I haven't had a drink in almost 45 minutes

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

After last season ended with lots of temporally ambiguous episodes, this season has come out swinging, with two straight episodes of references to days and recaps of the past and just the generally casual treatment of the space-time continuum that you know I love, because ever since I asked where we were in the semester, I haven't been able to let this go. Last time, we were getting close to the end of January. Where are we this week?

We begin with the revelation that there's someone coming to interview Sue in seven days. That gives us the standard one-week timeframe for this episode, so we just need to figure out what day the show begins. The last episode ended on Wednesday, so this week's episode could plausibly start that day, or Thursday, or Friday, and still only advance "show-time" by a single week. There are two clues:

1. After a weigh-in, Sue informs Mercedes that she has four days left to lose her weight before the pep rally. If said rally was on a Wednesday, that would place this scene some time during the weekend. Thus, Wednesday is out. Thursday is possible; the weigh-in could be Monday, and Sue could then have included Monday in the four days. More plausibly, the opening and pep rally could take place on Friday.
2. The pep rally happens, and the reporter offers to finish the interview the next day. This day includes a Glee club rehearsal, so one assumes it was a school day, which would end the show on a Friday. However, I lean towards the pep rally being Friday, and the interview being on Saturday; there are no hallway conversations to indicate that it's a school day, and there's no reason to think that a dedicated show choir wouldn't practice on Saturday. Either way, we're at the end of the first week of February, either the 5th or the 6th.

Two other temporal clues of note:

1. Kurt states that Parent-Teacher Conferences took place "about a month ago," but then states that he expects the Hudsons to be moving in with to the Hummel residence "before mid-terms." The first statement would put conferences smack at the beginning of January, but the second would make me wonder when else those conferences could have been. Thinking back to my own high school's schedule, January basically starts with first-semester final exams, so I guess right before finals is as good a time as any for conferences. Right?
2. April mentions that, right after her last appearance (yes, I've linked to the same post twice, so save your emails), she spent "a couple of months" in a bar. This, excitingly, lines up with the estimate that that episode ended in late October.

Finally, let's talk a bit about poor Quinn. It's lovely that she actually got a few lines in this show, but she's largely a forgotten plotline so far. She's seven months pregnant (she's due during Spring Break, which is the end of March/beginning of April) and barely showing, which has me concerned about the health of her child. Worse, she's no longer living with Finn, and there's been no word of her parents taking her back, so one assumes the poor girl is living on the street. This would, come to think of it, explain her sad-but-defiant stare in almost every scene, but the pregnant girl shouldn't be shunted aside like Ross and Rachel's baby. Free Quinn Fabray!

And that's what you missed on Glee. Next week... well, I forget what's happening next week. Maybe T-Pain shows up, and the show won't have to pretend it isn't using autotune for a week.

Tick Tock Tick Tock

WARNING: As usual, minor Glee spoilers ahead. If you haven't seen this week's episode, consider coming back tomorrow. It's okay, we'll still be here.

Sometimes, this whole chronology-of-Glee business can be a grind. Most of the episodes take place over the course of a week, and no amount of minute analysis (not that I'd ever be guilty of that) will move things more than a day, so it's easy enough to just move the date ahead six or seven days and say that's where we are and golly shouldn't they be talking about [insert holiday] here?

They shouldn't be talking about [insert holiday], of course. It is a goal of the show, at least implicitly, to be "timeless;" we rarely hear about actual dates, only days of the week. That way, it can be any time of year. More to the point, it can be any year; the show doesn't have to take place during the 2009-2010 school year. It does, however, take place during a school year, and that's the sort of thing that has schedules and rules, and that's what makes these posts worth doing to me. But I digress.

This week's episode, far from being a grind, is chock-full of juicy temporal clues. We begin with a reference to a Rachel-Jesse date, which took place on a weekend, and had to have happened after last week's episode. That gives us a starting point. If we assume that the opening conversation, and first Cheerios' routine, take place on Monday, then Will hands out the Madonna assignment on Tuesday. The Rachel/Finn duet and Will/Sue hair-joke-off take place on Wednesday. The "Vogue" video could be Thursday, or maybe Friday. None of this matters, of course, because we know that everyone has sex (or possibly not) on Friday night. We know this, of course, because Finn asks Rachel directly about her date with Jesse on Friday. This conversation, takes place on Monday, at which point New Directions gets a new member (snicker). Kurt and Mercedes see Sue on Monday, and presumably join the Cheerios then, but the "4 Minutes" number would then have to take place on Tuesday, which makes it Wednesday when the boys sing about being girls. That moves the show through almost two full weeks. If we allow the Wiggles concert to be on the first weekend after the last episode (i.e., January 15th or 16th), we are now at least at January 27. They really do have to start talking about Regionals soon, or Nationals won't happen until next Christmas.

And that's what you missed on Glee. I promise not to harp on the fact that they're not talking about Valentines Day next week.

It's Date Night!

Previously on Glee: The team won Sectionals the weekend before Christmas, then came to school that Monday (which we assumed was due to having to make up snow days), then took a long and well-deserved break. Will broke up with Terri, then made out with Emma, who, presumably, un-resigned. What shenanigans will we get into in 2010?

It's a new year and a new season as we head towards Regionals, but that doesn't mean we can't circle some dates on Glee's Crazy Calendar, now, does it? Let's review the chronology:

Actually, let's start with an aside, in which I point this out to the watchers, the writers, and most especially, the wardrobe personnel of this show: There are way too many outfit changes for the characters. This, in and of itself, provides fodder for this particular batch of posts. I can accept that people might change clothes when they get home from school, but, other than Rachel (who, I assume, keeps several lockers full of clothes around the school), if someone is wearing a different outfit, it is a different day. I get that it's TV, but even on TV there are rules to the non-cartoon world, and one of those rules is that time keeps moving forward, and there are only so many days that people actually spend in school. Suspension of disbelief can cover Rachel telepathically telling the backup band that she's about to sing "Hope it Gives You Hell," and the fact that everyone else in the already knows the words and has thought out the choreography. It cannot cover saying "Sectionals are in two weeks," and then having 30 days' worth of plot before Sectionals. If Regionals takes place in July, and then next season is an interminable march to Nationals, then... I suppose I'll have to write a very stern letter to... someone.

Anyway, thanks to the aforementioned outfit changes, and the fact that people can't be in two places at once and don't live at the school, we go through no fewer than six, and possibly seven, different days:

1) Sue's back! She'd have brought you coffee, but she doesn't like you. Will finds out that he has to place at Regionals.

2) Will hands out the week's assignment, Finn sings his Hello song. Brittany and Santana ask him out.

3) Finn goes on a date with Brittany and Santana. Meanwhile, Rachel meets Jesse.

4) The rest of the gang tell Rachel she can't date Jesse. Will goes to the VA performance and makes out with Idina Menzel. I cannot blame him for this.

5) Emma fixes Will dinner and runs into Terri. (This cannot happen on the same day as 4, by the way, because Will and Idina are clearly at his apartment.)

5ish) Sue brings Rachel to the "Old Maids' Club," Rachel sees Jesse and we find out he's playing her. These could be the same day, or they could be the same day that the Glee Club threatens Rachel, but in three scenes, she has—see previous rant—three different outfits. Not that it really matters, because they don't really have to bleed into a sixth day.

6) Emma confronts Will with a yearbook.

6ish) Final "Hello" sequence. Is it a fantasy sequence, or a rehearsal? I'll go with fantasy, because if it's an actual rehearsal, then this is day 7, because—you guessed it!—Will and Emma are in different outfits.

So, we've gone through at least a week of school days. However, we have a further temporal cue, because the scene with Terri and Emma takes place on Wednesday. Thus, we are at least at the second Thursday of the school year. In Ohio, that means it's January 14.

A few more temporal items of note:

• Jesse mentions having watched McKinley at Sectionals, which immediately struck me as odd, though I suppose it's not out of the question that their own competition would have been on a different day.
• By pointedly telling Finn that she's met "a senior," Rachel implies that Finn is not one. Thus, we have to assume that he is a junior, and by extension, so are Quinn and Puck. This makes little sense to me from the standpoint of high school social hierarchy, but at least they've figured out how to keep them around an extra season without going 90210 on us.

And that's what you missed on Glee. Next week: Madonna!

TWIFB Notes - April 11

Transaction Report:

On Thursday, red-hot shortstops Cliff Pennington and Edgar Renteria finally made it to rosters. Other pickups of note include St. Louis rookie hurler Jamie Garcia (6 IP, 1 ER and a win in his first start), and Joel Zumaya (notable mostly because Ryan Perry, who actually got the Tigers' first non-Valverde save, was dropped on the same day).

Sunday, though, saw the first closer replacement of the season, and the Athl3tics won the bidding war for Jim Johnson at a whopping $4. A month from now, they may wish they'd just used that money on a year-old Happy Meal, but maybe Johnson is this year's Ryan Franklin. We also saw the second closer replacement of the season, but because Neftali Feliz was (rightly) drafted, no bidding ensued.

Meanwhile, the league played Musical Middle Infielders, as Clint Barmes spent three days on waivers, and Jerry Hairston spent three days on a roster. The other Hairston (Scott) slipped on to Duke Silver's roster for $0, which will look good as the Padres' other outfield options continue to struggle. Also of note is Mouse Rat's acquisition of Jeff Mathis, as popular sleeper catcher Miguel Montero gets an early DL stint.

 

Jeff Clement Report:

I have proclaimed Pittsburgh 1B Jeff Clement "this year's Brandon Inge." Like Inge last year, Clement is a catcher-eligible player who is playing another position. In theory, the extra playing time garnered from playing a non-catcher position allows these players to be more valuable than your average catcher, because they can accumulate more counting stats. I'll be tracking this proclamation throughout the season.

Through week one, Jeff has accumulated one home run, two runs, and three RBI. While this does put him among the top 15 catchers in all three categories, he's a bit behind Brandon, who homered in his first three games last year, and had 7 runs and RBI by the end of the first week. The .294 OBP, though, is quite Inge-like.

This Week In Fake Baseball

The Preschool returns from our latest two-month hiatus with what I hope will be a weekly feature here (mainly, because it would give us some regular content). I am participating in ALOTT5MA's absurd 16-team, 400-player head-to-head baseball league.

Draft Recap

The major story of the draft is the drink I came up with: fresh-squeezed tangelo juice with bourbon and a splash of club soda. Serve it on the rocks. I call it a John McLaughlin, because I can't hear the word tangelo without thinking of a Saturday Night Live "McLaughlin Group" parody (the linked transcript is not the one with the tangelo reference).
The other story was the fact that this league uses on-base percentage as a scoring category instead of batting average, and a couple of teams didn't adjust to that (or, they planned to punt OBP in favor of other offensive categories). In a batting average league, Ichiro is usually goes by about the 25th pick, even though he's only a plus contributor in three categories. This is because he not only gives you a high batting average, his contact-heavy style means he does it over 50-60 more at-bats than the Pujolses of the world, which can more than cancel out a low-BA slugger later in your draft. In an OBP league, he still gives you runs and steals, but his OBP is much closer to average, and he doesn't give you the weight of a lot of extra plate appearances. There's nothing wrong with speedy leadoff guys, but they don't carry as much cachet in this setup. Carlos Peña, on the other hand, goes from a late-round utility player to just outside the top 10 first basemen (depending on what you do with guys who are eligible elsewhere).

I was all set to hand the "best pick" prize to Duke Silver's Moustache for snagging OBP-machine Jack Cust in round 19, until the A's went and designated him for assignment on Saturday. No one really wants to read 25 rounds of draft highlights, anyway, so let's move on.

Transaction Report

Most of this league hasn't played with an acquisition budget (each team has a mythical $100 to bid for free agents twice a week) before, and we're all getting used to it.

With Jose Reyes due to miss the first week, and Freddy Sanchez out longer than that, the Chicago American Giants needed some help at middle infield. Fifteen dollars on Juan Uribe, though, was probably not what they had in mind. Only two teams bid for John Bowker, so the Pawnee Possum Tacklers overspent a bit at $5. However, given the number of fourth outfielders (hello, Seth Smith!) on starting rosters, more teams should probably have been bidding for Bowker's services. Similarly, $2 for C.J. Wilson was $2 more than the Athl3tics needed to spend, but with only four healthy starters, this team was a rainout away from missing the starts minimum for the week.

Among the $0 bids, Vicente Padilla's two-start week makes him worth the pickup. Ziegler will be a nice add if Andrew Bailey steps awkwardly off a mound again. Kudos to the Bowling Alley Lawyers for snagging Chris Getz after teams with better bids or higher priority had already filled their rosters.

Coming Soon

Rather than continuing to fill up inches with the season preview, we'll save that for the next couple of posts. If you're not into the fake sports, there should be a new post about Scat this week.

Who are three people who have never been in my kitchen?

I've run into two separate discussions based on this article about Jeopardy! and baseball questions (er, answers). The relevant quote:

Let's say you're on 'Jeopardy!' and you're absolutely routing your two opponents. You have $40,000 going into the final round, while one of your opponents has, let's say, $15,000. You're guaranteed to move onto the next day, but the final category comes up and it has something to do with baseball, which is your favorite sport. How much — if anything — do you risk?

The discussions of this article go along two paths. One is to compare how many of the ten Final Jeopardy! answers everyone got right (most participants have gotten at least seven of the ten - both sites have a fair number of people who are very knowledgeable about baseball). The other is to loudly decry the article's premise that, since you should be fairly confident that you will respond correctly (the article, on the Yahoo! Sports site, assumes you're a baseball fan), you should bet your entire $40,000 haul, rather than the $10,000 that will guarantee a win. Since the Preschool is all about the math these days, let's look at this in a little more detail. Specifically, let's look at the following situation: you have $40,000, the second-place competitor has $15,000, and the Final Jeopardy! category is something squarely in your wheelhouse. How much do you bet?

This (like the Scat decisions I'm supposedly writing about) is a problem of expected value. Suppose that you estimate the probability you will respond correctly as p; for example, if p is 0.8, it means you think you're 80% likely to be correct. If you respond correctly (as you will do 100p% of the time), you gain whatever you've bet; call this number b. If you respond incorrectly (which will happen 100(1-p)% of the time), you will lose b dollars. Your expectation for this decision is then 40,000 + bp - b(1-p), which simplifies to 40,000 - b + 2bp. If your only goal is to maximize your expectation for this decision, then the answer is simple: if p is at least 0.5, you should bet everything; if it's less than 0.5, you should bet nothing.

However, this is not your only goal, because if you win the game, you get to come back tomorrow, and try to win more money. Thus, we must also factor in your expectation of future winnings. (Incidentally, the primary argument I'm seeing against betting everything is that "you're risking a lot in future winnings.") Let's assume, for the sake of argument, that the second-place competitor will bet his entire $15,000, and will respond correctly. Thus, if you bet $10,000 (or less), you will win (or tie), and come back tomorrow, regardless of your response. If you bet more than $10,000, and respond incorrectly, you will lose.

At this time, I should point out that we could make this a more complicated model, estimating the probabilities that the other competitors will respond correctly. Also, you could bet something other than $10,000 or $40,000. Even if you factor in the chance of the other competitors missing, betting something like $20,000 seems a doubly bad idea; you leave $20,000 "on the table" if you are right, and risk losing if you are wrong. You could bet slightly less than $40,000, thus improving the probability that you will win if everyone misses the final answer. In fact, I think $39,998 might be the best bet if you're "going for broke." That said, complicating the model takes time, and might be better suited to a paper than a blog post—any undergrads looking for a thesis, this might be a starting point! Once we decide to stick with a simple model, betting anything other than $40,000 or $10,000 makes the math work out less nicely, and a couple of dollars here and there isn't going to make much difference, nor will the tiny probability that you'll win with $2. Let's assume, then, that this is a binary decision: bet $10,000 and always win, or bet $40,000 and lose if you're wrong.

Assume that, if you win, your expected earnings from future shows is x (we could take some time and figure out what x might be, but let's wait to decide whether or not to deal with that). Now, we can look at two different expectations: one for playing conservatively, and one for betting everything. If you play conservatively, and bet $10,000, you will always win, thus adding x to your expectation. Using the formula above, your expectation is now 30,000 + 20,000p + x. If you bet your entire $40,000, you will lose everything, including future winnings, if you respond incorrectly. However, when you respond correctly, you double your money, and you still get to come back tomorrow. Thus, you add px to your expectation, which is now 80,000p + px.

An aside: I think this is what a lot of the folks making the "future winnings" argument miss—you aren't giving up your future winnings, just a portion of them. If you can win more today than you're likely to get on future days, there's a strong incentive to go for it.

Anyway, let's see what that incentive is. The difference between playing recklessly and playing safe is 60,000p - (30,000 + (1-p)x). Economists refer to this as an opportunity cost - by betting only $10,000, you are forfeiting whatever you could have earned by betting everything. If this cost is positive, then, economically, you are making a mistake by only betting $10,000; if it is negative, then playing it safe is the right call. Your decision is based on whether 60,000p is larger than (30,000 + (1-p)x).

Let's look at an example: Suppose you think you're 80% likely to get the Final Jeopardy! question correct. You are then comparing 48,000 to 30,000 + .2x. If x is less than 90,000, then 48,000 will be larger. In other words, you should bet everything unless you think you'll make at least $90,000 from future shows; given that $50,000 is a pretty hefty one-day total, this would require a fair amount of confidence on your part!

Does this mean Jeopardy! players should be wagering more recklessly? Well, perhaps. The example above is pretty solidly in favor of betting it all, but the big issue is estimating that probabilty. I managed to get 8 of the ten questions posed in the linked article (nine, actually, but I had to think about one for more than 30 seconds, so I'd have been hosed in Final Jeopardy), but that doesn't mean I'd be 80% confident I could answer a question based only the fact that it would be about baseball. Also, the necessary value for x in order to play conservatively goes down very quickly; if p = 0.7, then you're comparing $42,000 to 30,000 + .3x, and x only needs to be $40,000 to break even; considering that you've got $40,000 right now, this is getting to the realm of possibility. If we knew x (and we can probably come up with a statistical value, given the probability that the previous day's champion wins again, and the mean winning total), we could come up with a "break-even" value for p. I might return to this, but for now, I think the risk-averse behavior is better, unless you are very very sure of yourself.

Another Post Just To Prove I Still Post

The latest post on Scat strategy has been half-finished for over a week, as I have been working on presentations of old and new research. Also, I'm still trying to find a new trivia night; I may give up soon and just start posting my own stuff. For now, here's a fun definition:

Let A_k be the set {1, 2, ..., k}. A graph G has an optimal t-tone coloring if each vertex can be assigned a t-element subset of A_k in such a way that if vertices u and v are distance d apart (in other words, the shortest path in G connecting these vertices contains d edges), then their subsets have at most d-1 elements in common.

For example, vertices adjacent to each other can have no shared elements in their subsets, while if the distance between two vertices is greater than t, they may be assigned identical subsets.

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.