Sometimes, a pendant vertex is just a pendant vertex.

I had a dream last night, in which I saw the proof of a mathematical theorem. It involved copies of K₅, with pendant vertices arranged in such a way that all of the graphs linked perfectly together into a larger complete graph. It was a simple, beautiful proof, and I can still see the end of it now, hours later. A few of the details are fuzzy, but I'm sure I could work them out. There's only one problem.

I have absolutely no idea what it was I was proving.

You Know What I Mean

The Naboo half of Phantom Menace, stretched out to two and a half hours.

Total Recall, but without the cool perception-vs.-reality philosophy.

An evening of watching someone else play Halo and they won't let you play and they watch all the cut scenes.

An Inconvenient (and Ham-Handed) Truth.

Like Star Trek IV, but when they get back to the present, Kirk helps the whale-aliens blow up Starfleet Headquarters.

Courtesy of the w: A live-action remake of Ferngully: The Last Rainforest.

A late addition: Pocahontas in space.

Probabilistic Filler

Some important probabilities related to Scat:

As I mentioned last time, your chance of getting scat (no 1's) on your first roll is 3125/7776, or (5/6)⁵, which is just over 40% of the time. The probability, then, of having scat as your final hand is (5/6)¹⁵, or 6.5%, which works out to about twice every 31 turns.

The probability of rolling five 1's within three rolls is, in theory, about 1.3% (the exact probability simplifies to (91/216)⁵, which I think is kind of neat—not because that ratio has any significance, but because the otherwise-complicated calculation ends up simplifying so neatly), or about once every 75 turns. In practice, though, this probability is smaller, because of situations which dictate abandoning the all-1's strategy.

If there are k people still left to roll, then, the chance that at least one person gets scat is 1 - (1 - (5/6)¹⁵)^k; and the maximum probability that at least one person rolls all 1's is 1 - (1 - (91/216)⁵)^k. If, for example, four players were still to roll after you, there would be a 23.5% chance that at least one would roll scat (thus saving you from elimination, if you had a fairly low roll), and as much as a 5% chance that someone would end the game prematurely by rolling five 1's.

The main factor we must consider when deciding which dice to re-roll is the expectation of each decision; this expectation is specifically based on the probability of winning or losing the game after this decision. The probabilities above are part of calculating the expectation - you can make riskier moves (i.e., keeping only your 1's) when there are more players still to roll, because there is a higher chance that someone after you will get a bad hand, and because there is a higher chance that if you don't win right now, someone else will do so before you get another roll. I'll get more into the decision-making process next time.

Roll the dice, don't think twice

I'm a little behind on everything these days, and Scat strategy is one of them. Go back a month to review the rules.

When it is your turn, you have two strategies available:

1. Attempt to roll five 1's, thus winning the game.
2. Obtain a hand which is not likely to be the low hand for the round, thus avoiding elimination.

After your first roll, you have two opportunities to decide which dice to keep, and which dice to re-roll. This decision is a choice among three options:

1. Keep only the 1's, and re-roll all of the other dice.
2. Keep the 1's and your highest-ranking die or dice.
3. Keep the best hand you can make with the current dice.

For example, if your roll is 1-3-3-4-6, you can keep only the 1 (option 1), the 1 and the 6 (option 2), or the 1 and both 3's (option 3). There is no reason, ever, to keep the 4, or to not keep the 1.

In many cases, the decision is not even as complicated as the previous example; if the roll had been 1-3-4-6-6, or 1-3-4-5-6, options 2 and 3 would be the same. In fact, a little over 40% of the time (3126/7776, to be exact), there is only no decision to make - either you roll no 1's (3125 times), and must re-roll all five dice, or you roll five 1's (1 time), and win the game.

For those times when there is a decision to make, several additional factors should be considered:

• The current low hand. The low hand in each round is eliminated, so you will often (not always) choose the option with the best chance to beat the current low. More importantly, you will want to avoid options that have a very low probability of beating the low hand (keeping three 2's when the current low hand is four 4's, for example, is usually a bad plan).
• The number of players still to roll. Remember, you only need to beat one player each round, so if there are several players still to roll, you can choose a riskier option, based on the chance that not all of those players will beat you.
• The number of players still in the game. Without taking strategy into account, if there are n players at the beginning of the next round, your chance of winning by avoiding elimination this round is roughly 1/n.

I'll start getting into the probabilities of the game next time (and I hope that next time comes more quickly than this time did). However, I expect we'll find that, early in the game, the expectation gained by winning the game immediately will outweigh the risks of being eliminated early. In other words, rolling five 1's is the primary strategy (in my limited experience, most players make avoiding the low hand primary, and rarely go for the outright win unless they have already beaten the current low). This means that option 1 will frequently be the correct decision, because the other two options necessarily abandon the primary strategy. This will be particularly true after the first roll, because committing to any die other than a 1 restricts the ways for a hand to improve. I also expect that avoiding the low hand will become the primary strategy later in the game, when simply staying alive represents a greater increase in the probability of winning—my guess is that this happens when there are three players remaining, but we'll see.

Finally, and perhaps unfortunately, I expect we'll find that, while skillful play does increase one's expectation of winning, the overall effect won't be that great against people who do not play as well. In other words, you'll be able to win an n-player game against "normal" players more often than once every n games, but you'll likely need to play a few hundred games to see that result. Also, since each player's hand is independent of the other hands, we may find that there is a single pure strategy that maximizes one's probability of winning. In this case, the result of all players adopting that strategy makes the probability of winning an n-player game exactly 1/n. This is all conjecture for another time, though. Stay tuned.

A son! A son! A son! Hear the joyful celebrations in the street!

The trivia posts have been slowing down lately. The caller has moved to a new location, but hasn't set up trivia night yet, so this may have to hold you until 2010. Don't despair, though, I remain steadfast in my search for good trivia.

1. What early-'80s rock diva was originally classically trained as an opera singer?
2. Speaking of rock, and opera, what is Tommy's last name?
3. In the novel Treasure Island, what is the name of Long John Silver's parrot?
4. Who was the Girl from U.N.C.L.E.? (the character's name, not the actress)
5. What was the name of Carrie's high school (in the film)?
6. What is the only Van Gogh painting sold during the artist's lifetime?
7. Which NFL team won its first game of the season 17 years in a row, from 1973-1985?

Hit your marks, Stay in the light, and do the same thing every time, for continuity

WARNING: Minor Glee spoilers ahead. If you haven't seen this week's episode, consider coming back tomorrow.

When we last saw the gang at McKinley High, it was Wednesday, and Will had just stepped down as director. Sectionals are Saturday, and Saturday, just to remind you, is December 19.

We start with Emma taking over the glee club, and Mercedes showing us she ain't no Kelly Rowland. This might happen Wednesday, or it might be the next day, but that doesn't matter, because the team puts together its set list with just a few days to go, and then we jump to Saturday, and everyone's on the bus! Well, everyone except Mr. Schu, because he can't go, and Finn, because he's quit, and is busy cleaning out his locker, because football season has ended. Football season has ended? Really? The State Finals are Thanksgiving Weekend, and, as I have pointed out several times, McKinley wasn't making the playoffs. Football season has been over for this bunch for six weeks.

Anyway, we're over at Sectionals, and Frances McDormand and Lauren Hutton are rocking out to Jane Adams Academy and Haverhurst (no, seriously, watch it again, they're sitting right in front of the McKinley kids), and the set list has been compromised! That's okay, though, because Finn un-quits again (he's as good at this as Rachel), and the kids put together a song, complete with harmony and choreography, in one hour. This, I will point out, bothered the w quite a bit, but not me. I said this in a comment a while back, but a show like this is all about where you choose to suspend your disbelief. I'm okay with it being High School Musical, and everyone already knows the lyrics to every song and just starts dancing. What I'm not okay with is poor continuity.

And speaking of which, here comes Rachel and her amazing disappearing/reappearing band-aid (it's on her right knee... in about half of the first number)! Lea Michele is fantastic, but I gotta say, she has never looked less like a high school sophomore than in this number. Then we have the one-hour song, and we're off to judging. It's amusing and all, but then the Vice Comptroller mentions that she's only there because her boss got tickets to a NASCAR event at the last minute. And this is where I choose not to suspend my disbelief. It is December. Even if we give the show a couple of weeks of credit by just ignoring things like the football team's record (which gave us our first clue as to an actual date), it's the end of November at the earliest. Here is this year's NASCAR schedule. There isn't a NASCAR event in December. There isn't one remotely close to Ohio in November. On top of that, races are on SUNDAY! I know it's a throwaway line, it's just meant to be funny, but come on, people!

Anyway, the show ends, Emma resigns, her last day is Monday, and on Monday Sue is suspended and the kids sing Will a song and everyone is there despite the fact that it is December 21, and almost every public high school in the country let out for the year on the 18th. Maybe there was a snow day (and the snow completely melted away the next day, because the trees outside are still green), and now they have to make it up.

Anyway, that's what you missed on Glee. Join us in April for the Spring Semester. It had better be the Spring Semester.

Deep Blue Something, or possibly not.

We're a bit behind on trivia because of the holidays, too. We've only been going about every other week, as the popularity at this particular watering hole has slowed down; it's just not that much fun if there are only four teams, even if we do win more often. The trivia caller is, apparently, moving his gig to a new restaurant, so we'll see if that picks things up.

1. What occurred the day before the Passion of the Christ?

2. Which comedian famously stopped aging at 39?

3. Who was the last American man to win the Olympic marathon?

4. More Olympics: who was the only American to win a gold medal during the '68 Winter Games?

5. Who played Hazel?

6. Who sang the score for the film Breakfast at Tiffany's?

This show has 74 flaws as of yesterday...

WARNING: Minor Glee spoilers ahead. If you haven't seen this week's episode yet, you may want to come back tomorrow.

We missed last week's Glee recap because of the holiday, so it's time to pack two episodes into one post. This will work out well, because we absolutely must assume most of these episodes take place in parallel, as we're about to see.

Last week, we met the competition for Sectionals. We don't really know when Will met with their respective choirmasters, but we do know that the rehearsal with the School for the Deaf took place on Monday. That's not the same Monday as the previous episode, so it has to be the next Monday, December 14.

This week, we find out that Sectionals are "a week from Saturday." If it's already after the Monday from last week, that means Sectionals are on St. Stephen's Day, which doesn't seem likely. Therefore, this has to have happened the previous week. That's all very well and good, but the episode still has to end later than last week's episode, because Will is no longer the Glee Club director. It is, at the earliest, December 16 - the "Imagine" rehearsal is Monday, Will sleeps in the office on Tuesday (we need the day on Tuesday to resolve other subplots), Mattressgate hits on Wednesday. Sectionals are now set in stone as December 19, there is no further way around it.

And that's what you missed on Glee. Next week we finally see Sectionals, and figure out how Emma is there and at her wedding at the same time, and perhaps resolve once and for all the fake baby plotline.

There's no beating Judy's ham!

There's not much to say about this week's Glee. It's hard to stay with a timeline, as I can't find any point in this week's episode that specifically references an occurrence in the previous week's episode. There is a temporal clue, though, as we know that Finn is invited to the Fabrays' house for Sunday dinner. This does, of course, lead to more time-continuity issues, as Rachel appears to run into Susie Pepper in a restroom at school, immediately after dinner. That could be Monday morning, but then Rachel is wearing a different outfit before rehearsing with Mr. Schuester. Is that Tuesday?

Let's be generous, and assume that some of this week's episode happened in parallel with the previous episode's events. This seems unlikely, given that the gang would be spending a lot of time in wheelchairs, but if we don't start crediting the show a week here and there, we're going to hit Holiday Break in a hurry, and Sectionals (which are now "in a few weeks," by the way) will be happening around Valentine's Day. (This, as I alluded to last week, would actually be more in line with the schedule of show choir events in Ohio, but that's a whole other kettle of home-cured meat.) Sunday dinner is still going to happen after last week, but I'll assume that the final Schu-Rachel scene does take place on Monday (perhaps Rachel was slushied after lunch and had to change). That gives us a current date of December 7. Last day of school is the 18th, people, so sectionals better get here fast.

Back then, of course, if the fight lasted less than fifty rounds, we demanded our nickel back!

It's been a while since I put up trivia, and I've been stocking up on misses. Unfortunately, I've misplaced most of that stock, but here are a couple of the tougher ones from the last few weeks. Get these, and you too could be taking your local pub for a bottle of wine and a gift certificate every week.

1. According to Mattel, what is Barbie's last name?
2. Who is the only Major League Baseball player to hit a home run for both leagues in the All-Star Game? (Not in the same game, obviously)
3. What African dog breed is also known as "The Barkless Dog?"
4. What comedian introduced Johnny Carson as the new host of The Tonight Show on October 1, 1962?
5. What fruit forms the base for the liqueur creme de cassis?
6. Who was the only boxer to defeat John L. Sullivan by knockout?

It's beginning to look a lot like...

WARNING: Minor Glee spoliers ahead. If you haven't watched this week's episode, you may want to come back tomorrow. Also, there's a spot where I link to a blog post that links back here, so you could get caught in a Recursive Internet Loop if you're not careful.

Previously on Glee: The football team won its first game with three weeks left in the season, and then continued to practice for six more weeks. Meanwhile, Mr. Schuester announced that we were two weeks away from Sectionals, then led his merry band of singing, dancing teens through three weeks of mash-ups and throwdowns, all in preparation for the aforementioned Sectionals, which still haven't happened as of November 19.

They won't happen this week, either, as we find out that the team can't afford to rent the short bus to travel to Sectionals. There's cupcakes, and fighting, and more Artie than we're used to, which is a good thing, and more Brittany than we're used to, which is probably also a good thing. The plots all advance a little, but the main point of this episode is to showcase the Rachel-Kurt "Defying Gravity" performance, which is quite spectacular. Everyone feels good, everyone's happy, and there's no mention of fake pregnancy or any of the uncomfortable adult relationships or why anyone ever mentioned that Sectionals were in two weeks when most Ohio show choir competitions won't take place before next semester anyway.

So, what's the date? As usual, the show gives us one temporal clue, as Schu tells everyone to "be ready on Thursday" for the diva-off. We can assume, as usual, that all of the subplots and outfit changes take place over as short a time frame as possible, given the previous episode. The Thursday after November 19 would be the 26th, but that happens to be Thanksgiving. Thus, it has to be the following Thursday, December 3. We will note that the Cheerios have begun to practice indoors, so there is some attempt to keep the show in a universe in which Ohio actually gets cold in autumn and winter.

This week, it occurred to me that maybe the show wasn't totally serialized, and thus it could be earlier. After all, most of the subplots, particularly the various teen interactions, could easily have run in parallel. Kurt could join the football team and come out to his dad, and the whole Kristen Chenoweth thing could have been the week before, and that could still have been in a later episode, because they don't really have anything to do with each other. But every single episode has a moment, often just a few lines, that mark it as happening after the previous episode. This week, it's Quinn being kicked out of Cheerios. I get the need to tell the story as one continuous arc — it's easier for the viewer to keep track of everyone's relationships if we don't have to worry about whether today's events happened before or after the stuff we saw last week — but you have to account for the fact that time always moves forward. You can't just have everyone be a junior again for the second season.

Obviously, I'm more into the date-continuity issues of this show than most folks, but what really bothers me about them is how easy they would have been to avoid. If you don't say that the football team is 0-6, you can still claim it's early September. "Sectionals are in two weeks" was part of a throwaway line meant to show us that the kids weren't focusing, but details like that are exactly the sort of thing writers need to pay attention to. If you say "two weeks," you've committed yourself to a date, and that's a problem four episodes down the road, when you've had a month's worth of subplot, and still four more episodes until the actual competition.

And that's what you missed on Glee. Join us next week, as we'll find out whether women throw themselves at Will in spite of his v-neck cardigan/skinny tie ensembles, or because of them.

Why does it happen? Because it happens.

When I lived in Michigan, a friend of mine hosted Monday Night Football parties. I expect he still hosts them, I just don't live there anymore. Anyway, a bunch of guys would show up, and we'd spend the evening drinking beer and sort-of watching the game. Since we usually only had a passing interest in the game, we would turn quickly to other distractions, usually games. It was at these parties that I was introduced to a dice game my friends called "Scat." Scat is a variation of poker dice. The rules are as follows:

• The goal for each player is to make the best "hand" (actually, it's to avoid making the worst hand, but I'll get to that in a second). A hand is the best possible set of any one rank. Sets are ranked first by cardinality (the size of the set) and then by ordinality (the rank of the dice in the set); for example, three sixes is better than three fours, but four twos is better than three sixes. Only the set itself counts, the "kickers" are irrelevant, e.g. four twos and a six is the same as four twos and a three). Ones are wild, so a one and two sixes is really three sixes. However, a hand must have a one to be valid; a hand with no ones is "scat," and is worth nothing.
• Each player has three rolls to make a hand. After each of the first two rolls, the player may choose to stop rolling any or all of the dice. However, the player must roll a one before setting aside any dice, i.e., if the player has scat, he must re-roll all five dice. If a die is set aside after the first roll, it may not be re-rolled after the second roll.
• Play passes clockwise around the table. After each player has rolled, the lowest hand is eliminated, and the remaining players begin a new round. The honor of starting each round passes counter-clockwise: the last player to roll in the previous round rolls first in this round, the first player in the previous round now rolls second, and so on. When only two players are left, the game becomes best-of-three (the first player to have the low roll twice loses).
• If, at any time, a player's dice "stack" (i.e., one die lands on top of another), that player is immediately eliminated, and the round ends.
• If, at any time, a player rolls five ones, that player wins the game, regardless of how many players are remaining.

Obviously, this is largely a game of chance, so its main purpose was to shift small sums of money around a table. However, I noticed the occasional opportunity for skillful play. For example, a player who rolled two ones, two threes, and a six would usually keep the threes, thus keeping four threes instead of three sixes. However, if the low roll was four fours, that player would need to roll a one or three on the fifth die to avoid elimination. A player who kept the six in the same situation would still need to roll a one or six, but would have two dice, thus improving his odds.

This round of math lessons will focus on this game. I'll first point out some basic probabilities related to the game, and then delve a little into strategies in specific situations. Finally, I'll look at the game theory aspects, in an attempt to come up with a basic strategy.

Lesson 12: Why Bigger Isn't Better

In the final post on strongly asteroidal graphs, we look at asteroidal graphs other than S₃; namely, graphs III_n (for n > 2) and IV_n (for n > 1). Fortunately, changing these graphs into minimal strongly asteroidal graphs is not nearly as complicated as it was for S₃, for two reasons. The first, and most important, is that there is only one potential middle vertex (a₁) in the asteroidal triple, so only one construction need be applied. The second reason is that applying the category B constructions (and, by extension, category C constructions) fails to produce a light path. Here's why:

Suppose that X₁ has three consecutive a₁-heavy vertices x-y-z, with a₁-light vertices u between x and y, and v between y and z. If x and z are not adjacent, then this creates a copy of graph II (with u, v, and a₁ as the strongly asteroidal triple). If, on the other hand, we attempt to place a single a₁-light vertex adjacent to x, y, and z, then this creates a chordless cycle with a₁. Therefore, category B constructions are minimal only if X₁ contains just two consecutive a1-heavy vertices, so we need not apply them when n > 2. Further, applying any category B construction to graph IV₂ will create a k-sun, where k is 3, 4, or 5, depending on the adjacencies between the a₁-light vertices and the neighbors of a₁.

This means we only need to focus on the category A constructions. Applying construction A1 to any of these graphs creates a copy of graph I, except in the case of graph IV₂, in which case we get graph 9. Applying construction A3 to any of these graphs results in a copy of graph II. Only construction A2 creates new minimal graphs. For the graphs in family IV_n, the added vertex must be adjacent to both neighbors of a₁; if it is only adjacent to one of the neighbors, a copy of graph II is created. This gives us two families of minimal strongly asteroidal graphs; these are 55_n and 56_n below.

There is one final construction to consider for graph IV_n; adding a pendant vertex v somewhere along the path X₁ might create a new sAT, in which a1 becomes part of a v-light path between a₂ and a₃. This construction can only work for n > 2, because v cannot be adjacent to a neighbor of a₂ or a₃. However, for n > 3, this construction will create a copy of graph II. The construction does work for graph IV₃, but the graph it creates is graph 43. We are out of ways to modify the minimal asteroidal graphs, so we conclude that we have, at long last, found all of the minimal strongly asteroidal graphs.

We have also, at long last, concluded this topic. The math talk will continue, though. Next up is some probability theory, as I examine a dice game.

Lesson 11: From C To Shining C

Welcome to the penultimate post on strongly asteroidal graphs. We've seen category A constructions, in which a neighbor is added to one of the asteroidal vertices of S₃, and category B constructions, which modify the path between two asteroidal vertices. Today, we will look at category C constructions, which do both.

In a category C construction, a vertex c₁ is added which is adjacent to a₁, b₁, b₂ and b₃, and a category B construction is added to the path X₁. In addition, edges are added between b₁ and every vertex in the category B construction. The modified category B construction still creates an a₁-light path, because there will not be consecutive vertices adjacent to c₁. Equivalently, this can be thought of as adding a regular category B construction, and then making c₁ adjacent to every vertex except a₂ and a₃. However, it is more convenient, in terms of adding later constructions, to modify the category B construction so that all of the vertices are adjacent to b₁. Each category B construction has a corresponding category C construction, so the category C constructions can intuitively be labeled C1-C4. The constructions are in the figure below.

Because c₁ is adjacent to b₂ and b₃, removing b₁ from construction Cn will make a graph isomorphic to one with construction Bn applied. Therefore, in order to create a new minimal sAT, it is necessary for b₁ to be part of another light path.

In construction C1, this means that a category B or C construction must be applied to either a₂ or a₃, but in constructions C2 and C4, b₁ is already part of at least one light path, which includes one vertex from the category B construction (it is difficult to see, but these paths are colored red and blue in the diagram).

Construction C1 does not combine with construction A3 (a parasol is created), but we do get new minimal sATs by applying construction B1 or B2 to the path X₂, and applying constructions A1, A2, or B1 to a₃ (or X₃); these are graphs 37-41. In addition, construction C1 can be combined with constructions B1 and B3 to create graph 42; construction B4 can be used instead of B3 to create graph 44, which we'll get to in a second. In the above figure, note that several of these graphs have been rotated/reflected.

Applying C1 a second time can lead to several new graphs. If the vertices c₁ and c₂ are not adjacent to each other, then the vertices v₁ and v₂ become unnecessary. If c₁ and c₂ are adjacent, but v₁ and c₂ are not, then v₂ is still unnecessary. Finally, we can allow c₁ to be adjacent to c₂ and v₂, and c₂ to be additionally adjacent to v₁.

Adding construction B1 or B2 to any of these combinations, or B3 to either of the latter two combinations, will create a new minimal strongly asteroidal graph. Graphs 43-45 are those that occur using construction B2, graphs 46-48 use B2, and graphs 49 and 50 use B3. The category A constructions and construction B4 do not lead to new minimal graphs.

Finally, there is one way to apply construction C1 to all three asteroidal vertices. The vertices c₁, c₂, and c₃ are adjacent to each other, and only one additional vertex v is necessary; it is adjacent to b₁, b₂, and b₃. This creates graph 51. Other modifications of the adjacencies between c₁, c₂, c₃, and v₁, v₂, and v₃ do not create new minimal graphs; the most common result is a 3-sun.

Only a few minimal graphs arise from the other category C constructions. Construction C2 actually is a minimal sAT by itself, as the adjacencies between b₁ and the B2 portion of the construction create a₂- and a₃-light paths. This is graph 52. Construction C3 does not lead to any new minimal graphs; because the only added vertex in construction B3 is already adjacent to b₁, the addition of vertex c₁ is unnecessary to create an a₁-light path. Finally, construction C4, like construction C2, contains an a₂-light path. Category B constructions that modify the path X₃ do not combine with this, for the same reason they do not combine with construction B4. However, constructions A1 and A2 can be applied to a₃ to make graphs 53 and 54.

These are the final minimal sATs that arise from the graph S₃. Next time, we'll finish up by looking at the infinite families of asteroidal graphs, and seeing how to modify those to create strongly asteroidal graphs.

Zoot Suit Riotous

WARNING: Minor Glee spoliers ahead. If you haven't watched this week's episode, you may want to come back tomorrow.

It's rehearsal time at McKinley, and we acknowledge the fact that we've had Sue as co-director for a week, but now that's over, and we're back on track for Sectionals. Apparently, Sectionals have been postponed, because they should have been this past weekend. Remember that, writers? When you said that Sectionals were "in two weeks," and then you packed two weeks' worth of "show" time into the next two episodes?

That doesn't matter, though, because now there's trouble brewing at football practice. Coach Tanaka has his knickers in a twist, and has declared that the team must practice this Thursday, when the glee club normally rehearses. The issue of why the team is still practicing at all, given that the football season ended three weeks ago, is not brought up. Remember that, writers? When you said the football team was 0-6, and then ran through two months without showing any more football? High school teams don't play into December. We will note that the students are beginning to wear cold-weather clothing.

Anyway, back to the stated premise of these posts. Let's assume that the original rehearsal is on the Monday following last week's episode (so, November 16). The Thursday practice happens. Let's further assume that the Rachel-Puck scene occurs at the end of that particular practice (even though that means Rachel changed outfits post-rehearsal), which will allow Finn to return to glee club on Friday. This packs a heck of a lot of subplot into one week, but as usual, we are generous here at the Preschool. The date is Friday, November 20. We've missed Halloween, and are getting ready for Thanksgiving break.

And that's what you've missed on Glee. It sounds like there will be a few weeks without new episodes, which might allow real time to catch up with show time. Sectionals, no doubt, will be postponed without explanation again. See you in a month.

Eight Days a Week?

WARNING: Minor Glee spoilers ahead. If you haven't seen this week's episode, you may want to stop reading now. Of course, if you have normal human eyeballs, it's probably already too late. Sorry.

So... was "Keep Holding On" the sectionals song?

As you'll recall, last week's episode began with sectionals two weeks away, and ended a week (plus a few days) later. So, presumably, sectionals will be the next weekend.

This week, we begin with Sue and Schu fighting like Mom and Dad on payday, and then there's a flashback. A lot happens: Sue splits up the Glee Club, there's practices for a singing competition, Will flunks all the Cheerios (except for Quinn and Santana and Brittany, apparently), we find out that Puck is one of the Chosen People (shalom!), there's a big fight in Principal Figgins' office.

We could try to figure out dates for all of these. For example, assuming the previous episode ended on Thursday, the original meeting with Figgins has to be, at the very earliest, Monday - surely, he's not trying to "take the temperature" after a single practice? Fortunately, trying to figure out how all of these events shoehorn into a few days is totally unnecessary, thanks (as usual) to the B-plot - Terri's increasingly absurd fake pregnancy. Without going into any additional commentary on the ideas, writing, and character development surrounding this donnybrook, I'll just point out that the ultrasound appointment takes place (per Will) on Friday, just after the big shouting match with Sue. Assuming everything else did take place in the course of a single week (and, since there is no mention of sectionals, we have to assume that), we have a date for these events - it's Friday, November 13.

And this is where the show (finally) begins to lose date continuity. Will comes into Sue's office, and they apologize, and talk about the set list. We still haven't seen sectionals, so this scene takes place... when? Saturday morning? The sectionals were, per previous exposition, scheduled for this particular weekend. The gang sings us out, but it's unclear if this is the competition, or a practice, or a daydream. So, Will and Sue either make up Saturday, before sectionals, or Monday after sectionals which we didn't see (given the competition, plausible), or sectionals got postponed and no one told us, or most of the McKinley students and faculty own time-turners.

And that's what you missed on Glee. Tune in next week for football three weeks after the end of football season, and perhaps Halloween decorations just in time for Thanksgiving.

I also make a better door than a window

A while back, our regular trivia host took sick, and was missing for two weeks. One of the regulars got in touch with him, picked up the clues, and read them in his stead. In an effort to help out, I offered to write and read a round of trivia, and was allowed to do so the second week. I came up with ten questions that I thought were great. Unfortunately, it turns out I'm a better trivia-taker than trivia writer; the winning group in my round got one (1) question right, and they were the only ones to get that right. To be sure, I did a poor job. Hard questions are fine, but they're for separating the wheat from the chaff; you have to throw a couple of softballs out to keep people playing. Surely, though, you can do better than one.

1. Name the Pep Boys.
2. In Calvin and Hobbes, what is Calvin's favorite bedtime story?
3. Who was the 23rd President of the United States, serving between Grover Cleveland's non-consecutive terms?
4. What dog won the first three Frisbee World Championships, which are now named after him?
5. In the Velvet Underground song, "Waiting for the Man," how much money is in the singer's hand?
6. How many Canadian provinces border at least one Great Lake?
7. Who was the first African-American woman awarded the Nobel Prize in Literature?
8. On which island did Napoleon die, on May 5, 1821?
9. What is the name of the theme music for The Daily Show?
10. What Dutch driver holds the Indy 500 records for fastest practice lap, qualifying lap, four-lap qualifying time, and 200-lap race?

Lesson 10: All This Has Happened B4

It has been two months since I posted lesson 9. There are a few reasons for the delay, but one reason is that the final category B construction has been a thorn in my side for months. I originally thought that it gave rise to no new minimal graphs. Then, I found one new sAT, then two, then three, then went back down to two, and so on. Every time I thought that I had thoroughly examined the construction, another combination occurred to me. Over the last month, I have examined each combination of constructions that included B4, one by one, and have officially convinced myself whether each combination does or does not produce a new minimal sAT.

Construction B4: The vertex v₁ is adjacent to a₂. In addition, a vertex u₁ is added, which is adjacent to v₁ and b₃. This creates the a₁-light path a₂-v₁-u₁-b₃-a₃.

The great difficulty in construction B4 lies in the fact that it lacks symmetry. With the previous category B constructions, adding a construction to a₂ was no different than adding it to a₃. For example, two combinations produce graph 16. With B4, this is no longer the case; when we apply other constructions, it is important whether they are applied to a₂ or a₃. In addition, if we apply B4 a second time, it matters which of the asteroidal vertices is adjacent to the vertex v₂.

One important thing to note is that applying this construction, and then removing the vertex b₂, is equivalent to just applying construction B1. To create a minimal sAT, the other two constructions must be applied in such a way that either b₂ is part of every a₃-light path, or that every a₂-light path would be heavy if b₂ were removed. This means that the construction applied to a₂ cannot be a category A construction, or construction B3. In addition, applying construction B1 or B2 to the path X₃ creates either a graph in which b₂ can be removed, or one with a chordless cycle. Applying construction A3 to either a₂ or a₃ creates a copy of the parasol.

If we apply a category B construction to the path X₂, the added vertices must be adjacent to v₁, in order to avoid suns and parasols. Construction B1 will create new minimal graphs when constructions A1, A2, or B3 are applied to a3; these are graphs 33-35 below. Construction B2 results in a graph in which u₁ is removable; in fact, it creates a category C construction, which we'll get to next time.

Attempting to apply B4 a second time creates problems. If it is applied to the path X₂, we must decide whether the vertex v₂ is adajcent to a₁ or a₃. If it is adjacent to a₁, then we do not get a minimal sAT, as (depending on the adjacencies) the resulting graph will contain suns, chordless cycles, or a category C construction. Suppose, then, that v₂ is adjacent to a₃. Then applying a category A construction to a₃, or construction B3 to the path X₃ is no longer possible (for the same reasons they could not be applied to a₂ or X₂ earlier). However, allowing v₁ to be adjacent to u₂ and v₂, and v₂ to additionally be adjacent to u₁, creates the a₃-light path a₁-b₁-u₂-v₁-a₂, while also preventing the removal of b₂ or b₃. This is, therefore, a minimal sAT; this is graph 36 at left.

Applying construction B4 to path X3 does not create new strongly asteroidal graphs; in any strongly chordal graph where this is applied, either a1 or b2 will be removable.

Next time, I'll start on the category C constructions. It's all downhill from here, kids. Two more posts max, and then we can move to a new topic.

What Time Is It, Mr. Fox?

If you're like me, you spend your Wednesdays from 9-10 p.m. watching Glee (if you're not like me, you still watch it, but you have a DVR, so you watch it later and without commercials). I love the singing, love the one-liners, and I'd watch Jane Lynch read "Sinners in the Hands of an Angry God," because I think she'd read it in a sardonic tone that would just crack me up the whole time. The relationships (all of them) are AWK-ward, but I prefer to view that as intentional, like it's a whole school of Michael Scotts. One thing has been bothering me, though. A couple weeks ago, I made the following comment over at ALOTT5MA:

I'm having trouble with the tiimeline for this show. If there's enough time passing that it's noticeable that Mr. Shuster only shows up for rehearsal once a week, where are we in the semester?

(This was referring to the "Acafellas" episode. And yes, I've misspelled "Schuester.")

Fortunately, the next week, we got an answer to exactly where we were: towards the beginning of the episode, Ken Tanaka made reference to the football team being 0-6. Assuming (generously) that it took one week for the following:

1. Kurt joins the football team as its kicker (Monday)
2. Kurt and Finn convince the entire football team to take dance lessons (Tuesday)
3. The team choreographs and rehearses a dance routine to "Single Ladies" to the point of presentability (Wednesday, Thursday)
4. The football team, with the help of their newfound dancing skills, wins their game on Friday.

Then the team has run its record to 1-6. High school football in the North generally starts the last Friday in August, so, assuming there were no bye weeks, the game took place on October 9. Finally, we have pinned down a date!

(NOTE: At ALOTT5MA, I said this game would have taken place on October 16, based on the fact that football starts the week before Labor Day. However, with the late Labor Day this year, that was not the case. This jibes with my own high school's football schedule.)

Having figured out what day it is for a specific episode, we can keep track of the chronology by estimating how long each new episode takes. Again, I want to be generous with these estimates - a lot more happens in a typical high schooler's day than most of us remember. Anyway, "The Rhodes Not Taken" begins with Rachel having quit the Glee Club to join the school play (this happened at the end of the previous episode). Let's assume that this happened on Saturday, at a weekend rehearsal. Then the episode opens at Monday's reharsal. In the course of this episode:

1. Will talks to Emma about having found April's transcripts, then gets in touch with April.
2. April returns to high school and meets the gang. This, obviously, couldn't happen until Tuesday.
3. The Glee Club rehearses two songs. Along the way, April wins over the teenagers by contributing to their delinquency. This, plausibly, takes place over the course of the rest of the week. Given the number of outfit changes, this can't just be Tuesday and Wednesday. This is important, because:
4. They perform two songs at their invitational. This, presumably, takes place on Saturday - given the previous assumption, it can't only be Thursday, and the football team (and Cheerios) would have a game on Friday. April leaves after the first song. Fortunately, Rachel has gotten fed up with Sandy Ryerson (after, what, two rehearsals? Three?), and re-joins Glee for the second song.

After this episode, I estimate the date as being October 17. We're now caught up to this week's episode, which begins with Mr. Schuester saying that sectionals are in two weeks, and that the kids have been coasting since last week, when they got the notice that their only competition would be a school for the deaf and an all-girls alternative school. I had originally given the benefit of the doubt and decided this announcement took place before the invitational, despite April's absence, but Rachel's presence (and outfit - she did re-join Glee for a day in the last episode, but quit again partway through the rehearsal) indicates that this is post-invitational, and thus we begin on October 26 at the earliest. Will then pits the boys and girls against each other in a singing competition, to be started, and I quote, "one week from today." The boys sing on Tuesday (so it was actually the 27th at the beginning of the episode), the girls sing on Wednesday, a drug scandal hits the program on Thursday. Sue Sylvester is now co-coach of the Glee Club as of, at the earliest, November 5. Two things to note from this:

1. It is still sunny and warm in Lima, Ohio - as evidenced by none of the Cheerios wearing tights or long-sleeved shirts under their uniforms.
2. The Ohio high school football playoffs begin on November 6. Even if the football team has won out to finish their season, they are, at best, 4-6, and not making the playoffs. Football season is over.

And that's what you've missed on Glee. Stay tuned in the following weeks, as I keep everyone up to date on the date!

Back to Bedlam

We're back, after a two-month hiatus. I hope that I have a little more energy to keep this thing going this time.

Anyway, the last couple of graph theory lessons are ready (oh joy!), I've got a stockpile of trivia questions, and there's one new piece that needs to be somewhere other than ALOTT5MA comment threads. Stick around.