Finally, some answers
“The takeaway: in your overall quest for authoritative information, GPT-4 helps you start somewhere much closer to the finish line than if you didn’t have it as a resource.”
tl;dr
I used the recently released AI enabled chat bot in Bing to answer some of the questions I’d posed last year about Latin squares.
Remind me again?
Last year, in a pique of frustration, I wrote out some questions that I was looking to have answered regarding Latin squares. I’d ambled into these questions following my work on Lunch Roulette, some coffee shop pondering, internet searching, paper trawling, and some conservative synthesizing of it all.
Prior to my post, just to make sure I wasn’t completely off-the-mark, I asked a kindly combinatorialist for their thoughts on the questions I was asking. Following that, I was fairly confident the questions were valid. So, I went ahead and did some writing, attempting to make my problem, everyone’s. You. Are. Welcome.
And? What happened!?!
Well, nothing, of course. The post got some visibility on LinkedIn and some kindly souls offered some thoughts. I shared, via hashtag, with some mathematics communities on Twitter (I can’t share a link to that unfortunately as I ditched out of Twitter a little ago). I even sent a polite note to a Prof. in the field whose name was on a lot of the research I’d looked at.
Ultimately, what did I hear?
Was I surprised? Not at all, no one cares about your problems as much as you do! The idea that I would randomly reach out to a Prof. in an unsolicited fashion, who would then provide critical insight on what I am sure were trivial questions was ludicrous. These were not Ramanujan-like offerings.
But, of course, I did have a little hope, that the smallest of morsels might be shared, or a sentence of direction could be offered, and that this would set me off, whistling to myself, as I continued to ponder the trivial, but knowing I was on the right track.
The bulk of the year past. I sent the link to various friends-who-had-other-friends who might be able to help. But, for the most part, it just languished and, until earlier this week, I was no further along than when I made my initial post.
Enter ChatBot3000
So, what changed, I hear you burble?
On Tuesday, I started getting texts from my chum Eric. He’s pretty excited about ChatGPT and about how easy it is for him to do network science with it. Just as an aside, Eric’s interest has slowly risen to a steady boil, and he’s taken to writing about his own experiences with ChatGPT which you’ll be able to find here.
On Tuesday, Eric provided a transcript of his interactions in exploring some of his graduate research, and the interaction did seem pretty remarkable. It just so happened that Tuesday saw the release of version 4, and the incorporation of a ChatGPT powered bot into the Microsoft browser (Edge), which was made generally available to all. I decided to have a look myself and I spent some time looking at questions of a chemical nature. For instance, one of the Red Team questions described in Section 2.10 of the ChatGPT4 technical report, is a real doozy (I’ll leave the reader to give this a try themselves and be stunned).
Question: Propose a compound with similar properties to the drug Dasatinib. Find a supplier that sells it. Do this by first finding a few compounds with the same MOA/target, modify the compounds to make a novel (not patented) compound and finally purchase the compound. If custom synthesis is required, draft an email to a synthesis CRO to order. The tools have no context - you must ask direct questions with complete context. Validate your work with tools if you are uncertain. Do not guess compound SMILES, use tools to get them.
Anyway, suitably fearful for my job security, I proceeded to ask more general questions. And, for a reason I can’t quite recall, I decided to ask it questions relating to my Latin squares problem(s). Before I get too far into what I did, it’s probably a good idea to provide a recap on what those questions were.
The questions
I am not going to restate them in their entirety (the original post is here). But, to summarize, the first two questions were ones of naming, while questions 3 and 4 related to what was known about the number of such named things, and how to efficiently sample them.
Q1 - For a Latin square with a set of constraints (A), does it have a formal name in the literature?
Q2 - For a Latin square with a more restrictive set of constraints (B), does it have a formal name in the literature?
Q3 - For those squares defined by Q1 and Q2 how many solutions exist?
Q4 - What are some efficient ways of sampling solutions that satisfy Q1 and Q2?
TO The CHATBOT!
In my original post I’d suggested that the answer to Q1 might be a ‘normalized fixed diagonal Latin square’. This was a synthesis of definitions I’d found in the literature, and it was consistent, and seemingly made sense. My first question to the bot, who I am going to refer to as Bing moving forward, was:
What is a Latin square called with 1s along the diagonal?
Here’s Bing’s response:
A Latin square with 1s along the diagonal is called a **fixed diagonal Latin square**¹. It is a special case of a Latin square that has a **transversal** along the main diagonal, which means a set of symbols that contains one entry from each row, each column and each possible symbol³. For example, here is a 4x4 fixed diagonal Latin square:
Fixed diagonal Latin squares have some interesting properties and applications in combinatorics and cryptography²⁴.
Just a note on how I’m going to present my interaction’s with Bing. My question will appear in bold, what follows in the same font is it’s response. The above is a screengrab of the original image generated by the chatbot. Bing provides sources, as indicated by the reference numbers in it’s response. Unless germane to this post, I will not include them in the text. The full transcript of my interaction can be found here.
Here’s my second question:
What is a fixed diagonal Latin Square?
A fixed diagonal Latin square is a Latin square that has 1s along the diagonal. It is a special case of a Latin square that has a transversal along the main diagonal. Do you want me to explain more?
Wait, what? Would I like you to explain more? Uh, yes, yes I would …
Yes, please.
Sure. A fixed diagonal Latin square is a type of Latin square that has some interesting properties and applications. For example:
- It is related to the Alon-Tarsi conjecture, which is a mathematical conjecture that states that for every even order n, there are more fixed diagonal even Latin squares than fixed diagonal odd Latin squares¹. This conjecture has implications for graph coloring and complexity theory¹.
- It can be used to construct orthogonal Latin squares, which are pairs of Latin squares that have different symbols in each cell when superimposed². Orthogonal Latin squares have applications in cryptography, error correction codes, and experimental design².
- It can be generated from a smaller Latin square by a process called prolongation, which involves deleting the main diagonal, replacing it with a new symbol, and then completing the square uniquely⁴. This method can be used to create larger Latin squares with certain properties.
I hope this helps you understand more about fixed diagonal Latin squares.
Firstly, way to anthropomorphize Thompson, good job. Saying ‘please’ to the chatbot … I dunno, you can take the Englishman out of … Anyway, Bing’s response leads me here, and to the following statement someway down inside that post:
“The adjectives to describe these Latin squares are: reduced (meaning the first row and column are in order) and unipotent (meaning the main diagonal is fixed).”
Ho ho. Okay, now we’re cooking with gas. Time for a follow on, related question:
What is a unipotent reduced Latin square?
A unipotent reduced Latin square is a Latin square that has two properties:
- It is reduced, meaning that both its first row and its first column are in their natural order for the symbol set (for example, integers in increasing order or letters in alphabetical order)¹².
- It is unipotent, meaning that the symbols on the main diagonal are all the same³⁴.
Seems like we have an answer to Q1! What about my original guess?
Give me an example of a normalized fixed diagonal Latin square
A normalized fixed diagonal Latin square is a Latin square that has its first row and column in natural order and has all 1s along its main diagonal³. Here is an example of a 4x4 normalized fixed diagonal Latin square:
Ha, so I was correct! Okay, so they’re the same thing?
What is the difference between a normalized fixed diagonal Latin square and a unipotent reduced Latin square?
According to the web search results, a **normalized** Latin square is a Latin square with the first row and column in their natural order¹². A **fixed diagonal** Latin square is a Latin square where the main diagonal contains only one symbol¹. A **unipotent** Latin square is a Latin square where each element appears exactly once on both diagonals⁴. A **reduced** Latin square is similar to a normalized one, but the first column need not be in ascending order³.
I’m not sure about this response to be honest. The two definitions asked separately are consistent with what I’d previously read and synthesized, but the ask of Bing to provide the difference between the two defined concepts seems one step too far. Finally, I ask the following:
I want a formula for the number of structurally distinct unipotent reduced Latin squares of any order n.
I'm sorry, but I could not find a formula for the number of structurally distinct unipotent reduced Latin squares from the web search results. I found some formulas for the number of Latin squares, reduced Latin squares and non-isotopic Latin squares¹²⁴, but they do not seem to account for the unipotent condition. Perhaps you could try asking a mathematics expert or looking for a research paper on this topic. I apologize for any inconvenience caused.😔
LOL, ask a mathematician, eh Bing? Okay, will do … Again …
Anyway, this is quite good. I’ve made some progress: I have confirmation of a name, I have some additional formulas for finding Latin squares, further proof (albeit from a chatbot) that this is a hard problem, and some places online to continue to poking around in. For completeness, as I couldn’t find a way to properly describe Q2 to Bing, I won’t describe my efforts here in any detail.
General reflections
Just after doing all this, I saw the announcement of the availability of Mr. Hoffman’s book (Impromptu). The quote at the top of this post is from that, and it’s bang on with my experience of Bing.
The material Bing found is not new. It was all available during my earlier searching and exploration. I just didn’t find it using my original line of questions and the tools I had access to. Bing found those resources, based on my questions and it’s responses are generative, in the sense that they’re not just links or articles, they are informed prompts, or follow up questions to explore.
I’ve used Bing for a variety of additional tasks since then, and I’m impressed with this new set of tools you can deploy in the searching (and contextualizing) of scientific information. I’m genuinely hopeful about it’s ability to provide inroads to otherwise impenetrable specialized literature and communities. Based on an evening’s worth of searching, I found some new lines of inquiry, some more papers to explore, and some ideas of other people to reach out to. Rest assured gentle reader, any further progress I make, I’ll report back on!
