## How Much Might I Have Lost At Pinball?

After the state pinball championship last month there was a second, side tournament. It was a sort-of marathon event in which I played sixteen games in short order. I won three of them and lost thirteen, a disheartening record. The question I can draw from this: was I hopelessly outclassed in the side tournament? Is it plausible that I could do so awfully?

The answer would be “of course not”. I was playing against, mostly, the same people who were in the state finals. (A few who didn’t qualify for the finals joined the side tournament.) In that I had done well enough, winning seven games in all out of fifteen played. It’s implausible that I got significantly worse at pinball between the main and the side tournament. But can I make a logically sound argument about this?

In full, probably not. It’s too hard. The question is, did I win way too few games compared to what I should have expected? But what should I have expected? I haven’t got any information on how likely it should have been that I’d win any of the games, especially not when I faced something like a dozen different opponents. (I played several opponents twice.)

But we can make a model. Suppose that I had a fifty percent chance of winning each match. This is a lie in detail. The model contains lies; all models do. The lies might let us learn something interesting. Some people there I could only beat with a stroke of luck on my side. Some people there I could fairly often expect to beat. If we pretend I had the same chance against everyone, though, we get something that we can model. It might tell us something about what really happened.

If I play 16 matches, and have a 50 percent chance of winning each of them, then I should expect to win eight matches. But there’s no reason I might not win seven instead, or nine. Might win six, or ten, without that being too implausible. It’s even possible I might not win a single match, or that I might win all sixteen matches. How likely?

This calls for a creature from the field of probability that we call the binomial distribution. It’s “binomial” because it’s about stuff for which there are exactly two possible outcomes. This fits. Each match I can win or I can lose. (If we tie, or if the match is interrupted, we replay it, so there’s not another case.) It’s a “distribution” because we describe, for a set of some number of attempted matches, how the possible outcomes are distributed. The outcomes are: I win none of them. I win exactly one of them. I win exactly two of them. And so on, all the way up to “I win exactly all but one of them” and “I win all of them”.

To answer the question of whether it’s plausible I should have done so badly I need to know more than just how likely it is I would win only three games. I need to also know the chance I’d have done worse. If I had won only two games, or only one, or none at all. Why?

Here I admit: I’m not sure I can give a compelling reason, at least not in English. I’ve been reworking it all week without being happy at the results. Let me try pieces.

One part is that as I put the question — is it plausible that I could do so awfully? — isn’t answered just by checking how likely it is I would win only three games out of sixteen. If that’s awful, then doing even worse must also be awful. I can’t rule out even-worse results from awfulness without losing a sense of what the word “awful” means. Fair enough, to answer that question. But I made up the question. Why did I make up that one? Why not just “is it plausible I’d get only three out of sixteen games”?

Habit, largely. Experience shows me that the probability of any particular result turns out to be implausibly low. It isn’t quite that case here; there’s only seventeen possible noticeably different outcomes of playing sixteen games. But there can be so many possible outcomes that even the most likely one isn’t.

Take an extreme case. (Extreme cases are often good ways to build an intuitive understanding of things.) Imagine I played 16,000 games, with a 50-50 chance of winning each one of them. It is most likely that I would win 8,000 of the games. But the probability of winning exactly 8,000 games is small: only about 0.6 percent. What’s going on there is that there’s almost the same chance of winning exactly 8,001 or 8,002 games. As the number of games increases the number of possible different outcomes increases. If there are 16,000 games there are 16,001 possible outcomes. It’s less likely that any of them will stand out. What saves our ability to predict the results of things is that the number of plausible outcomes increases more slowly. It’s plausible someone would win exactly three games out of sixteen. It’s impossible that someone would win exactly three thousand games out of sixteen thousand, even though that’s the same ratio of won games.

Card games offer another way to get comfortable with this idea. A bridge hand, for example, is thirteen cards drawn out of fifty-two. But the chance that you were dealt the hand you just got? Impossibly low. Should we conclude from this all bridge hands are hoaxes? No, but ask my mother sometime about the bridge class she took that one cruise. “Three of sixteen” is too particular; “at best three of sixteen” is a class I can study.

Unconvinced? I don’t blame you. I’m not sure I would be convinced of that, but I might allow the argument to continue. I hope you will. So here are the specifics. These are the chance of each count of wins, and the chance of having exactly that many wins, for sixteen matches:

Wins Percentage
0 0.002 %
1 0.024 %
2 0.183 %
3 0.854 %
4 2.777 %
5 6.665 %
6 12.219 %
7 17.456 %
8 19.638 %
9 17.456 %
10 12.219 %
11 6.665 %
12 2.777 %
13 0.854 %
14 0.183 %
15 0.024 %
16 0.002 %

So the chance of doing as awfully as I had — winning zero or one or two or three games — is pretty dire. It’s a little above one percent.

Is that implausibly low? Is there so small a chance that I’d do so badly that we have to figure I didn’t have a 50-50 chance of winning each game?

I hate to think that. I didn’t think I was outclassed. But here’s a problem. We need some standard for what is “it’s implausibly unlikely that this happened by chance alone”. If there were only one chance in a trillion that someone with a 50-50 chance of winning any game would put in the performance I did, we could suppose that I didn’t actually have a 50-50 chance of winning any game. If there were only one chance in a million of that performance, we might also suppose I didn’t actually have a 50-50 chance of winning any game. But here there was only one chance in a hundred? Is that too unlikely?

It depends. We should have set a threshold for “too implausibly unlikely” before we started research. It’s bad form to decide afterward. There are some thresholds that are commonly taken. Five percent is often useful for stuff where it’s hard to do bigger experiments and the harm of guessing wrong (dismissing the idea I had a 50-50 chance of winning any given game, for example) isn’t so serious. One percent is another common threshold, again common in stuff like psychological studies where it’s hard to get more and more data. In a field like physics, where experiments are relatively cheap to keep running, you can gather enough data to insist on fractions of a percent as your threshold. Setting the threshold after is bad form.

In my defense, I thought (without doing the work) that I probably had something like a five percent chance of doing that badly by luck alone. It suggests that I did have a much worse than 50 percent chance of winning any given game.

Is that credible? Well, yeah; I may have been in the top sixteen players in the state. But a lot of those people are incredibly good. Maybe I had only one chance in three, or something like that. That would make the chance I did that poorly something like one in six, likely enough.

And it’s also plausible that games are not independent, that whether I win one game depends in some way on whether I won or lost the previous. But it does feel like it’s easier to win after a win, or after a close loss. And it feels harder to win a game after a string of losses. I don’t know that this can be proved, not on the meager evidence I have available. And you can almost always question the independence of a string of events like this. It’s the safe bet.

## What Pinball Games Are Turing Machines?

I got to thinking about Turing machines. This is the conceptual model for basically all computers. The classic concept is to imagine a string of cells. In each cell is some symbol. It’s gone over by some device that follows some rule about whether and how to change the symbol. We have other rules that let us move the machine from one cell to the next. This doesn’t sound like much. But it’s enough. We can imagine all software to be some sufficiently involved bit of work on a string of cells and changing (or not) the symbols in those cells.

We don’t normally do this, because it’s too much tedious work. But we know we could go back to this if we truly must. A proper Turing machine has infinitely many cells, which no actual computer does, owing to the high cost of memory chips and the limited electricity budget. We can pretend that “a large enough number of cells” is good enough; it often is. And it turns out any one Turing machine can be used to simulate another Turing machine. This requires us to not care about how long it takes to do something, but that’s all right. Conceptually, we don’t care.

And I specifically got wondering what was the first pinball machine to be a Turing machine. I’m sure that modern pinball machines are, since there have been computers of some kind in pinball machines since the mid-1970s. So that’s a boring question. My question is: were there earlier pinball machines that satisfy the requirements of a Turing machine?

My gut tells me there must be. This is mostly because it’s surprisingly hard not to create a Turing machine. If you hang around near mathematics or computer science people you’ll occasionally run across things like where someone created a computer inside a game like Minecraft. It’s possible to create a Turing machine using the elements of the game. The number of things that are Turing-complete, as they say, is surprising. CSS version 3, a rule system for how to dress up content on a web site, turns out to be Turing-complete (if you make some reasonable extra suppositions). Magic: The Gathering cards are, too. So you could set up a game of Magic: the Gathering which simulated a game of Minecraft which itself simulated the styling rules of a web page. Note the “you” in that sentence.

That’s not proof, though. But I feel pretty good about supposing that some must be. Pinball machines consist, at heart, of a bunch of switches which are activated or not by whether a ball rolls over them. They can store a bit of information: a ball can be locked in a scoop, or kicked out of the scoop as need be. Points can be tallied on the scoring reel. The number of balls a player gets to plunge can be increased — or decreased — based on things that happen on the playfield. This feels to me like it’s got to be a Turing-complete scheme.

So I suspect that the layout of a pinball game, and the various ways to store a bit of information, with (presumably) perfect ball-flipping and table-nudging skills, should make it possible to make a Turing machine. (There ought not be a human in the loop, but I’m supposing that we could replace the person with a mechanism that flips or nudges at the right times or when the ball is in the right place.) I’m wanting for proof, though, and I leave the question here to tease people who’re better than I am at this field of mathematics and computer science.

And I’m curious when the first game that was so capable was made. The very earliest games were like large tabletop versions of those disappointing car toys, the tiny transparent-plastic things with a ball bearing you shoot into one of a series of scoops. Eventually, tilt mechanisms were added, and scoring reels, and then flippers, and then the chance to lock balls. Each changed what the games could do. Did it reach the level of complexity I think it did? I’d like to know.

Yes, this means that I believe it would be theoretically possible to play a pinball game that itself simulated the Pinball Arcade program simulating another pinball game. If this prospect does not delight you then I do not know that we can hope to ever understand one another.

• #### John Friedrich 12:15 am on Saturday, 25 March, 2017 Permalink | Reply

I’m still struggling with the possibility that the entire universe is a computer simulation. No one has come up with a compelling argument against it yet.

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• #### Joseph Nebus 10:52 pm on Thursday, 30 March, 2017 Permalink | Reply

I’m getting pretty far outside my competence to talk about a problem like that. It really calls on the expertise of the philosophy community to work out well.

My poorly-considered thoughts about it run along these lines, though. Suppose our universe is a simulation run for whatever purpose in the super-universe. If there is no possible way for us to detect the super-universe’s existence, or to demonstrate that it is affecting our universe, then it’s hard to see what the difference is between the status of the simulated universe and whatever the universe being real might mean.

But suppose that there is a super-universe. Then it’s hard to see what arguments for our universe being a simulation would not also apply to our super-universe; why shouldn’t it be a simulation in a super-super-universe? But then why wouldn’t that be a simulation in a super-super-super-universe? And so on to an infinite regression of universes simulated within more computationally powerful universes.

That’s nothing conclusive, certainly. There’s no reason we can’t have an infinite regression like that. It feels wasteful of existence, somehow. But it also suggests there’s no point at which any entity in any of the super-(etc)-universes could be confident they were in reality. So either the infinite stack of simulations is wrong or there’s no such thing as “real”. Neither seems quite satisfying.

I expect the professionals have better reasoning than mine, though. And it might be something that produces useful insights even if it can’t be resolved, akin to Zeno’s Paradoxes.

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• #### fluffy 5:47 pm on Saturday, 25 March, 2017 Permalink | Reply

The thing with all those provable Turing machines is that they have dynamic and conceptually-unbounded amounts of storage space (Minecraft levels grow, Magic decks grow, etc.), whereas purely-mechanical pinball machines only have a finite amount of state. Which is to say that they are at best a finite state machine. I mean unless you want to go down the rabbit hole of considering every possible physical position of the ball to be a different state, in which case all you need for a Turing machine (given perfect nudge mechanics etc.) is a ball.

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• #### Joseph Nebus 10:59 pm on Thursday, 30 March, 2017 Permalink | Reply

This is true, although I take a forgiving view of storage space. If we can imagine a Magic deck growing as large as needed for the problem we’re working, why can’t we imagine a Pop-A-Card that has a six- or seven- or even 10100-digit score reel? (Taking the score reel to be how the state is stored.) At least it seems to me if someone can keep getting all this tape for their classical Turing machine then we can hook another reel up.

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## How Much I Did Lose In Pinball

A follow-up for people curious how much I lost at the state pinball championships Saturday: I lost at the state pinball championships Saturday. As I expected I lost in the first round. I did beat my expectations, though. I’d figured I would win one, maybe two games in our best-of-seven contest. As it happened I won three games and I had a fighting chance in game seven.

I’d mentioned in the previous essay about how much contingency there is especially in a short series like this one. My opponent picked the game I expected she would to start out. And she got an awful bounce on the first ball, while I got a very lucky bounce that started multiball on the last. So I won, but not because I was playing better. The seventh game was one that I had figured she might pick if she needed to crush me, and if I had gotten a better bounce on the first ball I’d still have had an uphill struggle. Just less of one.

After the first round I got into a set of three “tie-breaking” rounds, used to sort out which of the sixteen players ranked as number 11 versus number 10. Each of those were a best-of-three series. I did win one series and lost two others, dropping me into 12th place. Over the three series I had four wins and four losses, so I can’t say that I mismatched there.

Where I might have been mismatched is the side tournament. This was a two-hour marathon of playing a lot of games one after the other. I finished with three wins and 13 losses, enough to make me wonder whether I somehow went from competent to incompetent in the hour or so between the main and the side tournament. Of course not, based on a record like that, but — can I prove it?

Meanwhile a friend pointed out The New York Times covering the New York State pinball championship:

The article is (at least for now) at https://www.nytimes.com/2017/02/12/nyregion/pinball-state-championship.html. What my friend couldn’t have known, and what shows how networked people are, is that I know one of the people featured in the article, Sean “The Storm” Grant. Well, I knew him, back in college. He was an awesome pinball player even then. And he’s only got more awesome since.

How awesome? Let me give you some background. The International Flipper Pinball Association (IFPA) gives players ranking points. These points are gathered by playing in leagues and tournaments. Each league or tournament has a certain point value. That point value is divided up among the players, in descending order from how they finish. How many points do the events have? That depends on how many people play and what their ranking is. So, yes, how much someone’s IFPA score increases depends on the events they go to, and the events they go to depend on their score. This might sound to you like there’s a differential equation describing all this. You’re close: it’s a difference equation, because these rankings change with the discrete number of events players go to. But there’s an interesting and iterative system at work there.

(Points only expire with time. The system is designed to encourage people to play a lot of things and keep playing them. You can’t lose ranking points by playing, although it might hurt your player-versus-player rating. That’s calculated by a formula I don’t understand at all.)

Anyway, Sean Grant plays in the New York Superleague, a crime-fighting band of pinball players who figured out how to game the IFPA rankings system. They figured out how to turn the large number of people who might visit a Manhattan bar and casually play one or two games into a source of ranking points for the serious players. The IFPA, combatting this scheme, just this week recalculated the Superleague values and the rankings of everyone involved in it. It’s fascinating stuff, in that way a heated debate over an issue you aren’t emotionally invested in can be.

Anyway. Grant is such a skilled player that he lost more points in this nerfing than I have gathered in my whole competitive-pinball-playing career.

So while I knew I’d be knocked out in the first round of the Michigan State Championships I’ll admit I had fantasies of having an impossibly lucky run. In that case, I’d have gone to the nationals and been turned into a pale, silverball-covered paste by people like Grant.

Thanks again for all your good wishes, kind readers. Now we start the long road to the 2017 State Championships, to be held in February of next year. I’m already in 63rd place in the state for the year! (There haven’t been many events for the year yet, and the championship and side tournament haven’t posted their ranking scores yet.)

## How Much Can I Expect To Lose In Pinball?

This weekend, all going well, I’ll be going to the Michigan state pinball championship contest. There, I will lose in the first round.

I’m not trying to run myself down. But I know who I’m scheduled to play in the first round, and she’s quite a good player. She’s the state’s highest-ranked woman playing competitive pinball. So she starts off being better than me. And then the venue is one she gets to play in more than I do. Pinball, a physical thing, is idiosyncratic. The reflexes you build practicing on one table can betray you on a strange machine. She’s had more chance to practice on the games we have and that pretty well settles the question. I’m still showing up, of course, and doing my best. Stranger things have happened than my winning a game. But I’m going in with I hope realistic expectations.

That bit about having realistic expectations, though, makes me ask what are realistic expectations. The first round is a best-of-seven match. How many games should I expect to win? And that becomes a probability question. It’s a great question to learn on, too. Our match is straightforward to model: we play up to seven times. Each time we play one or the other wins.

So we can start calculating. There’s some probability I have of winning any particular game. Call that number ‘p’. It’s at least zero (I’m not sure to lose) but it’s less than one (I’m not sure to win). Let’s suppose the probability of my winning never changes over the course of seven games. I will come back to the card I palmed there. If we’re playing 7 games, and I have a chance ‘p’ of winning any one of them, then the number of games I can expect to win is 7 times ‘p’. This is the number of wins you might expect if you were called on in class and had no idea and bluffed the first thing that came to mind. Sometimes that works.

7 times p isn’t very enlightening. What number is ‘p’, after all? And I don’t know exactly. The International Flipper Pinball Association tracks how many times I’ve finished a tournament or league above her and vice-versa. We’ve played in 54 recorded events together, and I’ve won 23 and lost 29 of them. (We’ve tied twice.) But that isn’t all head-to-head play. It counts matches where I’m beaten by someone she goes on to beat as her beating me, and vice-versa. And it includes a lot of playing not at the venue. I lack statistics and must go with my feelings. I’d estimate my chance of beating her at about one in three. Let’s say ‘p’ is 1/3 until we get evidence to the contrary. It is “Flipper Pinball” because the earliest pinball machines had no flippers. You plunged the ball into play and nudged the machine a little to keep it going somewhere you wanted. (The game Simpsons Pinball Party has a moment where Grampa Simpson says, “back in my day we didn’t have flippers”. It’s the best kind of joke, the one that is factually correct.)

Seven times one-third is not a difficult problem. It comes out to two and a third, raising the question of how one wins one-third of a pinball game. Most games involve playing three rounds, called balls, is the obvious observation. But this one-third of a game is an average. Imagine the two of us playing three thousand seven-game matches, without either of us getting the least bit better or worse or collapsing of exhaustion. I would expect to win seven thousand of the games, or two and a third games per seven-game match.

Ah, but … that’s too high. I would expect to win two and a third games out of seven. But we probably won’t play seven. We’ll stop when she or I gets to four wins. This makes the problem hard. Hard is the wrong word. It makes the problem tedious. At least it threatens to. Things will get easy enough, but we have to go through some difficult parts first.

There are eight different ways that our best-of-seven match can end. She can win in four games. I can win in four games. She can win in five games. I can win in five games. She can win in six games. I can win in six games. She can win in seven games. I can win in seven games. There is some chance of each of those eight outcomes happening. And exactly one of those will happen; it’s not possible that she’ll win in four games and in five games, unless we lose track of how many games we’d played. They give us index cards to write results down. We won’t lose track.

It’s easy to calculate the probability that I win in four games, if the chance of my winning a game is the number ‘p’. The probability is p4. Similarly it’s easy to calculate the probability that she wins in four games. If I have the chance ‘p’ of winning, then she has the chance ‘1 – p’ of winning. So her probability of winning in four games is (1 – p)4.

The probability of my winning in five games is more tedious to work out. It’s going to be p4 times (1 – p) times 4. The 4 here is the number of different ways that she can win one of the first four games. Turns out there’s four ways to do that. She could win the first game, or the second, or the third, or the fourth. And in the same way the probability she wins in five games is p times (1 – p)4 times 4.

The probability of my winning in six games is going to be p4 times (1 – p)2 times 10. There are ten ways to scatter four wins by her among the first five games. The probability of her winning in six games is the strikingly parallel p2 times (1 – p)4 times 10.

The probability of my winning in seven games is going to be p4 times (1 – p)3 times 20, because there are 20 ways to scatter three wins among the first six games. And the probability of her winning in seven games is p3 times (1 – p)4 times 20.

Add all those probabilities up, no matter what ‘p’ is, and you should get 1. Exactly one of those four outcomes has to happen. And we can work out the probability that the series will end after four games: it’s the chance she wins in four games plus the chance I win in four games. The probability that the series goes to five games is the probability that she wins in five games plus the probability that I win in five games. And so on for six and for seven games.

So that’s neat. We can figure out the probability of the match ending after four games, after five, after six, or after seven. And from that we can figure out the expected length of the match. This is the expectation value. Take the product of ‘4’ and the chance the match ends at four games. Take the product of ‘5’ and the chance the match ends at five games. Take the product of ‘6’ and the chance the match ends at six games. Take the product of ‘7’ and the chance the match ends at seven games. Add all those up. That’ll be, wonder of wonders, the number of games a match like this can be expected to run.

Now it’s a matter of adding together all these combinations of all these different outcomes and you know what? I’m not doing that. I don’t know what the chance is I’d do all this arithmetic correctly is, but I know there’s no chance I’d do all this arithmetic correctly. This is the stuff we pirate Mathematica to do. (Mathematica is supernaturally good at working out mathematical expressions. A personal license costs all the money you will ever have in your life plus ten percent, which it will calculate for you.)

Happily I won’t have to work it out. A person appearing to be a high school teacher named B Kiggins has worked it out already. Kiggins put it and a bunch of other interesting worksheets on the web. (Look for the Voronoi Diagramas!)

There’s a lot of arithmetic involved. But it all simplifies out, somehow. Per Kiggins’ work, the expected number of games in a best-of-seven match, if one of the competitors has the chance ‘p’ of winning any given game, is:

$E(p) = 4 + 4\cdot p + 4\cdot p^2 + 4\cdot p^3 - 52\cdot p^4 + 60\cdot p^5 - 20\cdot p^6$

Whatever you want to say about that, it’s a polynomial. And it’s easy enough to evaluate it, especially if you let the computer evaluate it. Oh, I would say it seems like a shame all those coefficients of ‘4’ drop off and we get weird numbers like ’52’ after that. But there’s something beautiful in there being four 4’s, isn’t there? Good enough.

So. If the chance of my winning a game, ‘p’, is one-third, then we’d expect the series to go 5.5 games. This accords well with my intuition. I thought I would be likely to win one game. Winning two would be a moral victory akin to championship.

Let me go back to my palmed card. This whole analysis is based on the idea that I have some fixed probability of winning and that it isn’t going to change from one game to the next. If the probability of winning is entirely based on my and my opponents’ abilities this is fair enough. Neither of us is likely to get significantly more or less skilled over the course of even seven matches. We won’t even play long enough to get fatigued. But ability isn’t everything.

But our abilities aren’t everything. We’re going to be playing up to seven different tables. How each table reacts to our play is going to vary. Some tables may treat me better, some tables my opponent. Luck of the draw. And there’s an important psychological component. It’s easy to get thrown and to let a bad ball wreck the rest of one’s game. It’s hard to resist feeling nervous if you go into the last ball from way behind your opponent. And it seems as if a pinball knows you’re nervous and races out of play to help you calm down. (The best pinball players tend to have outstanding last balls, though. They don’t get rattled. And they spend the first several balls building up to high-value shots they can collect later on.) And there will be freak events. Last weekend I was saved from elimination in a tournament by the pinball machine spontaneously resetting. We had to replay the game. I did well in the tournament, but it was the freak event that kept me from being knocked out in the first round.

That’s some complicated stuff to fit together. I suppose with enough data we could possibly model how much the differences between pinball machines affects the outcome. That’s what sabermetrics is all about. Representing how severely I’ll build a little bad luck into a lot of bad luck? Oh, that’s hard.

Too hard to deal with, at least not without much more sports psychology and modelling of pinball players than we have data to do. The supposition that my chance of winning is fixed for the duration of the match may not be true. But we won’t be playing enough games to be able to tell the difference. The assumption that my chance of winning doesn’t change over the course of the match may be false. But it’s near enough, and it gets us some useful information. We have to know not to demand too much precision from our model.

And seven games isn’t statistically significant. Not when players are as closely matched as we are. I could be worse and still get a couple wins in when they count; I could play better than my average and still get creamed four games straight. I’ll be trying my best, of course. But I expect my best is one or two wins, then getting to the snack room and waiting for the side tournament to start. Shall let you know if something interesting happens.

• #### ksbeth 6:03 pm on Friday, 10 February, 2017 Permalink | Reply

Woo hoo! Good luck )

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• #### Joseph Nebus 4:43 am on Saturday, 11 February, 2017 Permalink | Reply

Thank you! I’m feeling good heading into tomorrow.

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• #### vagabondurges 7:33 pm on Friday, 10 February, 2017 Permalink | Reply

Best of luck! I am loving these pinball posts! And there’s a pinball place in Alameda, CA that you’ve just inspired me to visit again.

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• #### Joseph Nebus 4:45 am on Saturday, 11 February, 2017 Permalink | Reply

Thank you! I’m sorry I don’t find more excuses to write about pinball, since there’s so much about it I do like. And I’m glad you’re feeling inspired; I hope it’s a good visit.

The secrets are: plunge the ball softly, let the ball bounce back and forth on the flippers until it’s moving slowly, and hold the flipper up until the ball comes to a rest so you can aim. So much of pinball is about letting things calm down so you can understand what’s going on and what you want to do next.

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• #### mathtuition88 12:32 am on Saturday, 11 February, 2017 Permalink | Reply

Good luck and all the best!

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• #### Joseph Nebus 4:47 am on Saturday, 11 February, 2017 Permalink | Reply

Thank you! I shall be doing what I can.

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• #### davekingsbury 3:52 pm on Saturday, 11 February, 2017 Permalink | Reply

All this work and you’ll tell me you’re not a betting man …

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• #### Joseph Nebus 11:05 pm on Thursday, 16 February, 2017 Permalink | Reply

I honestly am not. The occasional lottery ticket is my limit. But probability questions are so hard to resist. They usually involve very little calculation but demand thoughtful analysis. It’s great.

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## Finally, What I Learned Doing Theorem Thursdays

The biggest thing I learned from my Theorem Thursdays project was: don’t do this for Thursdays. The appeal is obvious. If things were a little different I’d have no problem with Thursdays. But besides being a slightly-read pop-mathematics blogger I’m also a slightly-read humor blogger. And I try to have a major piece, about seven hundred words that are more than simply commentary on how a comic strip’s gone wrong, ready for Thursday evenings my time.

That’s all my doing. It’s a relic of my thinking that the humor blog should run at least a bit like a professional syndicated columnist’s, with a fixed deadline for bigger pieces. While I should be writing more ahead of deadline than this, what I would do is get to Wednesday realizing I have two major things to write in a day. I’d have an idea for one of them, the mathematics thing, since I would pick a topic the previous Thursday. And once I’ve picked an idea the rest is easy. (Part of the process of picking is realizing whether there’s any way to make seven hundred words about something.) But that’s a lot of work for something that’s supposed to be recreational. Plus Wednesdays are, two weeks a month, a pinball league night.

So Thursday is right out, unless I get better about having first drafts of stuff done Monday night. So Thursday is right out. This has problems for future appearances of the gimmick. The alliterative pull is strong. The only remotely compelling alternative is Theorems on the Threes, maybe one the 3rd, 13th, and 23rd of the month. That leaves the 30th and 31st unaccounted for, and room for a good squabble about whether they count in an “on the threes” scheme.

There’s a lot of good stuff to say about the project otherwise. The biggest is that I had fun with it. The Theorem Thursday pieces sprawled into for-me extreme lengths, two to three thousand words. I had space to be chatty and silly and autobiographic in ways that even the A To Z projects don’t allow. Somehow those essays didn’t get nearly as long, possibly because I was writing three of them a week. I didn’t actually write fewer things in July than I did in, say, May. But it was fewer kinds of things; postings were mostly Theorem Thursdays and Reading the Comics posts. Still, overall readership didn’t drop and people seemed to quite like what I did write. It may be fewer but longer-form essays are the way I should go.

Also I found that people like stranger stuff. There’s an understandable temptation in doing pop-mathematics to look for topics that are automatically more accessible. People are afraid enough of mathematics. They have good reason to be terrified of some topic even mathematics majors don’t encounter until their fourth year. So there’s a drive to simpler topics, or topics that have fewer prerequisites, and that’s why every mathematics blogger has an essay about how the square root of two is irrational and how there’s different sizes to infinitely large sets. And that’s produced some excellent writing about topics like those, which are great topics. They have got the power to inspire awe without requiring any warming up. That’s special.

But it also means they’re hard to write anything new or compelling about if you’re like me, and in somewhere like the second hundred billion of mathematics bloggers. I can’t write anything better than what’s already gone about that. Liouville’s Theorem? That’s something I can be a good writer about. With that, I can have a blog personality. It’s like having a real personality but less work.

As I did with the Leap Day 2016 A To Z project, I threw the topics open to requests. I didn’t get many. Possibly the form gave too much freedom. Picking something to match a letter, as in the A to Z, gives a useful structure for choosing something specific. Pick a theorem from anywhere in mathematics? Something from algebra class? Something mentioned in a news report about a major breakthrough the reporter doesn’t understand but had an interesting picture? Something that you overheard the name of once without any context? How should people know what the scope of it is, before they’ve even seen a sample? And possibly people don’t actually remember the names of theorems unless they stay in mathematics or mathematics-related fields. Those folks hardly need explained theorems with names they remember. This is a hard problem to imagine people having, but it’s something I must consider.

So this is what I take away from the two-month project. There’s a lot of fun digging into the higher-level mathematics stuff. There’s an interest in it, even if it means I write longer and therefore fewer pieces. Take requests, but have a structure for taking them that makes it easy to tell what requests should look like. Definitely don’t commit to doing big things for Thursday, not without a better scheme for getting the humor blog pieces done. Free up some time Wednesday and don’t put up an awful score on Demolition Man like I did last time again. Seriously, I had a better score on The Simpsons Pinball Party than I did on Demolition Man and while you personally might not find this amusing there’s at least two people really into pinball who know how hilarious that is. (The games have wildly different point scorings. This like having a basketball score be lower than a hockey score.) That isn’t so important to mathematics blogging but it’s a good lesson to remember anyway.

• #### elkement (Elke Stangl) 6:21 am on Monday, 22 August, 2016 Permalink | Reply

You are such a prolific writer – kudos! Sorry that I am hardly able to catch up in some months ;-)

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• #### Joseph Nebus 8:48 pm on Sunday, 28 August, 2016 Permalink | Reply

Aw, well, thank you, trusting that prolific is a good thing. I doubt I have time to read myself myself, as my problem with comments should prove.

It happens I’ve gotten into a slow stretch the past few weeks. I’m hoping that with the start of a new season I’ll be able to get to a better balance between twice-a-week and daily.

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## How Pinball Leagues and Chemistry Work: The Mathematics

My love and I play in several pinball leagues. I need to explain something of how they work.

Most of them organize league nights by making groups of three or four players and having them play five games each on a variety of pinball tables. The groupings are made by order. The 1st through 4th highest-ranked players who’re present are the first group, the 5th through 8th the second group, the 9th through 12th the third group, and so on. For each table the player with the highest score gets some number of league points. The second-highest score earns a lesser number of league points, third-highest gets fewer points yet, and the lowest score earns the player comments about how the table was not being fair. The total number of points goes into the player’s season score, which gives her ranking.

You might see the bootstrapping problem here. Where do the rankings come from? And what happens if someone joins the league mid-season? What if someone misses a competition day? (Some leagues give a fraction of points based on the player’s season average. Other leagues award no points.) How does a player get correctly ranked?

• #### sheldonk2014 3:26 pm on Tuesday, 18 August, 2015 Permalink | Reply

Interesting sounding

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• #### Joseph Nebus 9:13 pm on Saturday, 22 August, 2015 Permalink | Reply

Glad you’re interested. My love and I dropped in the rankings last pinball league meeting, although we were a bit overvalued going into it. So that was just part of the process of getting back to our detailed balance.

Still, there’s something terrible about putting up my best-ever game on Cirqus Voltaire and coming in third (of four) on that table. On the bright side I put in my best-ever game on Metallica too and came in first. And that was the final game of the night, so I could go out on a high note.

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• #### Joseph Nebus 3:13 pm on Wednesday, 20 May, 2015 Permalink | Reply Tags: adding machines, amusements, beach, games ( 6 ), Gottleib, pinball, secretaries

Last weekend I visited the Vintage Flipper World pinball museum just outside Ann Arbor, Michigan. Among the games there was Gottleib’s 1955 table Sweet Add-A-Line. It’s a peculiar table by modern standards, since nearly all the playfield is a bunch of lanes, channels through which the pinball might roll. But …

In this round of Sweet Add-A-Line I managed to get the 9, 7, 15, and 2 rollovers, lighting up the total of 33 on the lower-left adding tape. And scored overall 1,840,000 points. See the lights at the bottom. Unanswered question: so, “patio secretary” was a thing in 1955? I guess?

I apologize for the Coors sign reflected in the back glass. I didn’t even see it when I was taking the picture.

Each of the lanes is numbered. Rolling one down lights up that number in the backglass, as above. And if you roll all the numbers in one of the eight strips of tape, the game opens up bonus opportunities. It’s a fun game and certainly one of the top adding-machine-themed pinball machines I’ve ever played. I grant this is of marginal mathematical content, but, heck, I smiled.

The Internet Pinball Database has a scan of the game’s advertising flyer, which I like if nothing else for its defensive “Amusement Pinballs: as American as Baseball and Hot Dogs!” slogan.

• #### baffledbaboon 3:19 pm on Wednesday, 20 May, 2015 Permalink | Reply

A pinball museum? Sounds like heaven. In every arcade I have ever been to, I always find myself drifting towards the pinball machines.

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• #### Joseph Nebus 5:34 pm on Wednesday, 20 May, 2015 Permalink | Reply

It is indeed a great place. Unfortunately it’s only rarely open to the public (zoning problems). But there are wonderful and generally-open museums for pinball. The most famous is in Las Vegas — that one’s credited with leading the current happy revival of pinball — although the one I know best is the Silverball Museum in Asbury Park, New Jersey. Well worth getting to if you’re in the Garden State.

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• #### baffledbaboon 5:46 pm on Wednesday, 20 May, 2015 Permalink | Reply

Thanks so much! I am definitely adding this to my list of places to go.

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• #### Joseph Nebus 4:24 am on Friday, 22 May, 2015 Permalink | Reply

It’s certainly worth it. Well, the ones I’ve been to are. There are a couple arcades with a healthy number of pinball machines too. I’m happy to say Michigan is one of the centers of the current pinball renaissance.

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• #### sheldonk2014 2:56 am on Thursday, 21 May, 2015 Permalink | Reply

There’s nothing like a pinball machine,one wrong move and it titles,it doesn’t get any better then that

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• #### Joseph Nebus 4:27 am on Friday, 22 May, 2015 Permalink | Reply

No, there isn’t anything like that. The appeal of real things that move is a powerful one. Well, and making the right move where you slide the machine, save the ball, and get away with it. That’s really powerful.

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• #### Oscar Hills 12:17 pm on Thursday, 21 May, 2015 Permalink | Reply

I own one of these and love it. I got a Google alert on your post because I’m always on the lookout for a better specimen of SAAL. What makes it such a great game is the massive adrenaline rush of the “replay” that existed at the height of these games in the ’50’s combined with the unique fact that in SAAL if you roll over all the lanes, you score 26 replays – you max out the replay reel. That’s was a rush beyond compare “back in the day.”

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• #### Joseph Nebus 6:34 am on Friday, 22 May, 2015 Permalink | Reply

I’m glad to hear from someone who owns one! I wasn’t able to get more than one column of numbers completed in the limited time I had to play. (The museum is unfortunately open to the public only a few times a year, but it’s got a great selection of 1950s machines to play.) Just the thought of maxing out the replay reel is awesome, though. If I ever do it … well, wow.

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## At The Pinball Tables

A neat coincidence happened as our local pinball league got plans under way for tonight. There are thirteen pinball machines in the local venue, and normally four of them get picked for the night’s competition. The league president’s gone to a randoom number generator to pick the machines, since this way he doesn’t have to take off his hat and draw pinball table names from it. This week, though, he reported that the random number generator had picked the same four tables as it had last session.

There’s a decent little probability quiz to be built around that fact: how many ways there are to get four tables out of the thirteen available, obviously, and from that what the chance is of repeating the selection of tables from the last session. And there are subtler ones, like, what’s the chance of the same tables being drawn two weeks in a row over the course of the season (which is eight meetings long, and one postseason tournament), or what’s the chance of any week’s selection of tables being repeated over the course of a season, or of a year (which has two seasons). And I leave some space below for people who want to work out these problems or figure out similar related ones.

It’s also a reminder that just because something is randomly drawn doesn’t mean that coincidences and patterns won’t appear. It would be a touch suspicious, in fact, if the random number generator never picked the same table (or several tables) in successive weeks. But it’s still a rare enough event that it’s interesting to see it happen.

• #### abyssbrain 5:00 am on Thursday, 12 February, 2015 Permalink | Reply

Coincidences make life more exciting after all…

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• #### Joseph Nebus 9:20 pm on Friday, 13 February, 2015 Permalink | Reply

They make life exciting and also charmingly personal, somehow. They add intimacy to events.

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• #### Aquileana 12:00 pm on Tuesday, 17 February, 2015 Permalink | Reply

As Stéphane Mallarmé would say: “Un coup de dés jamais n’abolira le hasard” : “(Random) dice rolling will not abolish Fate”… Great post Joseph! :star: Best wishes, Aquileana :D

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• #### Joseph Nebus 11:15 pm on Tuesday, 17 February, 2015 Permalink | Reply

Thank you. And that’s a good quote.

The most wondrous thing to me is that it seems like dice-rolling is fate, at least in certain contexts. It’s amazing to see that work.

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• #### SASS-A-FR-ASS 11:22 pm on Monday, 23 February, 2015 Permalink | Reply

Good day! I hope you have remedied your eating of carrot cake?
In any case I have sent a well deserved Award your way my friend.

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• #### Joseph Nebus 3:20 am on Friday, 27 February, 2015 Permalink | Reply

You know, I haven’t got to the carrot cake yet, but I did have a doughnut today. And thanks kindly for the nomination.

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• #### SASS-A-FR-ASS 5:41 pm on Friday, 27 February, 2015 Permalink | Reply

Well, I suppose a doughnut will do as it’s still a nice sweet treat to be having after all. :)

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• #### Joseph Nebus 8:11 pm on Saturday, 28 February, 2015 Permalink | Reply

It so is. I should get one as we head out today, come to think of it.

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• #### SASS-A-FR-ASS 8:15 pm on Saturday, 28 February, 2015 Permalink | Reply

Yes indeed you should. Top priority possibly..

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## Pinball and Large Numbers

I had another little occasion to reflect on the ways of representing numbers, as well as the chance to feel a bit foolish, this past weekend so I’m naturally driven to share it. This came about on visiting the Silverball Museum, a pinball museum, or arcade, in Asbury Park, New Jersey. (I’m not sure the exact difference between a museum in which games are playable by visitors and an arcade, except for the signs affixed to nearly all the games.) Naturally I failed to bring my camera, so I can’t easily show what I had in mind; too bad.

Pinballs, at least once they got around to having electricity installed, need to show the scores. Since about the mid-1990s these have been shown by dot matrix displays, which are pretty easy to read — the current player’s score can be shown extremely large, for example — and make it easy for the game to go into different modes, where the scoring and objectives of play vary for a time. From about the mid-1970s to the mid-1990s eight-segment light-emitting diodes were preferred, for that “small alarm clock” look. And going before that were rotating number wheels, which are probably the iconic look to pinball score boards, to the extent anyone thinks of a classic pinball machine in that detail.

But there’s another score display, which I must admit offends my sense of order. In this, which I noticed mostly in the machines from the 1950s, with a few outliers in the early 60s (often used in conjunction with the rotating wheels), the parts of the number are broken apart, and the score is read by adding up the parts which are lit up. The machine I was looking at had one column of digits for the millions, another for hundreds of thousands, and then another with two-digit numbers.

• #### BunnyHugger 4:48 am on Sunday, 27 November, 2011 Permalink | Reply

I can’t quite imagine the kind of readout you’re talking about, even though I’ve been to the museum. I think I need a visual aid.

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• #### nebusresearch 7:22 pm on Sunday, 27 November, 2011 Permalink | Reply

I’ll try to find a picture of the backglass, or I’ll make that an excuse to visit there again.

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