphs. A puzzle graph takes the form of a graph of words with labelled arrows that connect the graph nodes. The labelled arrows are clues for building sentence components. The puzzle task is to turn each graph of words with its clues into a ‘solution’ that is a single sentence of English or multiple sentences. You can check your success by seeing if your solution matches the suggested answer.

    Goals for working through the puzzles are:

• to get a feeling for how words interact with each other (word dependencies)
• to learn about the components that go into building up connections within and between sentences

2    How to read a puzzle graph

In a puzzle graph, word content for a puzzle solution occurs outside of square brackets (‘[’, ‘]’).

    The content inside square brackets tells us about the grammatical roles of words. These are clues that inform where words should go in puzzle solutions.

    Square bracket content should not be included in a puzzle solution. In contrast, all word content should be included. You never have to modify the forms of words, so a word should occur in a puzzle solution as you find it in the graph.

    Words and/or square bracket content can be the label content for a graph node. They can also provide the label content for a directed arrow that connects graph nodes. Each arrow connects two nodes by pointing to one node (the head node) from another node (the tail node). It is typical for the label content of an arrow to be square bracket content to say the role of the head node with respect to the tail node.

    Most of the words needed for a puzzle solution are found in the graph, but sometimes you will need to add additional ‘invisible’ words. Some of the invisible words will turn out to be bound pronouns (e.g., he, her, themselves) which are used to reference already established graph nodes introduced because of earlier words. Note that pronouns that cannot be associated with something else from the discourse context (called unbound pronouns) do have graph presence.

    Other invisible words are words that have a formal role marking grammatical structure:

• relative pronouns, e.g., who, which, where, when, used at the start of a relative clause (Mr. McGregor came up with a sieve, which he intended to pop upon the top of Peter)
• dummy it, e.g., in extraposition constructions (Peter thought it best to go away), cleft constructions (It was the second jacket that Peter had lost) and weather reports (it is raining)

    It is also possible for a node to have no content. This indicates word absence from the puzzle solution.

3    Tips for reaching puzzle solutions

This section offers tips for reaching puzzle solutions. This starts with a simple puzzle. Then we gradually pick up skills to tackle more complex puzzles.

3.1    A simple puzzle

To form a puzzle solution, start by looking into a puzzle graph for an arrow labelled [arg0]. The tail node for such an arrow will be the main verb content for a clause, possibly with other supporting words, while the head node will be the logical subject for the clause, which is the ‘do-er’ or ‘be-er’ or ‘have-er’ of the clause.

    Consider Figure 1. This gives the puzzle graph for a single clause. From the arrow labelled [arg0], we can identify were as the main verb and four_Rabbits as its ‘be-er’ (logical subject).

When were (a form of BE) occurs with existential marker there, as in Figure 1, it will typically be followed by its logical subject. With this knowledge, we can construct (1).

(1)

there were four Rabbits

    With (1), we already have a full sentence of English! However, (1) is not a complete puzzle solution for Figure 1, since there are words that have yet to be used. One of these words comes as the head node of an arrow that has four_Rabbits as its tail node. This word acts as an attributive modifier ([attrib]) to tell us about the size of the four rabbits (little). In English, numeral four should occur before an attributive adjective modifier to give (2).

(2)

there were four little Rabbits

    Figure 1 also has an arrow labelled [nim] (unselected adverbial role) that has the main verb as its tail node and that has as its head node the complex adverb Once_upon_a_time. Connecting these words gives the final puzzle solution for Figure 1 of (3).

(3)

Once upon a time there were four little Rabbits

    As another example, consider Figure 2, which also has content for a single clause sentence.

Searching for an [arg0] labelled arrow, we find were once again as the main verb of the clause. The ‘be-er’ (logical subject) of the clause is names, which itself has an arrow labelled [gen] (genitive/possessive), which takes us to the genitive pronoun their. There is also an arrow labelled [prd] (predicative), which provides name content with head node and, which in turn acts as the tail node for four conjunct head nodes. Each conjunct is linked with a [conjN] labelled arrow, where N is a conjunct's place in an ordering of all the conjuncts. Turning all this into an English sentence gives (4).

(4)

their names were Flopsy, Mopsy, Cotton-tail, and Peter

    Now consider Figure 3.

This third puzzle graph brings together content from the two previous puzzle graphs with connections of material at two different graph nodes. First, there is connection from the very top node of the graph with the word and. Nodes corresponding to the main verbs of the two earlier graphs form the head nodes of arrows labelled [conjN], much as we saw with the connections that led to Flopsy, Mopsy, Cotton-tail, and Peter. We can therefore expect to reach a puzzle solution that is essentially the content of (3) connected by and to the content of (4).

    But there is one further point of connection that we also need to make sense of: The arrow labelled [gen] of Figure 3 does not have a node with content their as its head node like in Figure 2, but rather takes as its head node four_Rabbits, which corresponds to the four_Rabbits of Figure 1.

    We can interpret the [gen] labelled arrow as having an ‘invisible’ instance of their as its head node. This absence of their from the puzzle graph occurs because presence of their in the solution sentence is instead captured in the graph as a direct arrow connection (referential link) to the already occurring four_Rabbits. We are therefore justified to take ‘(3) and (4)’ — namely (5) — to be the overall puzzle solution for the graph of Figure 3.

(5)

Once upon a time there were four little Rabbits, and their names were Flopsy, Mopsy, Cotton-tail, and Peter.

3.2    An example with ellipsis

Figure 4 has a similar structure to Figure 3, in the sense that there are two verbs connected by [conj1] and [conj2] labelled arrows from and occurring topmost in the graph. However, while the verb of the first conjunct is overtly realised with lost, the content for the verb of the second conjunct is shown as a node without a label to indicate an absence from the puzzle solution (a lost lost!).

Following all the links of Figure 4, we can arrive at puzzle solution (6).

(6)

He lost one of his shoes among the cabbages, and the other shoe amongst the potatoes.

3.3    A complex puzzle

This section demonstrates how to approach a complex puzzle. We will break this into stages, first building content for simpler graphs, which then combine to give larger graphs, and so ultimately lead to the final graph for reaching the overall puzzle solution.

    To start, consider figure 5.

By finding [arg0], we can identify was as a verb with Mr._McGregor as the ‘be-er’ (logical subject). One other arrow extends from was. This is labelled on [clr] and connects to and, which takes us to two conjuncts via the labelled connections [conj1] leading to hands and [conj2] leading to knees. One more connection, labelled [gen] (genitive/possessive), originates from and and leads to Mr._McGregor to capture the information that the hands and knees are those of Mr. McGregor. By following these connections and including his to capture the genitive/possessive connection, we can reach (7) as the puzzle solution.

(7)

Mr. McGregor was on his hands and knees

    Now consider Figure 6.

An arrow labelled [arg1] (logical object) extends to cabbages having started from planting. In turn, cabbages leads via an [attrib] (attribute) labelled arrow to young. There is also a [clr] (closely related) labelled arrow from planting to out. By following these connections of the graph (and ignoring the ["arg0"] connection to an unlabelled node), we reach the sentence fragment (8).

(8)

planting out young cabbages

    Now consider Figure 7.

Figure 7 has connections to bring together node content from Figures 5 and 6. There is an unlabelled triangular node to capture how planting together with its connections acts as a restriction on the content from was. There is also the [arg0] connection from planting which now connects to Mr._McGregor, the ‘be-er’ (logical subject) of the main clause. A puzzle solution that meets these new criteria of Figure 7 is (9).

(9)

Mr. McGregor was on his hands and knees planting out young cabbages

    Now consider Figure 8.

Looking at jumped, this is linked to he with an [arg0] labelled arrow and up with a [clr] labelled arrow. Therefore, we can identify jumped as a verb with he as its ‘do-er’ (logical subject), to derive the simple sentence of (10).

(10)

he jumped up

    Now consider Figure 9.

Looking at ran, this is linked to he with an [arg0] labelled arrow. Also, ran is linked to Peter with an after [clr] labelled arrow. With these words and connections, we can derive the puzzle solution (11).

(11)

he ran after Peter

    Now consider Figure 10.

This combines Figures 8 and 9, with and to connect jumped and ran. Note how the single he connects as the ‘do-er’ (logical subject) to both predicates. A puzzle solution is (12).

(12)

he jumped up and ran after Peter

    Now consider Figure 11.

We can see waving is connected to rake with an [arg1] ‘done to’ (logical object) arrow. A puzzle solution is (13).

(13)

waving a rake

    Now consider Figure 12.

An imperative is an order or instruction for an unmentioned ‘do-er’ to do something, but also the ‘do-er’ can be addressed with a [nim] (unselected item) connection. Such an unselected item (called a vocative) can occur either at the front or end of an imperative, making (14) a possible puzzle solution.

(14)

Stop thief!

    Now consider Figure 13.

This is the puzzle graph for a clause with an embedding with the content of Figure 12. This embedding is connected by an [arg1] (logical object) arrow as an [utterance] (reported speech) that has calling as its content. A puzzle solution is (15).

(15)

calling out, ‘Stop thief!’

    Now consider Figure 14.

Figure 14 results from combining Figure 11 and 13. Here, and has conjuncts waving and calling. A puzzle solution is (16).

(16)

waving a rake and calling out, ‘Stop thief!’

    Now consider Figure 15.

Figure 15 has an unlabelled triangular node to combine Figures 10 and 14, much like we saw happen with Figure 7. Note how the verbs jumped, ran, waving, and calling all have [arg0] ‘do-er’ (logical subject) arrows that connect to the same he. A puzzle solution is (17).

(17)

he jumped up and ran after Peter, waving a rake and calling out, ‘Stop thief!’

    We are finally able to consider the overall puzzle graph of Figure 16.

This has but to combine Figures 7 and 15. Note how the verbs was, planting, jumped, ran, waving, and calling all have [arg0] ‘be-er/do-er’ (logical subject) arrows that connect to the same instance of Mr._McGregor. An overall puzzle solution for Figure 16 is (18).

(18)

Mr. McGregor was on his hands and knees planting out young cabbages, but he jumped up and ran after Peter, waving a rake and calling out, ‘Stop thief!’

4    Square bracket content

This section presents tables for content placed inside square brackets together with illustrative examples.

4.1    Relation labels

(19)

four little Rabbits

(20)

Mr. McGregor's garden

(21)

very big

(22)

Baroness Burdet Coutts founded the village.

(23)

The village was founded by Baroness Burdet Coutts.

(24)

He was put in a pie.

(25)

Peter asked her the way.

(26)

It was a hoax.

(27)

I find this interesting.

(28)

Nasha went to the window.

(29)

He was safe in the wood.

(30)

the idea to go

4.2    Conjunction

(31)

Janet was praised by Rodney, criticized by Sidney and ignored by Gregory.

(32)

He said Beware! His eyes flash!

4.3    Operators

(33)

Merchandise orders came from every client.

(34)

A player wins if they reach the mousehole.