I don't know, but . . .

The kids across the street finally broke their pinata; a good time apparently had by all. One just now gave up on trying to see if he could shake any more candy from the broken green donkey.

But from my PoV, there was a great moment earlier when, after a few hard but ineffective bashes, a wind gust blew through the tree the pinata hung from, and hundreds of small yellow leaves showered everyone.

Just noodling for fun. This is speculative, not authoritative.

And, on inspection, I clearly have run Blogger's TeX interface way past its limits. I don't know if splitting this up will help--I doubt it. Ah well. Where you see boxes around things, just read the TeX inside, or skim over it. I may have to try to embed gifs, or maybe make a pdf. It's hard to see if I made a typo. And I don't know how to put large parentheses around the matrix--Blogger doesn't handle some of the TeX options.

What does a world look like if it has two dimensions for time instead of just one?

Your intuition undoubtedly says--"That's silly, it would have to look very different from ours, so why bother?"

• Because it is an amusing way to spend time on the bus
• Because some string theorists have come up with modest arguments that we do have two time dimensions. Bars, Vongehr, and Gogberashvili have been looking at systems with 2 time dimensions. I hold no brief for string theory, but hey, it's an excuse.
• In one of his speculative moods, Eddington wrote about interfaces between (3,1), (2,2), and (1,3) spaces, where the numbers in parenthesis are the number of space and time dimensions respectively. The notion fascinated me ever since.
• Because there are serious problems with invisible mass in cosmology, and visibility of matter on other timelines is probably going to be a problem—just as an intuitive guess.

I just want to take a preliminary look right now--not trying to figure out what quantum mechanics would look like, for instance. And by macroscopic time dimensions I mean large enough to use a wall clock or a calendar to measure.

---

There are a few questions about visibility, the speed of light, and causality that don't have obvious answers. I'll try to keep it logical. Assume that the speed of light always appears constant. I'll also assume that objects on different timelines can interact--at some point they had to, so why not now also? Also, if A measures B's relative timeline angle, it should be the same as B measuring A's.

Wait, what do I mean by a "timeline?"

Assume that the two time directions can be viewed as a Euclidean plane, with time-1 in one direction and time-2 at right angles. As an undisturbed object ages, it will assume time-1 and time-2 ($t_1$ and $t_2) values which lie along a line in the plane.

Here are two examples. In the left the upper line has the object moving more along $t_1$ than $t_2,ドル and the lower line tilts more along $t_2$. Where zero is is arbitrary, by the way.

The right-hand image shows a complication that we need to keep in mind. Timeline A is kind of banal. Timeline B, relative to A, also seems ordinary. It has positive components of its timeline direction both parallel to A and perpendicular to A. So from A's perspective B will not go backwards in time.

However, when B meets C, it will appear to be going backwards in at least one time component.

Should we allow that in our initial study? We can work non-causality in if we rely on small interaction rates, or demand that it only work on small distances, but that seems ad hoc. Let's pretend it isn't going to happen and plo

The obvious first approach is to modify the Einsteinian formalism. In some coordinate system, denote points by $(t_1, t_2; x, y, z),ドル where I separate space and time components with a semicolon. Use the same convention for momentum: $(E_1, E_2; P_x, P_y, P_z)$. For two points $a$ and $b,ドル assume an analogous invariant to Einstein's: $(t_{1a} - t_{1b})^2 + (t_{2a} - t_{2b})^2 - (x_a - x_b)^2 - (y_a - y_b)^2 - (z_a - z_b)^2$. Assume that a transformation to a different frame of reference will be linear.

To keep things simple, just ignore $y$ and $z$ for now, and use $\delta {t_1}^2 + \delta {t_2}^2 - \delta x^2$ as the separation.

A linear (and symmetric) transform can be parameterized as

Invariance requires that ${{t_1}^'}^2 + {{t_2}^'}^2 - {x^'}^2 = {t_1}^2 + {t_2}^2 - x^2,ドル from which we can derive equations which specify $A,ドル $\tau,ドル $\epsilon,ドル and $\lambda$ in terms of $\alpha$ and $\beta,ドル where the latter act like the $\beta$ in the usual 1-time dimensional theory, just for the two different time axes. Think of them as $\beta_1$ and $\beta_2$. By looking at the limit $\alpha=0$ we can determine the right sign for the square root.

For ease in reading the result, define:

$\gamma \equiv {1 \over{ \sqrt{1 - \beta_1^2 - \beta_2^2}}}$

$A=\gamma$

$\epsilon = {{\beta_1 \beta_2} \over {\beta_1^2+ \beta_2^2}} (1 - {1 \over \gamma})$

$\lambda = 1 - {{\beta_2^2} \over {\beta_1^2+ \beta_2^2}} (1- {1 \over \gamma})$

$\tau = 1 - {{\beta_1^2} \over {\beta_1^2+ \beta_2^2}} (1- {1 \over \gamma})$

And of course $\beta_1$ and $\beta_2$ are the speeds as a fraction of c along the $t_1$ and $t_2$ axes respectively. Their squared sum will never exceed 1, and so $\gamma$ is always real. Yep, this assumes that nothing exceeds the speed of light in any frame. And if you define a rotation in the $t_1:t_2$ plane--a rotation to a different timeline--you can turn a simple boost with one time direction to one with a mix, and it matches the parameterization here ($R^{-1} B_{1,0} R = B_{\beta_1,\beta_2}$), where $\beta_1$ and $\beta_2$ are the original $\beta^'$ times the sine or cosine.

Recalling that energy is non-negative (skipping QM subtleties), an object with momentum $(E_1, 0, p_x, p_y, p_z)$ will not break into objects with non-negative $E_2,ドル since there's no existing energy in that "bin" to give them. It could break into $(E_1^', 0, p_x^', p_y^', p_z^') + (E_1-E_1^{'}, 0, p_x-p_x^{'}, p_y-p_y^{'}, p_z-p_z^{'}),ドル but not something with a positive $E_2$ component.

In practice that means that if you don't have any local $t_2$ activity, you won't get any. The situation is stable.

Is there any way to detect the other time dimension?

Typically you and everything about you is going along the same timeline--what is there to make it change? Your best bet would be something distant, or something from a great distance that comes to pay a visit.

Suppose that you always measure the speed of light as the same, no matter what timeline it is on or came from.

Suppose you have two objects, with the same timeline direction, but one starting from a different $t_2$ time, $T_2$. It is at some nearby position $x$.

Suppose the first object has interacted with something in its past so that it has a certain amount of $E_2$ energy available to emit a photon that can reach, and bounce off, that second object. Without that, you'll never emit anything with $t_2$ component non-zero, so you'll not hit the second object. OTOH, assume that the first object has enough $E_1$ that the total $E_2$ is negligable. That way the bulk of what you measure is along the $t_1$ line.

In this thought experiment, what you would measure is the time between the emission and absorption of the photon along the $t_1$ axis, since by assumption you're not measuring anything in $t_2$. That time will be twice the travel time $\tau,ドル as projected onto $t_1$ axis, or 2ドル \sqrt{x^2 - {T_2}^2}$. (Observe that if $T_2 = x,ドル the photon will only reach it going along $t_2,ドル without any $t_1$ component. It will go there and back again in 0ドル$ $t_1$ time; not detectable. If larger, the separation is time-like, but in a time direction invisible to you; again it will seem dark.)

Since the speed of light is always measured as the same, you will predict $\tau^2 - {x^'}^2 = 0$; that the $x^'$ will appear to be closer to you.

OK, so far so curious.

If you receive a photon from a different timeline, and only measure the energy component parallel to your own, you will underestimate the photon's energy.

But if you measure a foreshortened distance, and can also measure how many wavelengths away the object is, you will predict the photon's wavelength to be less, and therefore its energy higher. If you know a priori what kind of photon you emitted, and measure the energy of the returning one, the energies won't match.

This little contradiction might offer a way to indirectly detect other timelines--one of the assumptions won't hold.

Some time back I proposed that the University institute a mandatory "Fill in the Gaps General Studies" course for undergrads, consisting of an Oxford-model(*) tutor/student (open doors) meeting in which the tutor figures out what the student doesn't know and assigns readings/watchings to – not quite fill in the gaps, but learn the outline for things they ought to know about.

They could shanghai time from researchers and postdocs to help fill up the tutor count (they take overhead from the grant money already, which gives a precedent). It would need a full-time team to review the tutors, especially if they try to vet the tutor's proposed general readings. If they imposed a set of readings I think I'd be apt to ignore it. It's a pass/fail course, of course.

Imagine that I came out of retirement for a while (deeply unlikely in the near term for health reasons) and participated in such a program.

Suppose I had a student for my proposed "General Studies" course in my office twice a week. A semester is 15 weeks, so that's 30 sessions. Yes, I doubled the number for this exercise, to make it easier.

I figure the students will come in several types:

1. literate and relatively well read and grumpy about having to take the class
2. literate and interested
3. relatively illiterate and likely grumpy.

The second type sounds like a delight, and the first type one might be able to work with. I figure "tell me about the last book you read" will show me in less than two minutes which of the three types I have.

Unfortunately I cannot trust the student in the third category to read assignments outside the office, so for some of them I would have to rely on in-office readings (30 minutes). The illiterate (who nominally can read and write but rarely exercise the skills) can maybe do 20-40 pages an hour, which translates to only 10-20 pages in my office. If I ask, as I probably should, that they write down quick notes as they go about words or ideas they don't understand, that slows them down more.

What can I assign the third group?

I figure since this is the USA, part of the West, informed by Christendom, other cultures are important but have to be lower priority.

For reference, in the Bible on the desk the gospel of Mark is 21 pages—at the high end for a slow reader in half an hour. A short dialog by Plato is about 26 pages (in one edition). (A life of Buddha can be found in 100, and of Muhammad in double that—I'd have to find children's versions of those, and they're lower priority anyway.)

• Chapters of Morte D'Arthur are sometimes short, but those don't give much of the flavor.
• Lots of poems are short. I think I'd want those read aloud: one or two for each era.
• Thumbnail history of Europe—not sure where to find one—needs to have a map. Maybe a video instead of reading?
• Thumbnail history of the USA—probably a video. Careful, some are invidious and some not very clear.
• For the math-deprived: There are some good videos out there that could be useful. Animations help illustrate some concepts. It's not the same as learning and doing, but if it gives a feel…
• Music: I probably stand in need of a little background reinforcement here too. My music theory is … um, deficient, and history has a lot of gaps.
• Science: I'd be tempted to lecture, but there are some good videos out there too.

You may notice a tendency to rely on videos with the semi-literate.

Figuring a list for the type 2 students – curious and ready to read – seems like more fun. What would you pick to make sure your student had a rudimentary background of the important stuff?

---

(*) I know, Oxford uses the American plan these days. Probably cheaper.

Use literary or allusive references to describe chess. Queen's Knight could be Lancelot (Mallory be the pawn?), Becket could be the King's Bishop (Eliot be the pawn?). Windsor could be the Queen's rook (Howard?). Ranks could be partridge, dove, hen -- or maybe that's not elegant enough.

I need to think about move descriptions; "Howard showed Mallory eternity?"

We learned, so long ago we've probably forgotten when, about the Taylor Series. \begin{equation} f(x) = f(x_0) + (1/2)(x-x_0) f^{'}(x_0) + (1/6)(x-x_0)^2 f^{''}(x_0) + ... \end{equation} If the derivatives are large, this might not converge very quickly (if at all). If the function $f$ is reasonably well behaved, and positive, we can try looking at products instead. Never mind the complex logarithms for now. \begin{eqnarray} f(x) = e^{log(f(x))} \\ g(x) \equiv log(f(x)) \\ g(x) = g(x_0) + (x-x_0)g^{'}(x_0) + (1/2)(x-x_0)^2 g^{''}(x_0) + \dots \\ f(x) = f(x_0) e^{(x-x_0)g^{'}(x_0)} e^{(1/2)(x-x_0)^2 g^{''}(x_0)} \dots \end{eqnarray} Will this converge any faster? For a distance use the difference between the approximation so far and the true value, divided by the true value. Pick a couple of simple examples: $f(x)= e^{x}$ and $f(x) = x^2$. The first one converges much faster with a "Taylor Product" \begin{eqnarray} f(x) = e^x \\ g(x) = x \\ g(x) = x_0 + (x-x_0) 1 + 0 + 0 + 0 \dots \\ f(x) = f(x_0) e^{(x-x_0)} \times 1 \times 1 \dots \end{eqnarray} The "distances" for the approximations are \begin{eqnarray} (f(x)-f(x_0))/f(x) = 1 - e^{x-x0} \\ (f(x)-f(x_0) e^{(x-x_0)})/f(x) = 0 \\ 0 \\ \dots \end{eqnarray} The second function example is, of course, much easier to approximate with a Taylor Series; you only need three terms for it to be exact. \begin{equation} f(x) = x_0^2 + (x-x_0)\times 2x_0 + (1/2) (x-x_0)^2 \times 2 = x^2 \end{equation} But never mind that; let's use the "Taylor Product" anyway. Here $g(x) = 2\log(x)$ If we let $x=x_0+1/2,ドル and let $x_0 = 1$ Suppose instead that $x=x_0 + 1/2$ but $x_0 = 1000$. It probably won't surprise you to see that it converges faster, using the given distance measure. \hline I don't know what this is actually called, and search engines turned up reams of irrelevancies. On a related note, MathJax in Blogger doesn't understand tabular mode.

If you haven't read Simak's Goblin Reservation, do. It's a fun sci-fi book set in Madison in an era with interplanetary teleportation and also time travel--which sets the scene. The Time Department has brought William Shakespeare forward to give a lecture. (There's a saber-tooth and a ghost and a Neanderthal as well, but read the book for yourself.)

Take that notion as the what-if. You have Shakespeare available for a couple of weeks. He's an astute businessman, and will be happy to write whatever play you commission, and might be happy to write fewer than five acts. (Producers and directors would trample each other for the chance to produce/direct/film it. I suspect Shakespeare would love the "Take-2" capability of film--and also the ability to work with women instead of boys for the female parts.)

Would you risk a comedy? He might need a crash course in "what's funny this year"; as AVI noted, humor doesn't always age well. You might feel overawed and leave it up to him, but just for laughs, try to think of something--maybe a fairy story?

He did some historical plays. Their Finest Hour might be too huge a canvas even for him, but WWII seems like an inevitable choice. Unless you wanted him to try Apollo? Or if you wanted to keep it to things he knew about already, King Harold? Odyssus might not be a good fit, but Achilles might.

Or a tragedy. I brought the topic up at the table, and Youngest Daughter suggested Yamamoto: facing Fate in the form of the death cult militarists and the Emperor, and Nemesis in the form of the US Army Air Force. I don't think Nixon's story would be dramatic enough for Shakespeare. Maybe something classical?

Just for fun, what would you suggest?

During a boring interlude, I wondered what sequences of continuous functions you might get with $f(x),ドル $f(f(x)),ドル $f(f(f(x))),ドル etc, and when would the sequence converge? Restrict ourselves to $x \in [0,1],ドル and require that the function also stay in that range. Figuring out the convergence is not a tough problem: if the sequence converges to some $f^{\infty}(x),ドル there's only one function that works.

The sequence might not converge at all: For $f(x) = 1-x$ the result jumps back and forth between two graphs. I don't see why you couldn't generate longer chains of repeating graphs.

Or it might converge, but not to something continuous. If $f(x) = x^2,ドル the graphs get flatter and flatter, and $f^{\infty}(x)$ would be 0ドル$ for $x \neq 1$ and 1ドル$ for $x = 1$

If you parameterize the function and play games with the parameters, I'd bet you get chaotic behavior in there somewhere.

I used pari/gp to create the base images and gimp to create the animations. I know, all the cool kids use python.

FWIW, I noticed that the high school math classes used the same graphics calculators over the time all our kids were in school (12 year age range), and marveled that there'd been no improvements over that time. Well, there have been, and the ubiquitous cell phones can easily download an app that will do their algebra factoring etc for them. The math classes don't dare use anything more recent, or the kids would use their shortcuts and not get the hang of algebra themselves.

There are some popular "documentaries" which mix interviews with paleontologists with "live action" dinosaurs interacting with people and things like automatic doors in the modern world. In one I was shown today, my "people wouldn't act like that" alarm rang. e.g. They'd devise improvised weapons to protect the kids while staging their retreat.

Which, of course, had me wondering: if you took a dozen young adults with pocketknives and tossed them back into the Cretaceous, how many generations would it take before they decided they'd reduced the risks enough (and could build sturdy enough pens) that they could start trying to domesticate some of the dinos? So far, at least, reptiles in general have been at best tamed a little, but only as pets and not for food or for assistance. But dinos weren't exactly reptiles--maybe it would work.

It might help if you started with a pack-animal type (wolf to dog)--footprints tell us some of the dinos moved or hunted in packs. Most modern reptiles don't seem to--though some move in family groups and some crocs seem to coordinate hunting.

Trying to picture the result gets biased by what happens when we domesticate mammals (rounder muzzle, floppy ears). There were already relatively small dinos about. Breed herbivores to not freak out around omnivores and be small enough for a human to conveniently slaughter. Breed some velociraptor to accept humans as alphas and help herd herbivores.

DALL-E-2 gave me this in response to "Herd of domesticated albertadromeus with big eyes and rounded muzzle kept in line with two domesticated velociraptors with giant cycads in the background." A bit more colorful than clear... I think there's some Dall-E fu I haven't mastered.

I was doing a literature search and my eyes started glazing over. I took a break by skimming through a journal I'd never heard of before: Elemente der Mathematik "publishes survey articles and short notes about important developments in the field of mathematics; stimulating shorter communications that tackle more specialized questions; and papers that report on the latest advances in mathematics and applications in other disciplines. The journal does not focus on basic research." In other words, suited for both more applied and more random stuff. The discussion section is in German, unfortunately.

"The irrationality measure of $\pi$ as seen through the eyes of $\cos(n)$. A student asked: "What's the limit of $\cos(n)^n$?" Since it's always less than 1 in absolute value, as n becomes large, the result should go to zero. Except it doesn't. In fact, it "oscillates", because larger and larger fractions come closer and closer to approximating $\pi$. They go on to discuss qualitative irrationality, but that's more for specialists.

I suppose one conclusion to draw from this is: pay attention to student questions. Sometimes there's something weird hiding that nobody noticed before.

Drat. The article is too recent to qualify for the open access. I could read the paper in the library, but not from home. In order to cut down on the number of points, I didn't draw anything with abs() less than .01--pretend there's a line across the middle. Done with python, and I'm trusting that their $\cos$ function handles large numbers well.

Consider the sequence

• 0
• 1
• 0.1
• 1.1
• 0.01
• 1.01
• 0.11
• 1.11
• 0.001
• ...etc...

The pattern is easy to see and extrapolate.

What's the most compact way you can describe this?

This won't surprise many of us, but these kids have a great uncle. Watch the video.

No, it doesn't take the fun out of things to test them systematically.

Aye aye, I eye aye-aye eye.

When I was in middle school I was trying to play Solitaire when Dad took the deck from me and showed me what the casinos did. You paid so many dollars for the deck, you got to play through just once, and they paid you a dollar for every exposed card. I looked at those numbers and figured I wanted no part of that game. I don't recall the initial cost anymore, and it doesn't matter for my life anyway.

I needed some practice with object-oriented aspects of python, so I set myself the task of writing that Klondike solitaire player I threatened. Some of what I found is about what you'd expect to see. I don't claim my algorithm is the best; one could be more strategic, especially as you learn what's in the deck. But it's an OK first approximation.

Yes, of course I wrote it so you can vary the number of ranks, number of piles, number of suits, and so on. Why bother otherwise?

I ran 10000 games with the different parameters. I kept 4 suits and 2 colors, but had 13 ranks and 7 piles (standard) and also 13 ranks with 6 piles, and 3 variants of 14 ranks (Tarot deck): 6, 7 and 8 piles. You expect that the more piles there are, the more cards are buried and the more will be left unplayed at the end of the game. And so it was.

With the usual game it looks like you win about 1 time in 20. (BTW, these are from random number generators, and are not exact.)

That's the easy part. What's a bit harder is quantifying the "frustration factor": how close you feel you got to winning. I figure the oftener you feel like you almost won, the more likely you are to start a new game--provided you actually do win from time to time. And the game can't be too easy. I'm not sure how to quantify that either. But from the fact that the standard game uses 7 piles instead of 6, I suspect that 6 makes the game feel too easy.

Notice the spike near 0 for the standard game. Most of the time you lose by a significant amount, but you come close oftener. I list the numbers in the table above, but I show the plots below.

For what it's worth, even a game with 2 suits and 3 ranks (and 2 piles) doesn't always wind up with every card face up. And the game above goes through the deck repeatedly.

Have you noticed that kibitzers of a solitaire game seem as engrossed as the player? As my officemate noted, suggestions from the crowd are unwelcome, so the kibitzer isn't really interacting. Perhaps a second-order kind of interaction; interaction by proxy?

There's a way to check. Create a computer solitaire game that plays itself while you watch. If it is popular, then just watching is good enough for involvement. If not, then you have to identify somehow with the player. I'm pretty sure some percentage of people who hear about it will just watch: as said officemate noted there are people who pay to watch other people play video games--there are channels devoted to this.

I thought about it for a few minutes and figured I could code and test the Klondike play logic in about a day (with variants for number of cards in a suit--use a Tarot deck, for example), but the graphics to go with it would be much more time-consuming. Maybe I could piggy-back on pysol and shorten the development time. Counting the downloads from sourceforge would be a proxy for measuring popularity. A web page with a java or javascript app would be better (count the connections and connection time), but I don't control my own server and I don't think it quite appropriate to host a game on the experiment's machines. One factor that's hard to control is publicity.

Too much work, too many uncertainties. Though I'd like to see people's reactions :-)

I won’t be bothering with the movie Noah. They didn’t succeed in interesting me enough to part with my time and money. Even worse, reports suggest that the movie is something of a turkey.

But it got me wondering what sorts of problems Noah would run into with other people...

---

”So, have you checked everything?”

The chubby inspector wiped his face and reseated his crow-feather hat. “Yes, I think so. We may want to go back and look at some things later. I need to settle down and draw up my report. You should join me to discuss it.”

”Will that help?”

”Sometimes details can be … um … explained, or negotiated.”

”I’ll be happy to help. What do you need?”

”I need a good smooth table in the shade, some clean water and clean fine rags, and a jug of good wine. And I fancy a silver cup to drink it out of—it helps me see better.”

”The rags will take a little while, and we can borrow a cup, but we’ll have the rest here in a few minutes. We’ll set up by the left ramp base.”

The inspector nodded and headed for the latrines.

Twenty minutes later they stood around a tall wooden table as the inspector spread out his vellum sheets and his brushes.

”Let’s start at the top,” he said. “You have no stanchions along the sides. You cannot tie up to a dock without them.”

”We don’t plan to use the docks. This is longer than any of the city docks—we couldn’t fit there.”

”The stone quarry dock is long enough. You built this to ferry stone for the new palace for King Caleb (may he live forever), but you didn’t think it through. You need to build stanchions in—how else can you tow it?” The inspector rapidly and accurately filled in the outlines of the upper decks as he spoke. There were little frowns drawn above the dotted images of stanchions.

”We got the license to start building this 85 years ago from King Jucline, and Ham has gotten that renewed by every king since. He asked for and got a waiver from King Clileb for upper deck furnishings.”

The inspector turned and spat. “Clileb, may his name be forgotten, was a usurper and a scoundrel, and any waiver he gave is invalid.”

”That’s what his inspector told us about King Cable before him. And Cable’s about the king before him. In fact, I think you were there that time. ... King Calib is certainly living longer than his predecessors.”

”Yes, he is (may he live forever), and since he is celebrating the first anniversary of his eternal reign with a new palace, your ark will do very well for transporting stone—with some deck modifications.” He paused. “Though I admit I don’t know how you plan to get it down to the river.”

”You no doubt reviewed the design plan before you came. This is designed to carry food, animals, and plants. The decks aren’t built to handle large stones. And we expect the river will rise to this level and float us off.”

The inspector looked up at him and pointed at the drawing. “Yes, I saw pictures of the statuary and how you meant to distribute the smaller work around the boat. You didn’t make the decks nearly strong enough—the beams need to be 3 times closer together. That’s not negotiable. And since decks 7 and 8 will be carrying stone slabs they need even more reinforcement. Plus new ramps. I don’t know how you got this approved with only one mid-deck loading port. It would take months to unload even with day and night slave crews.”

”Didn’t Ham explain that this is for live animals, not statues? They aren’t as heavy and they move under their own power. Look—would we need gaps and vertical shafts for ventilation if this was just stone? Or the dung shafts and the bucket chain to drain the sump?”

”No, Ham was waiting for an audience with the undersecretary for the Secretary for the Guardian of the Sublime Portal. And anyway, that sump is obviously for retrieving the lubricating slurry you need to drag the statuary.” He put his brush down and stepped back from the table to face Noah. “I’ve been an inspector for fifteen years and I know how boats work.”

He turned back to the table. “By the way, I really like the style of that cup.”