The Number Field

Yesterday, I described a simple construction that led to a very interesting result: a sequence of rectangles whose aspect ratios (i.e. width divided by height) approach \( \frac{\pi}{2} \). Today, I'll show why that happens. A disclaimer before proceeding: I want this explanation to be accessible to a wide audience. It won't be as rigorous, abstract, or terse as some would like, while it will be too technical for others. If you want additional explanation for anything, please ask in the comments.

To make the problem more precise, we need to introduce notation. We'll call \( w_n \) and \( h_n \) the width and height, respectively, after \( n \) steps. Just to get used to the language, what we know is that \( w_1 = h_1 = 1 \), and we want to show that \( \lim_{n \to \infty} \frac{w_n}{h_n} = \frac{\pi}{2} \). First, we need to find the right expressions for \( w_n \) and \( h_n \) when \( n \gt 1 \).

Adding a new rectangle to the side corresponds to forming \( w_{n+1} \), which is the width of the existing rectangle plus the width of the new addition. The height of the new addition is \( h_n \) and its area is \( 1 \), so its width must be \( \frac{1}{h_n} \), which shows that \( w_{n+1} = w_n + \frac{1}{h_n} \). Likewise for the height, but since we add to the side before we add to the top, there's a small difference: \( h_{n+1} = h_n + \frac{1}{w_{n+1}} \).

Let's see how it works: \[ \begin{aligned} w_2 = 1+1 = 2,\ & h_2 = 1 + \frac{1}{2} = \frac{3}{2} \\ w_3 = 2 + \frac{1}{3/2} = \frac{8}{3},\ & h_3 = \frac{3}{2} + \frac{1}{8/3} = \frac{15}{8} \\ w_4 = \frac{8}{3} + \frac{1}{15/8} = \frac{16}{5},\ & h_4 = \frac{15}{8} + \frac{1}{16/5} = \frac{35}{16} \end{aligned} \] and so forth.

Sometimes when it doesn't look like there's much we can do, we might as well try whatever presents itself. In this case, it looks like we could at least factor all of the numerators and denominators: \[ \begin{aligned} w_2 = 2,\ & h_2 = \frac{3}{2} \\ w_3 = \frac{2\cdot 4}{1\cdot 3},\ & h_3 = \frac{1\cdot 3\cdot 5}{2\cdot 4} \\ w_4 \frac{2\cdot 4 \cdot 2}{1 \cdot 5},\ & h_4 = \frac{1 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 2} \end{aligned} \] I've thrown in a few extra \( 1 \)s to make it look pretty, because that's always allowed, but there's something not quite right with the last row. That's because we wrote everything in lowest terms, which means that some \( 3 \)s cancelled out. If we put them back, it looks much better: \[ w_4 = \frac{2\cdot 4 \cdot (2\cdot3)}{1 \cdot 3\cdot 5} = \frac{2\cdot 4 \cdot 6}{1 \cdot 3\cdot 5},\ h_4 = \frac{1 \cdot 3\cdot 5 \cdot 7}{2 \cdot 4 \cdot (2\cdot3)} = \frac{1 \cdot 3\cdot 5 \cdot 7}{2 \cdot 4 \cdot 6}. \]

Looking at that, and taking care with the variable \( n \), that's enough to conjecture the general form for every \( w_n \) and \( h_n \), specifically: \[ \begin{aligned} w_n & = \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 5 \cdots (2(n-1)-1)} \\ h_n & = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2(n-1))} \end{aligned} \]

If that's enough evidence for you, just skip the next few paragraphs, where we prove that these formulas work. But you shouldn't, because this is the best part--we're about to use induction. Specifically, we'll assume that \( w_n \) and \( h_n \) both follow the pattern (which is a good assumption, at least when \( n \leq 4 \) ), and show that \( w_{n+1} \) and \( h_{n+1} \) also follow the pattern. We'll start with \( w_{n+1} \): \[ \begin{aligned} w_{n+1} & = w_n + \frac{1}{h_n} \\ w_{n+1} & = \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 5 \cdots (2n-3)} + \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 5 \cdots (2n-1)}. \end{aligned} \]

Most mathematicians would not even attempt to do the next step in public, because it's just arithmetic, so you look dumb if you do it at all and even dumber if you do it wrong. It's just adding fractions with different denominators! But I'll press forward. First, we factor out like terms: \[ \begin{aligned} w_{n+1} & = \left( \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 5 \cdots (2n-3)} \right)\left( 1 + \frac{1}{(2n-1)}\right) \\ w_{n+1} & = \left( \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 5 \cdots (2n-3)} \right)\left( \frac{2n-1}{2n-1} + \frac{1}{2n-1}\right) \\ w_{n+1} & = \left( \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))\cdot(2n)}{1 \cdot 3 \cdot 5 \cdots (2n-3)\cdot(2n-1)} \right) \end{aligned} \] which is exactly what we need, although we do have to remember that \( n = n +1 -1 \) to make everything work out.

Fortunately, there's a way to leverage this and simplify the calculation of \( h_{n+1} \) by going back to the idea of the ever-growing rectangle. When we get to \( h_{n+1} \), we've added \( 2n \) rectangles, each with area \( 1 \), in addition to the starting unit square, so the area of the big rectangle, which is \( w_{n+1} \cdot h_{n+1} \), must be \( 2n +1 \). If we divide both sides by \( w_{n+1} \), which we just calculated, we find that the equation given for \( h_n \) is correct!

Now it's time for the big finish. With formulas in place for \( w_n \) and \( h_n \), it's time to put them together in the ratio \[ \frac{w_n}{h_n} = \frac{ \frac{2 \cdot 4 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 5 \cdots (2(n-1)-1)} }{ \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2(n-1))} } \]

When we simplify the fraction and line everything up from smallest to largest, we get: \[ \frac{w_n}{h_n} = \frac{2 \cdot 2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdots (2(n-1))}{1 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \cdots (2(n-1)-1)}, \] which, in the limit (as \( n \) gets larger and larger), is known as the Wallis product . There are two ways of showing that it converges to \( \frac{\pi}{2} \)--one is slick and short, but the other requires only integral calculus. I originally planned to derive at least one of them here, but the Wikipedia article does a good enough job, and this post has gone on long enough.

So there you go. The explanation is always more work and less exciting than the mystery, isn't it? I guess from that point of view, Lost got it right. You weren't expecting that, were you? That's right, I went there:

P.S. Hey redditors, thanks for your interest ! I have plans to start a dedicated math blog with lots of content like this at some point in the future, but the pieces for that are not yet in place, and for the time being I post mathematical musings here on my personal blog only rarely. You're welcome to visit whenever you want, but you'll probably only find stories about my cute kids. I'll be sure to let you know when I get the math blog going. If you need help with anything math-related, please leave a comment on this post and I'll get back to you as soon as I can.

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Start with a square. To keep things simple, I'll say its area is exactly 1 square unit. Since it's a square, that means each side is 1 unit long.

I'll describe a procedure for growing this square into a very large rectangle. The first step is to place a second square (also with area 1) adjacent to the first, so that together they form a rectangle with width 2, height 1, and area 2.

That was easy enough, but what comes next is more complicated, because we're going to start adding rectangles instead of squares. However, at each step the area of the new addition will always be equal to 1, but one of its sides will be long enough to match the existing rectangle. This time, we place a rectangle on top of the two squares, so it must have width 2 and height \( \frac{1}{2}\) to meet the side length and area requirements.

Now we make an addition to the side, this time matching the existing height of \( 1 \frac{1}{2}\), so the addition must have a width of \( \frac{2}{3}\).

Repeat this process of adding rectangles, alternating between top and side, ad infinitum . The total area grows and grows, but each added rectangle is longer and skinnier. If the viewing window keeps zooming out so that the big rectangle always has the same width, we see something interesting.

After 1,000 steps, it looks like this:

After 10,000 steps, it looks like this:

After 1,000,000 steps, it looks like this:

The fact is that these all look pretty much the same--a mathematician might say that the ratio of width to height approaches a limit.

What is that limit? Well, it's approximately 1.57001 after 1,000 steps, but grows to 1.57072 at 10,000 steps and 1.570795 at 1,000,000 steps, which probably pretty close to its true value. That number may look unremarkable until we multiply it by 2 to get 3.14159.

Happy Pi Day!

(The explanation of this phenomenon requires Calculus. Is anyone interested?)

Update : Here's an explanation .

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Jenny, Elena, and Roman are spending a week in Seattle with Carrie, Matt, and their kids. I joined them for a long weekend, which we filled with fun activities and lots of time to play.

A visit to the Pacific Science Center was the centerpiece of our weekend. When we first arrived, walking through the hall of dinosaurs, Elena seemed overwhelmed by all the things to look at--she bounced from one place to another, not ever stopping long enough to take anything in. She calmed down eventually as we worked our way through the exhibits. Roman took everything in stride, not minding too much as we handed him back and forth between the four adults, depending on who was nearby and who needed to run off to chase the others.

We touched sea anemones, explored water machines, looked at snakes and lizards, and made funny shapes with our bodies against the shadow wall. All that happened before we made our way to the main attraction: the butterfly house. Jenny didn't enjoy running the gauntlet of yucky bugs before we got to the butterfly house's controlled access entrance. When we got inside, Elena wanted a butterfly to land on her but it wasn't destined to happen. I'm glad that we managed to keep her from taking matters into her own hands, because we were under strict instructions not to touch the butterflies. After we exited the butterfly house, we caught the second half of a demonstration about air. The best part was the big finish: a whirlwind of fire.

After lunch we all had a good time playing in the funhouse mirrors and learned more about the human body. For instance, I learned about my center of gravity by falling right on my nose while trying to complete one of the activities. I'm not sure if that means I have a high center of gravity or a low one, but at least everyone got a laugh out of me.

On our way home we stopped to see one more sight: the Fremont troll . The children were almost asleep, so they stayed in the car while Matt and I snapped a few pictures. It looks pretty odd, so I don't think everyone else missed out too much by not getting a closer look.

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Roman is everywhere, or at least that's how it seems. Yesterday when I was on the phone with Jenny, she had to interrupt our conversation to ask "Elena, did you drag him over here? No? Well, he got over here awfully quickly." Every day his peak speed increases, and we have to be cognizant of that changing capability.

There's another factor in Roman's seeming omnipresence: like a cat under foot, he always knows where to be. He pays close attention to our patterns of behavior, to the things we like, and goes after those things. For example, he loves to play with Elena's toys, especially the ones that she likes the most. But he loves electronics even more. He makes a beeline for my laptop whenever he sees it lying on the floor. It's always a race to see if I can make it over there first to move it out of his reach, and someday soon he'll probably win a round. When I hold him in one arm and am surfing the web on my phone with my free hand, he waits until I drop my guard and my hand strays just a little bit closer to him, then executes a precision lunge and grab for it. Maybe he wants to write an email to his new cousin about everything he's learned in his first six months?

Roman still crawls army-style, but he's already showing signs that he might be an early walker. He wants to crawl the normal way so he gets up on his hands and knees, but so far all he can do is rock back and forth--it will still be a while before he figures out how to lift and move one hand or leg at a time. Until he reaches that milestone, there are other things he can do instead, including the "downward dog" yoga position with impressive form. That's what causes us to think he could be on his feet soon--the good balance and the desire to get his feet under him.

There's one very nice thing about all of Roman's movement and exploration--he plays well on his own for long stretches of time. When one toy no longer interests him, he just scoots over to the next thing that catches his eye. Perhaps most interesting is what happens when he starts to fuss, as he occasionally does. One of us will pick him up and soothe him for a few minutes, but often he'll start squirming after being held for just a few minutes because he's ready to start playing on his own again. It's off to the races just as soon as his hands and feet touch the floor.

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Economics is the study of the allocation of scarce resources. Ever since Freakonomics , it's become popular to apply it to non-monetary aspects of life. I first heard about Spousonomics when Jenny recommended that I listen to the Thanksgiving episode of Marketplace Money . The problem with books like this one is that the treatment of the problems they address is necessarily shallow--they generally cover only topics that might be found in an entry-level college course. Furthermore, economics is just one small discipline in the universe of academic study, and certainly not the only one that can be applied to real life problems, which are so much more complicated than any formal (or semi-formal) study can completely explain. It's the difference between a map and real terrain--economics gives us one map, but maybe it's a road map when a topographic map would be more appropriate, and in both cases it would be foolish to mistake looking at the map for exploring the land in person.

For instance, the authors of Spousonomics suggest that the principle of competitive advantage be applied to household chores--if spouse A is better at washing dishes and spouse B is better at something else--then assuming that the time commitment is similar, A should always was the dishes and B should always do the other thing. But this exposes one of the major flaws of economics ¹ , because it encourages trading of long term systemic benefits, like flexibility, for short term quantifiable gains, like reduced time for chore completion. Returning to the example, the downside is that when I don't wash the dishes enough, my dish washing skills atrophy to the extent that when I do need to wash the dishes, as I inevitably will, I do a much worse job than I would have if I had at least helped out once in a while. It's possible that the total loss may outweigh the incremental gains realized by focusing on competitive advantge. There are almost certainly more advantages to domestic cross-training than those described in simple analysis, such as potential skill transfer between disciplines, but those may be unknown unknowns and I think I've made my point.

A story gives us another conundrum that economics can't solve. Roman takes naps in a swing, but at night he sleeps in his car seat--that is, when he does sleep. This is not an option that will continue to work forever, as he continues to grow. He eventually will sleep in a crib-like area; we would prefer that he learn to sleep in the pack-n-play that we have set up for him. Jenny mercifully decided that last week was a good time to begin the transition, because I would be away three nights in a row. Things didn't go well. It was like my dad's old joke: "I slept like a baby last night; every twenty minutes I woke up crying." I'm afraid that I didn't help much when I returned home--the first night, I stayed up for the first shift and tried to comfort him, but Jenny naturally wakes up when Roman cries, so it didn't help her sleep much, especially since I'm not as good as she is at comforting him ² . When faced with a similar but apparently less intense situation, one of the spousonomocists wrote about stated versus revealed preferences , but didn't even try to solve the core issue of lack of sleep, which is the truly scarce resource in this situation.

As a mathematician, I always try to state a problem abstractly ³ . I am drawn to a half-remembered musing on philosophy by Douglas Hofstadter, arguably one of the great thinkers of our time. He wrote columns for Scientific American that were eventually collected into a book which I read as a teenager. In many of the articles he posed thought questions and invited responses from readers. Here's one of them: suppose you are in a decision-making position for a whole culture or society ⁴ . Your wisest and most trusted advisors come to you one day with an important decision. If you choose the first course of action, the entire world will be better off for more than a hundred years--GDP will be greater, lifespans will increase, happiness surveys will be off the charts--but at the end of that time (long after almost everyone currently alive is dead), an inevitable catastrophe will ensue, setting the entire society back more than 500 years. If you choose the second course of action, none of that happens: none of the boost, but also no catastrophe. What do you choose? At this time scale and magnitude of consequences, traditional economics is worthless ⁵ . I don't think that Hofstadter ever came down definitively on one side or the other, I but remember a follow-up column that explored both sides further.

With Roman, that situation is reversed and the time frame is shortened. To what extent are we willing to go through pain now in order to get to the long term steady state solution? Although it may appear that the situation is not comparable in scope to the previous example, I argue that when dealing with needs at the base of Maslow's pyramid (in this case, sleep), even short period of deprivation can lead to drastic effects. I don't know of any ten year olds that still sleep in car seats, but we run the risk that every day of delay makes it harder for him to make the change. While I sit here philosophizing, Jenny soldiers on, sacrificing her sleep and health.

This is the unacknowledged heart of the news stories that pop up every few months about whether parents are less happy than non-parents. It's a good question to ponder instead of counting sheep as I try to calm myself down on a sleepless night, but it's impossible to actually give an answer . There are buildings full of very smart people that can't even accurately forecast next year's tax revenues, much less long-term economic trends. Surveys of parents and non-parents may purport to have statistical significance, but they have no more actual significance than reductivist economics arguments about real life. It's like calculating the length of a coastline from a map's depiction ⁶ --you can try to do it, but it's subject to an uncertainty principle as real as Heisenberg's, rendering results meaningless regardless of their claimed significance.

1 This is presumably the kind of thinking that leads to the banana monoculture and banking crises.

² Another instance of competitive advantage gone awry.

³ I also attempt to reduce problems to those that have been solved, or at least stated, previously.

⁴ Let's pretend for now that this is an episode of Star Trek , where leadership structure is always simple and well-defined, and often immutable.

⁵ Regardless of your position on climate change, this is the fundamental disconnect between the two warring camps. Hofstadter was writing in the early 1980s, so he was probably motivated by the specter of the atom, both split and fused.

⁶ Hint: it's fractal .

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For my entire adult life, my pants have not had a main point of failure--they just wear out slowly until Jenny forbids me from wearing them. In the past several months, I've lost two or three pairs of pants to threadbare, ripped knees. At least one pair was only about a year old, and seemed otherwise to be in good condition.

There must be something going on, but there are a couple of distinct factors, and I'm not sure whether one thing is the primary culprit, or if both are significant contributors. The factors' names, of course, are Elena and Roman. In her fourth year, Elena is more active than ever, and although she's never heard of Hop On Pop , she figured out on her own how fun it is to crawl all over me. I spend a lot of bent-knee and on-the-ground time playing with her. Now that he's moving around, Roman also causes me to bend, kneel, and crawl a lot, possibly enough to induce pant knee failure.

When I was a boy, I blew out the knees on many pairs of pants. Back then, my mother would iron in a patch and I could keep wearing them. Sadly, I don't think that would be acceptable now with my workplace attire. Maybe I should start preemptively putting patches in to reinforce the knees. At least the ripped knees help me feel young again.

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One evening Elena took an interest in a book that she had seen lying around many times, the Pocket Guide to Familiar Birds of Lakes and Rivers , and asked me to read it with her. It wasn't a book with a story, so we just looked at the pictures and I told her all the birds' names.

At the beginning of the book, just after the loons and grebes, there was a picture of two white pelicans. For whatever reason, the pelicans both fascinated and distressed Elena. "Daddy, what's that bird called?...Daddy, I don't like that 'elican...it's not a friendly 'elican." But after every few pages, she wanted to go back and look at the pelicans again. No other birds got close to the same reaction from her.

A few days later we took advantage of the unseasonably warm weather to take a walk by the river after nap time. At first it was going to be just Elena and me, but Jenny and Roman woke up in time to join us. There are lots of fun things to do by the river, and we did most of them: We said hello to the birds and the other people on the trail, picked up sticks and rocks to throw in the water , and took lots of pictures as we meandered down the path.

Then Jenny noticed something in the distance--one bird was much bigger than the others. Could it be? Did we dare to hope? I looked at it through the camera with my lens on maximum zoom, but still couldn't be sure, so we went closer, as close as we could. It was a pelican! He spent a few minutes hanging out with the seagulls before slipping into the water and paddling downstream towards his home.

On the walk back to the car, Elena talked about how much she liked pelicans now, because the one we saw was so friendly and said hi to her.

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Elena had been looking forward to her birthday for a long time, at least since Jenny's birthday last October. For the past few weeks, she's been counting down the days. I was sad to miss her waking up on Friday morning, so I got home as early as I could in the afternoon.

I brought gifts home, things that we definitely knew she wanted: a helium balloon and a new doll. She always likes to get balloons, and she had asked Jenny specifically for a new doll. We gave her the balloon as soon as she woke up from her nap, and Grandpa gave her two more balloons when he got home from work. We saved the other presents for after dinner.

We played for a little while, then video chatted with my parents, but had to get to dinner quickly, because according to Grandpa and Grandpa, the Red Robin gets busy early. It was especially busy that evening, filled with a surprisingly large number of fathers and daughters dressed up in their Sunday best. Curiosity finally got the better of me, so I asked one of the fathers what was going on, and found out that it was the night of the Fifteenth Annual Father Daughter ball. There were a lot of birthdays, so we kept seeing and hearing the waitstaff singing and clapping from different corners of the restaurant as they delivered birthday sundaes. Elena liked the festivity when it was far away, and wanted them to come sing to her, but they came up from behind when she wasn't quite ready, which scared her a little bit. The ice cream helped her get over any lingering nervousness once the singing was over.

We waited for the cousins to meet us back at home before opening presents. Among the things she received were: a new dress, money to buy some new books, a soccer ball, a Doodle Penguin, and the aforementioned doll. She liked all of them, but Jenny and I were happy to hear her say "it's just what I wanted" when she first saw the doll. After presents, we had another round of dessert, this time an ice cream cake care of Grandma and Grandpa. And of course we had lots of energetic playtime before bedtime. It was a happy day for all of us!

Although last Friday was Elena's big day, I couldn't help thinking about how far our whole family has come since that stormy night when she was born. It's been a wild ride for all of us, but she's been a great companion through all of our adventures and projects. We're happy to have her in our family.

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I was playing with Elena and Roman on the living room floor. It would probably be more accurate to say that I was roughhousing with Elena while Roman rolled around on a blanket nearby. Elena likes to wrestle--which means that she pushes me down, then I tickle her for a while before we reset and do it again. I try to encourage her creativity and problem solving skills by not always going down on the first push, so she has to try different approaches.

During one short break in the action, I noticed that Roman was a lot closer than he had been before. He had been working on rolling and squirming to move himself closer to things that he wanted. This time it was different: he had moved farther, faster, and more directly than would be possible by rolling. I halted Elena's next advance so we could observe Roman. He soon showed us an army crawl, pushing up with his right hand and muscling forward with his left forearm. A video is worth more than the hundred and eighty four words I've written so far:

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Now that we've been here for a few weeks, we're settling into a schedule. I stay close to work Mondays and Tuesdays, but make a quick trip to see Jenny and the kids on Wednesday evening. Thursday mornings I wake up extra early so I can make it to work on time, then head back to them for the weekend as soon as my work week is over on Friday afternoons. It's not ideal, but at least I am with my family more evenings than not, and when I'm with them I'm more aware of my desire to be present and involved, not distracted.

Jenny, Elena, and Roman are adjusting well to their new living circumstances, although the transition has not been completely smooth. Elena, in particular, took some time getting used to living with her grandparents. Now she loves to play with them, and Grandma has clearly spent lots of time and effort acquiring just the kinds of toys and items that grandchildren love. Every time I go back there, it seems like she's produced some fun new thing from a closet or storage area. It's amazing, because I've looked in those storage area and know that there are still plenty of surprises in store.

The opportunity to spend time with Jenny's family and friends was one of the main things we were looking forward to about this trip. Thus far, it's certainly living up to our expectations. Most importantly, Jenny, Elena, and Roman get to spend time with Grammies (Jenny's grandmother) at least twice a week. Since Grammies lives with Jenny's brother Paul, that also means that we get to see him and his family on a regular basis; Elena always loves getting together with cousins. As we get in touch with other people, mostly Jenny's friends from when she was younger, our social calendar is filling up with other plans as well. Many of them also have young children, which means more friends for Elena and Roman!

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