The phrase “GNU Terry Pratchett” originates from one of the Discworld novels, “Going Postal,” where it is used as a code or message passed between characters to remember the deceased.

Specifically, there is a communication system known as “the clacks”, and there part of the way that system works involves additional metadata messages being sent (known as “the overhead”).

I only recently discovered that people have been embedding that message in all sorts of things as a way to memorialize Terry. One of the places they do this in the headers of websites.

So, you will now see an additional header in all the HTTP responses on this site:

X-Clacks-Overhead: GNU Terry Pratchett

After investigating Star Trek episode titles with interesting characters, lets look at ones with interesting words.

Numbers

I already included this as a side note in the previous post, but many episodes have numbers as words, with each series having at least one.

Ordinals

Non-English words

This is probably the collection that caused me the most difficulty in deciding what should be included.

I settled on:

• Actual words (no 11:59 or E²)
• Real word languages
• Not proper nouns (especially names of characters)

Considering all that, there are in fact only two non-English languages used in the titles of Star Trek episodes:

French

Latin

Honorable mentions

One Klingon word, and one Vulcan word that is only probably a proper noun.

Character names

Who has the privilege of being named in an episode title? Again, there are surprising edge cases so I’ve split it up into several groups.

• Q appears in the most titles.
• Two titles mention two characters: Datalore and Ménage à Troi (which I assumed, in the spirit of the episode, is a reference to both Deanna Troi and Lwaxana Troi).

Direct character references

Definite direct references to characters in Star Trek.

Other characters

References to people, either fictional or real that were not actually characters in Star Trek.

Indirect

Indirect references to characters that are awkward in some way.

Accidental

Character names that are not intended to be references to Star Trek characters. Mainly The Doctor from Voyager, but also Q2 which was the script name for the member of the Q Continuum in the TNG episode Deja Q.

One

There is one special character name that is clearly not intended to be a reference, but comes up a lot - One from Voyager.

Q

And finally, the letter Q appears a few times while clearly not being a reference to Q.

Just the character

One bonus list considering all the above - titles that are just a character name.

New audio versions of all* of the Discworld books are being produced. I’ve made a few posts tracking the release dates. Since then I’ve updated the blog to support nice sortable tables.

So here are the the books listed on Audible as of 21st May, 2023 in a single table (now with primary narrator). All columns should be usefully sortable.

If you don’t already have an Audible membership, you can get a 30-day free trial (UK).

New Discworld audiobook release dates

* There are 41 Discworld books. The website says “40 magnificent new recordings”. The one that seems to be missing is “The Last Hero”.

While doing other more systematic investigations of Star Trek episode titles, I made notes of interesting things I spotted.

Here are some collections of episode titles with interesting characters.

Q puns

Most episodes featuring Q had a pun in the title:

Digits

Punctuation and other interesting characters

Containing '

There are so many titles with apostrophes I decided to break it down by use:

Possessive

Part of an alien word

Contraction

Clarifying a plural

Containing -

Several episode titles contain a hyphen. One episode contains two. The producers and/or writers probably intended an em dash or similar but were likely limited to hyphens for technical reasons.

Containing ,

Several episode titles contain commas. One had the option to include two with an Oxford comma.

Containing ?

Containing …

Containing :

Containing !

Containing à (and é)

There are a pair of episodes with accented characters (which have recently been joined by Võx from Picard).

Containing ²

Star Trek Picard season three has just completed. It was not perfect but was probably the best live action Star Trek since Enterprise.

In honor of this, here is a quick analysis of the episode titles.

I’m not including Picard in the main analysis for consistency for now. There is also a small chance there may be more episodes (or possibly something else #StarTrekLegacy).

Title length

Picard does not challenge for overall shortest or longest title.

Shortest

Like other series, it has several single word titles:

Longest

Single word titles

The overall single word title stats are very similar to Enterprise.

When split by season there is a lot variation though. I expect this to be true of most Star Trek series after Enterprise - the seasons are generally shorter and the serial nature means there are more likely to be patterns within seasons that don’t hold between them.

Ionic episodes

Two new “ion” titles (but no “ions”):

Image copyright © Paramount Global and used under fair use.

With nearly 700 episodes, there were bound to be some that are similar to each other. But which are most similar? And what do we mean by similar?

Summary

Before I go into the more technical analysis, I shall begin with some headlines.

Not technically part of this analysis but there are two Star Trek movies that share a title with an episode (if you ignore the “Star Trek” part of the title):

There are three pairs of titles that differ by only one character. Two of them are just a final “s” as part of a plural:

There are three pairs of titles that differ only by a leading “the”:

One of the more significant lines in Star Trek has both its variations appear as titles:

There is also a TNG episode that is specifically a sequel of sorts to an earlier episode with a similar title:

A pair of episodes with weirdly similar titles, but in a way that isn’t easy to define:

There is an interesting prefix that appears three times in TNG, and only in TNG:

Four incidents worthy of naming:

Technical details

Below I present the top 10 pairs of similar episodes, with four different measures of similarity:

• Levenshtein distance - Informally, the number of single character edits needed to change one title into another.
• Prefix - The number of characters at the start of the titles that are in common (ignoring case).
• Suffix - The number of characters at the end of the titles that are in common (ignoring case).
• Contains - If one title is contained entirely within another (possibly at the start or end), then the length of the contained title (0 otherwise).

Additionally each one is presented with an absolute version, and another normalized to the length of the longest title.

Most similar titles

Levenshtein (absolute)

Levenshtein (normalized)

Prefix (absolute)

Prefix (normalized)

Suffix (absolute)

Suffix (normalized)

Contains (absolute)

Contains (normalized)

Most similar titles without leading “the”

Levenshtein (absolute)

Levenshtein (normalized)

Prefix (absolute)

Prefix (normalized)

Suffix (absolute)

Suffix (normalized)

Contains (absolute)

Contains (normalized)

In my previous post, I described an extension to Elo that could handle multiple players. The next restriction to overcome is that Elo assumed players of equal skill have an equal chance of winning.

In most games, players don’t actually have an equal chance of winning a single game. Chess overcomes this by having a match consist of several individual games with players switching who goes first. This is a good solution for two player games but gets awkward for multiplayer games. It is also inconvenient if having a multi-game match is undesirable for any reason.

Adjusted rating

Elo is fundamentally designed to handle players of different skill levels and produce a probability of them winning. The approach is therefore to determine an adjusted rating for someone such that the probability of winning is as desired.

The formula to work out an Elo expected score is:

In our case we have an expected score (the win probability) and want to work out a rating difference, so we can just rearrange to get:

We then subtract this adjustment from the player’s actual rating to get their effective rating to use in the rest of the Elo calculations.

The reason we subtract it is that for players A and B, the rating difference is calculated as R_B - R_A which is the opposite way round to what we want.

Here are the rating adjustments for some sample win probabilities:

If you try to calculate the adjustment for a win probability of 0 or 1, you get -∞ and +∞ respectively.

This means, for example, if you beat someone of an equal rating at a game you only have a 10% chance of winning, it’s the same as beating someone rated 382 points higher at a fair game.

Putting it all together

After explaining the theory of how to expand an Elo rating to support all the variations in Tic-tac-toe Collection, it’s time to put it into practice.

While writing this I’ve also been working on an implementation.

As well as adding it to Tic-tac-toe Collection, I’m planning on creating some kind of site and/or app just to work out Elo ratings for various results (and hopefully even to maintain ratings for players in any kind of competition you might want to run).

My earlier post included a list of all single-word episode titles. That list is long enough that it could be analyzed in isolation and still produce some interesting results. Firstly, here is a summary:

Single-word titles by Series

It seems the number of single-word episode titles started low and increased over time, peaking with Voyager (although Enterprise was very close behind by percentage of total episodes).

The Original Series

The original Star Trek is the only series to have a season with no single-word titles at all (and it has two if you follow the convention that the original pilot, “The Cage” is part of its own “Season 0”).

Many titles from the original series were intentionally unusual, in the hope it would generate interest from people browsing TV listings (as explained more by Bruce Bennett). This generally led to longer titles and is part of the reason there are so few with just one word.

The Animated Series

The animated series followed a similar naming style to the original, although in this case even the single world titles are pretty odd.

The Next Generation

TNG was clearly a transition point in the history of Star Trek. It was obviously more modern and naturally led into the later series with most of the new production staff staying with Star Trek.

It was also still led by Gene Roddenberry early on, along with some of the original writers. This meant that although there were a plenty of long and unusual titles, the preference for simpler ones was becoming established.

Looking at the list of titles we can see TNG has the first that might stretch the definition of “single word”, and that maybe 11001001 is not even be a word. By my established definition though, it counts.

Deep Space Nine

DS9 followed a similar pattern to TNG. Many single-word titles, but plenty of longer ones too.

It also has another title that is questionable as a single word (that is yet another Q pun) in Q-Less.

Voyager

By the time Voyager was written, the preference for single-word titles is well established, peaking in season 2 in which 20 out of 26 have only one word, and includes two separate runs of seven consecutive episodes with single-word titles.

There is another Q pun with Q2 and a bit of controversy with 11:59.

Enterprise

Enterprise titles continue the trend set in Voyager, despite dipping slightly.

If you wanted to be picky you might question the single-word status of Kir’Shara since not only does it contain an apsotrophe, but also a capital letter in the middle.

What about “the”

My anlysis has so far treated a leading “the” as a word. Repeating this while ignoring “the” may produce interesting results, and at the very least will increase the count in Voyager season 2.

Ions

Finally, while browsing the list one interesting pattern stood out. There are 8 episode titles ending in “ions”, including the longest Transfigurations, and a further 22 ending in “ion”.

Sadly, there are none from the animated series.

One of the features I have planned for Tic-tac-toe Collection is a player rating system.

One such common rating system is the Elo rating system, most famously used in chess. There is a lot of discussion about how good it is, but for my purposes it seems fine - at least as a starting point.

There are two fundamental restrictions on Elo that I would have to overcome though:

• It is limited to two players.
• It assumes each player has an equal chance of winning if they have equal skill.

Elo

First an introduction into how Elo rating works. There are two parts:

Given two player ratings, calculate an expected result.

The expected result is essentially the chance of winning, given as a decimal between 0 and 1, in which 0 represents a guaranteed loss and 1 a guaranteed win.

Given a player rating, an expected result, and an actual result, calculate how much the rating should change.

If you do better than expected then your score will go up. If you do worse then your score will go down. With a bigger difference there will be a bigger change.

Multiplayer

Simple Multiplayer Elo

I found an existing implementation of Elo for more than two players by Tom Kerrigan called Simple Multiplayer Elo.

In SME, players are ordered by performance, and then a simulated match is played between pairs of consecutive players, with the players’ ratings updated using the normal Elo method for two players.

This is certainly acceptable, and Tom includes some tests to show it works well. One oddity is if you get a victory over someone with a much higher rating than you, but also beat them by more than one place, you essentially don’t get the benefit for it. For example, consider the following result:

Here, Player A with a rating of 1000 has beaten Player C with a rating of 1500, but essentially hasn’t gotten the credit for it.

My unnamed Elo

My solution is to conceptually run the algorithm on every pair of players (in this case there would be A v B, B v C and A v C). There is a straightforward optimization so that the estimating part of the operation does not directly depend on the other players, just the difference between a single player’s expected and actual scores. So the algorithm is actually:

1. Calculate the expected scores for each pair of players.
2. Sum the expected score for each player.
3. Divide each score by the number of players to normalize back to the range 0 to 1.
4. Calculate the rating change for each player using their actual score and the combined expected score.

The details

With the same data as above, the results are as follows:

Which is made up of the following score pairs:

This results in rating changes of:

• Player A has gained more which is good. That was basically the goal of the change.
• Player B has now changed a small gain into a moderate loss. That’s a little odd and probably not desired, after all the victory against C should be more significant than the loss against A
• Player C has changed a big loss into a slightly smaller (but still big) loss. After some thought, that is probably reasonable. Although C is expected to win just having more players should generally reduce your expectation of winning and therefore how much you are penalized for failing.

Improvements

So why did these results happen? It might be better to look at a table also including expected and actual results, which reveals a choice that has to be made that I have not yet mentioned:

Notice the actual scores used. Player A has a score of 1 indicating a win. Players B and C have a score of 0, which indicates a loss, but an equal loss.

If we instead use different values for the actual score we get a more sensible result

In this case, the “actual score” for a player placed Nth out of C is:

Handling ties

To be as general as possible (and because it actually happens in some of my multiplayer Tic-tac-toe variants) we need to handle arbitrary ties.

The simplest way is to evenly distribute the score to all the players that are tied. So if the top two players in a three-player game tie, their scores of 0.67 and 0.33 are summed and split between them (0.5 each).

As a more complex example, consider:

Final thoughts

I’ve addressed the first restriction of Elo, only supporting two players. As for unfair games, that will have to wait as this post is long enough as it is.

I wrote a post about calendar reuse that included an offhand comment about Easter. The common definition of when Easter is, is:

the first Sunday after the first full moon that falls on or after the vernal equinox

That isn’t quite right, and I made a vague mention of the correct definition.

Since then, someone has made a video explaining the the correct definition in depth with details about the history and why it is this way.