Hey guys, it's Nate, AKA The Foot Accountant. Welcome back to the channel. It's Monday, market outlook time. You know what that means. We're gonna take a look at this week in FIFA 21 Ultimate Team and talk about what market movements we could see, what type of content, and how we might be able to make some coins this week in FIFA, in this Road to the Final Team number one promo. So yes, right now it's Road to the Final Team one, as we know, and there is a few things that I wanna point out. We're looking at the market tonight on early Monday morning, late Sunday night, squad battle rewards timeframe. I don't know how many people are playing squad battle rewards anymore or are playing squad battles and getting rewards, but the market still is impacted every single week by the rewards that come out. So there still must be people playing. So it is late Sunday night. And again, we're gonna take a look at this market right now. Rewards were a couple hours ago. You again, have seen a bit of supplies come onto the market. Some of the gold cards have dropped. And the first thing that I'm gonna mention today is doing a club stock is gonna be a really good idea, in my opinion, in the near future if you have not done one already. This is gonna set you up for the next couple of weeks. Honestly, we have a lot of SBCs on the horizon with Player of the Months, more Road to the Final content, and of course Icon SBCs on the horizon. So we'll be talking about that Icon SBC thing a little bit later. But 84s look to be really cheap right now. A lot of these cards are usually around 2.6, 2.7, and 3K. They're down in the low twos. Grimaldo 2.5K is a Spanish left back. What I would do is just go through, pick some of these up on bid or on snipe. Even today, like if you're looking to do something with your coins, you don't mind setting aside a couple hundred K possibly, or maybe even a hundred thousand coins, stacking up on some 83s, good leagues, good nations. 84 is 85s even. 86 is maybe not quite yet, but 82s to 85s would be kind of the range right now that I would focus on doing a club stock, because I think these players are low and getting them in the club, there's gonna be more SBCs that are coming out in the future that are gonna make these cards go up. So just getting yourself ready for those SBCs could be big. Now if you, of course, if you're investing for a player of the month, like a Harry Kane or a Hyunmin Sun that could possibly come out this next Friday, then maybe you would wanna dip in at some of the higher ratings, right? Maybe some 86s, maybe some 87s, 88s. But just be careful with those, because of course, player of the month so far this year, none of them have really been cheap. I don't think we've had like a super good value player of the month SBC yet. Even like the league one player of the month SBC, the Nianne, I think is his name that's out right now. Calvert Luhn was overpriced, you know, Kramerich was okay. Nianne was kind of overpriced for the player that he is, the leagues and the links. And honestly, like Oyarsabal might be the best value in terms of just like statistics on a player of the month card that we've gotten so far. Ansu Fadi wasn't bad, but we haven't really had a good value player of the month card yet. I think that just makes sense because EA is, I don't know, I don't know if EA wants us to be able to complete a really good SBC, a really good card SBC this early on in the game and they're not really requiring really expensive stuff. So we haven't had like any huge names win player of the month yet. So Hyunmin Sun, if he does win, of course player of the month Mbappe is on the horizon. Some people are asking about Mbappe and Mbappe can't come out because the vote yet has not been released and there's still 13 days left on this Neane SBC. The Calvert Loan SBC is going away in four hours or something like that. And I think we're gonna be getting Premier League player of the month this Friday. So this upcoming Friday should be Prem player of the month. That is something that we will see. The Bundesliga player of the month has seven days remaining on the SBC. The voting for that should be open. If you Google EA Sports player of the month, you can go on there and vote for the Premier League one right now and probably Bundesliga voting will open up very, very soon. So that's kind of some player of the month. Maybe do a club stock this week. That's probably something that would be a smart thing to do. I think we're gonna get more SBCs as a part of this promo. And again, taking a look at the webpage about this promo, they continue to say here and Foot Economist even tweeted this out himself, the Foot Economist who works at EA. He highlighted this sentence right here on Friday when the promo was released and said, stay tuned for additional UEFA Champions League and Europa League themed content throughout the campaign. And as of right now, we don't have any Europa League cards in the game. It's only Champions League cards. So I'm gonna have a feeling that some of those cards might come through objectives, might come through squad building challenges. And then even, I don't even know if they're gonna put more Europa League cards into packs or what they're gonna do. But that's kind of like the biggest question mark for this week even as we head on to the week, like what is this gonna look like in terms of just, are we gonna be getting UEL cards? Are they gonna put more cards into packs? Are they gonna try to crash the market and stuff like that? So of course that was like the, that's the biggest question mark for this week in my opinion is what is EA gonna drop content wise related to Europa League stuff? You're related to more Champions League contents because we do have two SBCs out right now and one objective player. But I feel like we're gonna get at least two or three more because this is a big promo for EA, and so far the player selection, as you guys know and we've talked about, is kind of lackluster. But the SBC content, the SBC player selection has been decent so far. So I'll be curious to see, especially if we get the Champions League SBC card being used, not a road to the final, but the Champions League like SBC card. It's like a green with purple on the bottom. So that is something that could be used even this week or we could have another Forsberg type SBC for his UCL player moments version for a UCL player that perform in a game in the past. So that could be something we see this week as well. Of course, this is the SBC that people are talking about right now, the 81 plus UCL upgrade because this is what was released yesterday on Sunday and it did impact the market a little bit. In my opinion, what you did see is right away when this SBC dropped, you saw players like Son, like Joe Gomez, like Veran, they were panic sold. Now panic sold a little bit, but what happened was, Footbin, something is up with Footbin right now. I don't know if you guys are having the same issue, but Footbin is having issues with updating player prices because today they were saying that Hyunmin Son was dropping down like 20,000 coins. Footbin had Veran listed at 261,000 coins. And he technically was getting listed there because people were seeing the Footbin price and listing them up there. He was still actually like 290, 300K. So just be careful with Footbin right now. If you're gonna list a card, don't just go look at Footbin and like, okay, that's the price. Actually check the card price on the game because that extra checking might save you a lot of coins. In this case, it saved people a lot of coins with Veran today and Bape had the same issue. Footbin yesterday was listing him at one point. 27 and he was selling at one five. That's crazy. So Footbin, please get it together. I don't know what the issue is with that, but there was panic selling right away today when the 81 plus UCL upgrade came out. It costs 8K to do the pack. Is it worth it? It's really not. Maybe do like one or two packs just to see if you get anything sick from it. The player pool is kind of small, right? Because there's only the UCL cards in there. Actually on stream today, we packed somebody in Neymar from these packs. So I can't just sit here and tell you to not do any of them because we packed a Neymar yesterday, but they're really not a good value because you have to get a walkout every time to make your money back from the SBC because 85s are going for like 5K and 86s are 11K. So it's got to be either a OP, lower rated card like it made a Furlan Mendi, an Usman, an Embele. You need a card like that to be able to get your money back from this SBC or you have to get a 86 plus every time if it's a fodder type player. So that's just something to keep in mind. I would not do many of these. Maybe do a couple, test your luck, but it's really not worth it. So this is notable though. This is the first like repeatable upgrade SBC that we have had this year. I know we have the gold upgrades that are always out, but this is different because it's an 81 plus. This is the first real upgrade SBC that we have had this year. So again, speaking about the market and what is kind of going on right now, it looks like some of the squad battle supply is rolling in. Some of these prices still look kind of decent to me at the moment. Now I have bought some cards. I have some cards on my transfer list and stuff. Some stuff looks like it's rebounded slightly already after rewards, but one thing that I am noticing is I'm scrolling through some of these past team of the week cards and some of these are still pretty low. Zaha 220 is pretty low for me. I don't have to get involved with that one. Scrolling down through even some of these older rule breaker cards. These guys are pretty low still. Gerente is really low at five, three, 28, excuse me. Andres 69K is really low. He was 80K. Maybe that's a foot bin undercut. That could possibly be the case there. Yeah, again, another foot bin undercut. It's telling you Andres is 69K, but he's actually 78, 79. So just be careful with these foot bin prices, man. They're all over the place. But I do think you will see a little bit of a market rise into the week this week, Monday, Tuesday, Wednesday, kind of like last week. Now of course it could very easily stop and be halted depending on the content that EA puts out. They could easily put out some sort of SBC or something to make that stuff drop later this week, whether it's a repeatable SBC, pack supply SBC. Alan St. Maxman is very cheap right now. That could be a flip. Courtois, not even that bad. Let me check once to watch this. What is Partet, 185, 210 for Bale? Now again, it's international break this weekend. By the way, if you have this, the Etch wants to watch, please sell it because he's not getting an inform. He had an informant last week's team of the week and he had one assist. His card is up 20,000 coins because of an assist. So just be careful with that one. But some of these informs are low and I do think they will rebound into Monday, Tuesday, and timeframe. They are out of packs. If they're still low when you're watching this, take a look at the graphs. Now be careful, right? This Bruno Fernandez, I don't think I would really invest in him because he is a front runner for team of the week this week because of his performance against Everton over the weekend. That's why this card is down like 70, 80,000 coins. Now if for some reason Bruno does not get in, this card will shoot right back up over 350 on the PlayStation and probably like to 350 plus, 370 maybe on the Xbox as well. But I think he is gonna get in a team of the week because EA is not gonna miss out on a chance to get a little GGMU action in the team of the week. So just be careful with that. If you do have that card, that'll be something to keep an eye on. And one thing that I actually shouted on stream, when was it? Friday night. You can't make this investment anymore because the card's already risen. But we were looking at O'Para. O'Para is 46,000 coins. And this is just one teaching point that I wanna talk about in this video because we're here right now.  And on Friday and Saturday, this guy was about 37,000 coins after going all the way up to 75K when the objective came out. Of course, that happened midweek. On the weekend, people sell their objective cards and they sell their cards and they go play week in league, right? So on Friday and Saturday, this guy went all the way down to 35, 36,000 coins. I picked one up, I should have bought more.  So this card has risen all the way back up to 46,000, 50,000 coins almost is what I'm hoping to get for O'Para. So that's one thing that we can kind of take note of for the future is that people that are doing objectives during the week, those cards that are expensive that rise and out of packs in form or out of packs card, if it does drop over the weekend because people are playing week in league, we can kind of watch that card for a rise back up when people go to complete that objective again in the midweek. So that's just something to think about that was a money maker today. But again, watch through some of these out of packs in forms. D Maria, 351, right? This guy was almost 400K earlier this weekend on Sunday even. Yeah, he had 379,000 coins, so 380K. And then on Saturday, he was 390 on both consoles. So the fact this guy's down in the 350s, that's kind of interesting to me. So just, I do think you will see a little bit of a market rise into the early part of this week. Now again, speaking of content, what do we have laid out for us this week? It's really another question mark, right? We're not guaranteed. There's no UEFA Marquis matchups this week. The thing that's gonna carry the content in FIFA Ultimate Team this week is going to be the player SBCs and whatever other road to the final SBCs that they put out. Now again, according to the website, it looks like that the cards that are in packs right now are road to the final team one, road to the final team one. Players are in packs right now. And there are in packs for again, four days and 12 hours. So that is Friday, 6 p.m. UK. Rewards this week is gonna be interesting because these cards are not very packable, if I'm being completely honest. Their prices are dropping. And last year, we looked at buying these cards on Sunday night into Monday. I think I would hold off on these cards still because they're still dropping, they're still getting packed and they don't have any Champions League games for another three weeks, two weeks actually. The 24th of November is the next time we have Champions League. So I would hold off on buying these cards in my opinion. Some of the really meta ones might rise up, but I think Thursday rewards might be kind of the timeframe to buy these cards. I think you might wanna wait just because there's not as much hype with these as there were last year and we don't have games on the horizon right away. So I would wait it out if you wanna buy any of these road to the final cards, let them drop Griezmann's down a lot from where he was, Klosterman is down, Salah's down like 400, 300,000 coins from where he was at one point. So just let these guys keep dropping in my opinion. And especially if set two comes out and it's more hype, that's usually not what happens, but I guess it could be the case this week because set one is definitely like not that hype. So that is what it is. Again, this week for content, it's kind of just a question mark, but there is gonna be ways to trade, of course, trading with the new cards, they're gonna fluctuate up and down on a daily basis. The market's probably, if we don't get any content, it's gonna chill and swing upwards from Monday into Wednesday before rewards. I do expect more content than last week though, so I think we might have some interesting movements maybe on Tuesday or Wednesday of this week just with whatever content EA wants to drop. So we'll definitely keep an eye on that, but should be a good week to FIFA boys. Road to the final is a good promo, really good content that's come out so far in my opinion and hopefully continues throughout the rest of this week. If you enjoyed this video, smash your thumbs up on it, comment down below if you have any questions and subscribe to the channel for new. It's been Nathan with Account and catch you guys later. Peace, out.


Hey, for Apple. Apple at a computer views and our videos here at DCTV. B is for Buntful Broadcast. This is Colin. C. E is for Dominos. E is for everyone. M is for friends who broadcast together. Hi, I'm Jack and G is for Grape Prizes. H is for Histiocytosis. I is for I Love MPS. J is for Junkers who work broadcast. K is for Disca. M is for Long Night, which has had my all. M is for the money we raise for charity. N is for Nonsense, which Flan takes none off. Oh, it's for Artichus, which takes your drug. P is for pipe. I have one right here. K is for Quilt. R is for Rudolph. S is for Studio. T is for Tomas, who's right there. That's me! U is for Unbelievable Fun! V is for Very Disappointed, because today I asked for a 1,050 cup of cappuccino, and I got a cup of milk. That won't keep me away for the 24 hour broadcast, now will it? W is for Waterfountain, which we finally have in the Henry Gratton. X is for X-rated content, especially Babe Station. Y is for Yonkete. Give me 5 minutes. How many words? Z is for Zomp.


I'm often asked what my opinion about a diet or disease is. Who cares what my or anyone else's opinion is? All we should care about is what the science says. What does the best available balance of evidence publish to the peer-reviewed medical literature have to say right now? Welcome to the Nutrition Facts Podcast. I'm your host, Dr. Michael Greger. Did you know that a single meal high in saturated fat can impair artery function for hours after consumption? Here's our first story. We are only as old as our arteries. What can we do to preserve artery function as we age? A poor diet and sedentary behavior can lead to adverse aging processes like impairment of the little power plants in our cells, which can result in free radical formation, oxidative stress, and inflammation, which leads to artery dysfunction that can end in the cardiovascular disease that ends us. In a series of videos I did about a decade ago, I discussed this landmark research showing that a single high-fat meal could cripple artery function within hours of consumption compared to no change in the low-fat meal. The high-fat meal that so crippled artery function included the sausage and egg McMuffins from McDonald's. How do we know the sausage, egg, or cheese was to blame? What about the crappy carbs and the biscuits? We know because of the low-fat meal that didn't impair artery function was a sugary mass of carby frosted flakes. And just when your artery function finally starts to recover five or six hours later, lunchtime, when your arteries may get whacked with another load of meat, eggs, dairy, or oil. Why does it matter so much what happens after a meal within your body? Because most of us spend about 16 hours a day in that after-a-meal state, constantly hammering our arteries. No wonder cardiovascular diseases are our number one killer. That doesn't just inflame the arteries in our heart, but our lungs as well. A high-fat challenge increases airway inflammation and impairs bronchodilator recovery in asthma. Have asthmatics cough up sputum from their lungs four hours after the same kind of high-fat meal, and inflammatory cells shoot up in the high-fat meal group in terms of lung function, give them two hits of their inhalers, containing a drug called albuterol or ventolin, and their airways open up as they should after the low-fat meal, but after the high-fat meal, the same inhaler doses don't work as well, crapping out after a few hours because of all the extra inflammation in their lungs. What you eat can determine how well you breathe. OK, but that was asthmatics. But even in non-asthmatic subjects, you get the same spike in inflammatory cells and sputum coughed out of your lungs four hours after eating what was, in this case, a Jimmy Dean's Meat Lovers Breakfast Bowl. And there aren't only more inflammatory cells. There's also a doubling in the amount of pro-inflammatory oxidized bad LDL cholesterol sucked up by the type of white blood cells that go on to form foam cells. Those are the cells that build up the inflamed pus in your artery wall that leads to heart attacks. And all this happens within just hours of eating pizza, in this case.  Endotoxins are components of bacterial cell walls, and foods like meat can be so contaminated with bacteria alive and dead that they accumulate endotoxins. And we're talking about both red meat and white meat. But recent research, published in 2020, suggests the main culprit may not be endotoxins, after all, but the fat itself. The saturated fat floating in your blood after an unhealthy meal may be inducing the inflammation more directly. Either way, we are accountable for what we eat meal by meal for the modification of the risk factors for chronic metabolic disease. In our next story, we look at the window of time in which sufficient physical activity can help mediate some of the damage caused by eating an unhealthy meal. In my last video, I discussed studies that show a single meal high in saturated fat can impair artery function in men as measured in the arm, but what we're more concerned about is blood flow to the wall of the heart. Research has randomized men to eat either a high-fat meal that was more than 60% fat, half of it saturated, with more than an egg's worth of cholesterol, or a low-fat meal that was mostly carbs with less than 10% fat and 50 times less cholesterol. If part of a coronary artery is blocked for any reason, the surrounding vessels expand. That extra expansion capacity is called the coronary flow reserve, and it's clamped down within hours of eating a fatty meal, undermining the heart's ability to compensate for clogged arteries. That's how a high-fat meal affects blood flow to the heart. What happens if you exercise though right after that high-fat meal? After the meal inflammation following the prolonged elevation of fat in the blood that occurs when you eat high-fat meals, provides a likely explanation for increased cardiovascular disease, but substantial evidence has shown that acute exercise is an effective modality for clearing out some of that fat after a meal. However, the benefits of acute exercise for post-prandial lipemia, for after-the-meal fatty blood, appear to be relatively short-lived, going a few days for that exercise. You may completely negate any benefit, no matter how fit you are. The time window appears to be between 18 hours before the meal, up to around 90 minutes after the meal. And how much exercise do we need? About an hour of moderate intensity exercise should do it. In this study, though, it only took 20 minutes of stair-climbing, broken up into 5 minutes every hour for 4 hours after a McDonald's breakfast of hash browns, eggs, pancakes, muffin sausage, and a milkshake. Following such a meal, artery function significantly decreased when the subjects just sat around after eating, but not when they did the hourly stair-climbing exercises. In conclusion, hourly exercise may attenuate the negative effects not only of prolonged sitting, but also of high-fat meal intake, suggesting that stair-climbing should be incorporated as an easily accessible lifestyle strategy to protect artery function. Of course, it goes without saying that the other way you can protect artery function is not the breakfast at McDonald's in the first place. Such a meal would also have more than 2,000 milligrams of sodium, more than the 1,500 the American Heart Association recommends you stay under for the entire day. Give someone a meal with even less salt, a third less, and that alone can still impair artery function within an hour of consumption, even independent of the increase in blood pressure.  Their blood pressure doesn't really depend much on their salt intake, so for these people, is salt okay? No. High dietary sodium intake reduces artery function regardless of whether your blood pressure is salt sensitive or salt resistant. Your artery function is impaired either way, going from a low salt diet to a high salt diet. See, there's an influence of dietary salt beyond blood pressure. In spite of the seemingly unanimous consensus, some researchers too often funded by the salt industry claim that it's actually not good to cut down on salt, but the evidence is against these dissenters. Like the saturated fat meat, dairy, and junk, the science indicates that sodium, not sodium, and reduction is the real villain. Finally today, if you're going to eat something unhealthy, is there anything you can pair it with to help mediate the damage it may cause? If you compare the artery function of those who don't eat meat to those who do, the healthy ability of arteries to dilate and let more blood flow is significantly better among those eating vegetarian, and not just by a little. We're talking four times better. Well, duh, vegetarians tend to be younger, smoke less, be slimmer, and have lower rates of diabetes. High cholesterol, high blood pressure, and heart disease. But the researchers control for all that. They only let healthy non-smokers into the study and recruit a group of meat-eaters who are just about as slim on average, and have about the same blood pressures, even practically the same cholesterol, a really healthy cohort of omnivores. Yet they still got their arteries handed to them by the vegetarians. And the longer someone was meat-free, the better. The degree of superior artery function correlated with the number of years eating meat-free. Instead of their artery function worsening over time as they aged, it got better the longer they ate that way. This suggests that vegetarian diets by themselves have a direct beneficial effect on an artery function and may help to account for the lower instance of atherosclerosis and cardiovascular mortality. Since the researchers were able to control for other known risk factors, they figured it must be the food. But what aspect of the food? Is it simply the lack of the deleterious effect of meat? Or could it also be because the vegetarians may simply be eating more whole healthy plant foods, for example, up to serving a day, more vegetables? When researchers compared two crappy meals, sausage and egg McMuffins to frosted flakes, and found that the fatty meal impaired artery function within hours, but the sugary meal didn't, they blamed the fat. But it may just be the animal fat, since high fat whole plant foods like nuts don't have the same effect. In fact, if you look at a systematic review of all the randomized controlled trials on the effect of nut consumption on artery function, you find that nuts actually make things better over time. Enough to counter the artery crippling effects of a salami sandwich? Let's find out. And the answer is yes for walnuts, but no for almonds, just like there are some fruits that are better than others, like blueberries over bananas. There are some nuts that are better than others, and walnuts appear to be the blueberries of nuts. What about the blueberry of berries? Blueberries themselves are randomized controlled crossover-trived cooked blueberries, raw blueberries, or no blueberries at all. If you feed people buns made of white flour, eggs, butter, and salt, and fill them mostly full of sugar and eggs, you get a gradual drop in artery function over the next six hours. But add the equivalent of a cup of wild blueberries to that same bun, and you instead get a big boost in artery function, almost as if you would just mix the blueberries with water. But the same amount of strawberries failed to rescue artery function from the likes of two cheese blinces with whipped cream, a sugary syrup, egg, and bacon, but that is quite the heavy load to bear. What about acai berries versus a meal with a similar amount of fat? One and a half frozen smoothie packs with half a small banana in water were able to significantly rehabilitate arterial function compared to a control smoothie with the same banana and water colored to look like the acai one, though obviously would have tasted differently. This group of researchers went all out, though, and performed a double-blind randomized control trial with raspberries, measuring artery function after two hours, then 24 hours after drinking about three cores of a cup of frozen red raspberries blended with water, or about a cup and a half versus a placebo drink meant to match both color and taste. The fake berry drink had no effect on artery function, but the other two did. Note the three-quarter cup dose seemed to work just as well as the cup and a half dose, which is what you see with blueberries. The benefits plateau after about a cup. The bottom line is that consumption of dietarily achievable amounts of red raspberries acutely improves artery function for up to 24 hours. You say, yeah, but by the end of the day, you're only up like one percent. Ah, but at a population level, each one percent increase is associated with a 12% reduction in risk of a cardiovascular event like a heart attack or stroke, all from just having a berry smoothie. What about berry juice? Five different concentrations of cranberry juice were used, along with a placebo control, evidently indistinguishable in color and taste. The 25% cranberry juice drink gave a little bump at two hours. The 50% juice was still working eight hours later. The 75% juice, the one that was nearly pure juice, and the ultra-concentrated juice also improved artery function within hours of consumption. But this, like that last raspberry study, just involved straight berries without some artery-crushing meal. Would berry juice be able to stop artery dysfunction caused by high-fat meal this unhealthy? Squeezing down artery function within hours? Researchers created a cocktail of grapes, lingonberry, blueberry, strawberry, and black aronia berry, and no significant change after the high-fat meal. Of course, if you had just drunk those berries alone, you'd probably get an improvement, but it's better than nothing. Well, what about something a little less exotic than black aronia berries? What about OJ? Participants were provided a high-fat meal of ham and cheese croissants, along with either a cup of water, orange juice, green tea, or red wine. The arteries didn't much like the croissants, and OJ was useless, as was the green tea and red wine. So it's probably best not to eat ham and cheese croissants in the first place. In fact, drinking orange juice with a fatty meal could actually make things worse. If you give people bacon and cheese muffins with or without orange juice, the OJ can lead to a prolongation of elevated fat in the blood, as your body preferentially burns for energy, although rapidly absorbed free sugars in the juice, meaning sugars not encased in cell walls, as in those in whole fruit. We would love it if you could share with us your stories about reinventing your health through evidence-based nutrition. Go to nutritionfacts.org slash testimonials. We may share it on our social media to help inspire others. To see any graphs, charts, graphics, images, or studies mentioned here, please go to the Nutrition Facts Podcast landing page. There you'll find all the detailed information you need, plus links to all the sources we cite for each of these topics. For a timely text on the pathogens that cause pandemics, you can order the e-book, audiobook, or hard copy of my last book, How to Survive a Pandemic. For recipes, check out my second to last book, My How Not to Diet Cookbook. It's beautifully designed with more than 100 recipes for delicious and nutritious meals. And all the proceeds I received from the sales of all my books goes to charity. NutritionFacts.org is a non-profit science-based public service where you can sign up for free daily updates on the latest in nutrition research via bite-sized videos and articles. Everything on the website is free. There's no ads, no corporate sponsorship. It's strictly non-commercial. I'm not selling anything. I just put it up as a public service as a labor of love, as a tribute to my grandmother, whose own life was saved with evidence-based nutrition.


This lecture is part of Berkeley Math 115, an introductory undergraduate course on number theory, and it will be about numerical calculation. So, let's have some typical examples of problems you might want to calculate the answer to. So, first of all, we might want to solve ax plus by equals c, where a, b, and c are given. And this is, of course, just something to do with Euclid's algorithm. Another problem we might have is n prime. And if it's not prime, we might want to factorize it. Thirdly, we want to solve f of x is congruent to 0 modulo p for some prime p where f of x is a polynomial. For instance, we might want to solve x squared plus 1 is congruent to 0 modulo p. Or, fourthly, we might want to work out what is a to the b modulo n. Now, all of these problems, if you look at them, they're, in some sense, trivial to solve. We can solve all of them just by doing a lot of case by case checks. So, if n is prime, we can factorize it just by checking all factors up to the square root of n and so on. With all of these, there's no problem in principle of solving them. The problem is we want to be able to solve them fast. So, the idea is that these numbers a, b, and c, and the prime p, might be very large numbers, maybe tens or hundreds of digits. And we want to have an algorithm that's reasonably fast for solving them on a, assuming we've got a computer, of course. I mean, if these numbers are hundreds of digits, you don't want to do them by hand. So, we want to actually find reasonably fast algorithms. And we also want to estimate how fast the algorithms are. So, let's start with a very simple example. Problem, let's find a plus b.   So, we're going to use the usual algorithm for that. And what we want to do is just estimate how long it takes. Well, the usual algorithm for adding up a and b, well, the number of operations is going to be about the number of digits of a and b, or maybe twice the number of digits or something. So, we're not interested in the exact amount to multiply the number of digits by. So, what we do is we use something called the O notation. You say it's O of number of digits. So, what O means is it's some constant times the number of digits. Or, more precisely, we might say it's less than or equal to some constant times the number of digits. So, it might be twice the number of digits or three times the number of digits. But we don't really care very much because that just depends on minor details of how you implement it. And we can ask, is this best possible? Well, obviously, yes, because it takes O of the number of digits even to read in the number a. So, you can't even look at the number a without using at least this number of operations. So, we have a precise, reasonably precise estimate for how long addition takes O of the number of digits. And if we write the number of digits n, we would just say this takes O of n steps. Notice that n is not the size of a and b. It's not the size of the numbers a and b. It's their length. So, it is the length of them in decimal notation. So, you've got to be a bit careful about this. More generally, if we've got a problem, if you've got an algorithm, we might say it takes, we'll see some examples where it might take O of n squared or O of x of n. So, this means it takes at most some constant times n squared steps. And again, you've got to be careful. n squared, the number n is not one of the numbers a and b in the input. It's the length of the numbers written in decimal form. In other words, it's the length of the input to your algorithm. So, this is the sort of standard way of estimating running time of algorithms. Use this big O notation. And we want to know what is a fast algorithm and what is a slow algorithm. Well, a very rough guideline is that polynomial time algorithms tend to be reasonably fast and anything that takes more than polynomial time is tends to be a bit slow. So, polynomial time means something like O of n to the k for some fixed k where n is the length of the input. People use polynomial time a lot because, I mean, the exact running time of your algorithm depends on what computer you've gotten, exactly how you implement it and so on. And, you know, details of your computer might change from O of n to the 5 to O of n to the 6 or something. But if something is polynomial time, then it's probably polynomial time no matter what computer you're using. So, polynomial time is the kind of stable property of your algorithm. So, now let's do another example. Let's try and multiply a times b and try and figure out how long it takes. Well, the usual algorithm, you know, for multiplying 13 by 17 or whatever, you take 3 by 7, which is 21, you take 1 by 7, which is 7, you take 1 by 3, which is 3, you take 1 by 1, which is 1, and then you add them all up. And as you can see, the number of steps is the number of digits of a times the number of digits of b times some constant. So, the number of digits of a and b is going to be bounded by the length. So, this is an O n squared algorithm. If n is the length of a and b, then it will take some constant times n squared steps, and I don't really care what this constant is. I mean, it might be 2 or a half or something. And it seems kind of obvious that this is best possible. I mean, if you've got a 10-digit number times a 10-digit number, it's very hard to imagine any way of multiplying them that's any better than taking every digit of a times every digit of b. Estonishingly enough, there are much faster algorithms. I mean, I just didn't believe this when I first heard it because the usual algorithm is so simple and obvious. It's very hard to see how you could possibly be improved. Well, there's not just one faster algorithm. There are several faster algorithms. So, I'll sketch two of them. The first of them is something called the fast Fourier transform, which sort of revolutionized signal processing. And what the fast Fourier transform is, is very roughly, you take these numbers a and b, and you apply something called a Fourier transform to them. Well, you don't really, but so this is something called a Fourier transform. And the fast Fourier transform, as its name suggests, is very fast and takes o of n log of n steps, roughly speaking. Then you do point something called pointwise multiplication, which takes o of n steps.  And then we do something called an inverse fast Fourier transform. Actually calling it an inverse fast Fourier transform is a bit misleading because the inverse fast Fourier transform turns out to be really the same as the fast Fourier transform. And if we do this, we get a times b. And this takes o of n log n steps. Well, the problem with this is I need to tell you what the fast Fourier transform is. And that's a little bit complicated for an introductory number theory lecture. So, I'm not going to. And instead, I'm going to give, describe a somewhat, a sort of slightly different method of doing very fast multiplication. Now, this method I'm going to describe is easier to understand than the fast Fourier transform, but in practice, it's usually a little bit slower. So if you really want to multiply numbers fast, you probably need to go and learn the fast Fourier transform. So the idea of this is to use the Chinese remainder theorem. So you remember the Chinese remainder theorem says that we can write the integers, the integers modulo ab is the same as looking at the integers mod a times the integers mod b. In other words, if you know an integer modulo a, and you know an integer modulo b, then you know it modulo ab. Now suppose you want to multiply two numbers m times n. What we do is we choose a lot of small primes, two, three, five, seven, etc., whose product two times three times five times seven is bigger than a times b. And let's say the product, let's call this product p. And now what we're going to do is we're going to work out mn modulo big p.  So that product should have been mn. So how can we do this? Well, we can find mn mod p as follows. What we do is we find mn modulo two, three, five, seven, etc., and this is really fast because these are going to be fairly small primes and all we have to do is to reduce mn mod the prime and multiply them. So how many steps is this going to take? Well, it's going to be roughly the number of primes. And how many primes do we have? Well, the number of primes will be about the logarithm of m, or the logarithm of n, which will be about O of n. It might be a bit bigger than that. It might be a little bit smaller than that. We're just going to give a very rough estimate. So we're going to, the number of primes you need is about the length of the input. And multiplying mn modulo, each of these primes is again going to take very roughly that number of steps. And then what we can do is we can reconstruct mn from mn modulo two, three, five, and so on by the Chinese remainder theorem. And you remember this uses Euclid's algorithm and Euclid's algorithm is pretty fast. And if you put all these three steps together, you find the algorithm takes about n times log of n, or n times log of n squared steps or something, you depend on exactly how you implement it. Incidentally, this is a bit like the fast Fourier transform. Both of them, you do some preliminary transform, you either reduce m and n modulo lots of primes, or you take the Fourier transform. And then you reduce to doing lots of lots of individual multiplications. So you do multiplication, mod two, mod three, mod five, and so on. And here, you also have to do a lot of separate multiplications to work out the point-wise product of the Fourier transforms. So although the fast Fourier transform and the Chinese remainder theorem method look rather different, they're somehow both using the same key idea in the middle of them. So in practice, there are really three different cases you have to think about when multiplying two numbers. First of all, the case of numbers that will fit into one computer word, which these days is about 64 bits. And in that case, you just use the hardware multiplication routine built into your computer. Another case is when the numbers fit into a few computer words, and then you just use the usual high school algorithm for multiplying, except that you work that the base you use is two to the 64 or something. And the third case is very big numbers when you use some sort of fast Fourier transform algorithm, or maybe this algorithm, if you can't remember how the fast Fourier transform works. So let me just give an example where using this Chinese remainder theorem method is actually possibly the best one. So suppose you want to have the following problem. I want to compute some determinant and it's maybe going to be a 10 by 10 determinant where all these numbers of say 100 digits and this is maybe 10 by 10. And you could do this by just sort of multiplying everything up using maybe even a fast Fourier transform method. But that's going to be a bit complicated. The easy method is we compute determinant modulo p for lots of small primes p two, three, five, seven. And again, we want the product should be the product, all these primes should be bigger than the absolute value of your determinant. So you have to estimate what's the absolute maximum possible value this determinant could possibly be. And you might say, well, you're going to take 100 digit numbers and you're going to multiply 10 of them together. And you're going to have a sum over 10 factorial of those or something. So you can you can estimate it's going to contain most, you know, 2000 digits or whatever. So you find 1000 primes and you work out this determinant modulo each prime. And I should say in order to compute determinant, please use Gaussian elimination. So you remember from a linear algebra course, there's a fast method of computing determinants using Gaussian elimination. And there's a really slow method where you sum over all ways of taking a product of one entry from each row and taking one entry from each row and multiplying and adding is a is a really stupid way to compute determinants because it's incredibly slow. So you have this fast way of computing each determinant modulo p. And then we can reconstruct using the Chinese remainder theorem. In fact, sometimes all you want to know is whether this determinant is zero or not. And in that case, you don't even need to use the Chinese remainder thing, you can just check whether the determinant is modulo zero modulo p for all these primes p. So in cases when the the reason why this works so well is is that computing determinants only involves addition and multiplication and subtraction and reducing numbers modulo lots of primes is really good for that. It's much worse if you want to sort of do division with a remainder or test whether one number is bigger than another. To do that, you've got to convert these back to proper integers using the Chinese remainder thing, which is a bit slow. So the reduction mod p method is really fast if you're just using addition, multiplication and subtraction. Another reason why this is good is it allows parallel processing. So what this means is that instead of having one computer that works it out mod p for 1000 different primes, you have 1000 computers all working together. And each of these computers is just doing one prime and the calculations modulo all these primes are also independent of each other. So the computers can all do this calculation simultaneously. So some of modern very fast computers have many thousands or millions of processes all working together. So if your calculation allows parallel processing, then then you can sometimes speed it up a lot. You can't always you can think of the analogy suppose you're digging a hole. Well if your hole is a big trench, then that's easy to do by parallel processing. You can have 500 people all digging a little bit of a trench at the same time and it's very fast. However, if your hole is is something like an oil well that goes very deep into the ground, then then having 500 people all trying to dig in at the same time is not really possible. There's only room for one guy at the bottom of this hole doing the digging. So that's something where you can't do parallel processing. And problems in number theory are the same. Some of them are very good for parallel processing and some of them just aren't. So I should also have a bit of a warning about log n factors. So you'll sometimes see people say the running time of an algorithm is o n or o n log n or o n log n cubed or o n log log n or various other things. And what I want to warn you about is a lot of these estimates are kind of meaningless. In particular, factors of log log n are generally meaningless in practice. The point is that log log n is almost constant. In fact, I've been doing calculations for a long time and I can tell you that log log n is nearly always approximately three, unless your number is very big indeed. So for example, if you've got two algorithms, one of which takes log log n times n steps and the other which takes 100 n steps Now you might think that this algorithm is going to be better because if n is large, 100 n is going to be much less than log log of n. But 100 n will only be less than log log of n when log log of n is bigger than about 100. So what you would, so this would only happen when n is approximately e to the e to the 100. And this number is so ridiculously large that it, you know, you can't write it down in the in the visible universe. So in practice, the apparently slower algorithm will be better. And that's because as I said, log log of n is in practice, it just doesn't get big. Even these exponents here, in fact, as a log event should be taken with a big pinch of salt, because the actual running time of an algorithm will not be the running time given by by your theoretical estimate. The point is, there are always extra time delays that theoretical analysis does not take into account. For example, if you've got a if you're running a big computer, it might fill a room or something. And now, during your calculation, you've got to retrieve numbers from storage and and storage is going to be on the other side of the room. Well, that's not a big deal because your your signals are traveling at the speed of light. But light's going to take several nanoseconds to cross that room. And on computers, operations are often just taking one nanosecond. So if you've got to retrieve something from across the room at the speed of light, that's going to slow down your calculation by quite a large factor of 10 or 100 or something. And these delays are not taken into account by these theoretical estimates. And that will give you extra factors of log of n or maybe even an extra factor of the cube root of n or something.  So all these estimates of running time are rough guidelines. So an algorithm that takes O of n steps is probably better than an algorithm that takes O of n log of n cube steps.  So we should also ask how many operations are possible in a calculation. This honestly varies with time. So I can give you a historical example. In the 1960s, the English mathematician Peter Swinnett and Dyer wrote a paper on computational number theory. And he gave an estimate for this at the time. He said that a million operations was trivial. A billion operations was possible but needs thought. In other words, you should rather carefully write your computer program to be as efficient as possible. And a trillion operations is about the limit of what it's possible to do. And he said that it could only be justified by major science if we advance. Well, we're now half a century beyond that and 10 to the 12th operations is now trivial. I mean, your screensaver probably burns through more than a trillion operations. 10 to the 18 operations. Well, here it's possible. So large supercomputers, I don't know what the current record is, but a few years ago they were doing 10 to the 15 operations a second. At least that's what they claim to be doing. So they could get through 10 to the 18 operations in about an hour or so. So that's certainly possible.  What is the limit of what we can do? Well, I really don't know.  So somewhere between 10 to the 24 and 10 to the 30 operations is probably around about the limit of what we can do. Of course, this might speed up quite a lot some time in the near future because we have people trying to build quantum computers. And at the moment, quantum computers can't do all that much. But if the size of quantum computers got speeded up by a few, sorry, if the size of quantum computers increases by a factor of a few thousand, then they will be able to do things that are beyond the ability of any imaginable classical computer. Another thing you can do to speed up computers quite a lot is build specialized hardware. So building specialized hardware can give you another factor of a few thousand. For instance, the deep blue chess playing computer, which was the first computer to beat a world chess champion about 20 or 30 years ago, that use specially designed chips that for playing chess. Another example that's very common these days is we have specially designed chips for mining Bitcoin, which are thousands of times faster than an ordinary laptop, which is why you're probably not going to get very far mining crypto currency just by using your your laptop or cell phone. So now I'm going to discuss some other algorithms. First of all, I want to recall the so-called Russian peasant algorithm. This is, it's supposed to be called the Russian peasant algorithm because there are claims that when travelers went to went to Russia in the 19th century, they found Russian peasants multiplying numbers using this algorithm. I don't believe this for a moment because I'm sure Russian peasants are better things to do with their time than do long multiplication of integers. But anyway, it's got this catchy name of Russian peasant multiplication. And let me explain how to do this. So let me try and multiply 13 by 5. And apparently Russian peasants knew how to multiply and divide by 2. And they knew how to do addition, but didn't know anything else, which is a game one of the reasons I don't believe this was done by Russian peasants. So what you do is you keep dividing this by 2. So we have 13 divided by 2 is 6. And 6 divided by 2 is 3. And 3 divided by 2 is 1. And 1 divided by 2 is 0. And you notice sometimes we had to throw away a remainder. So we'd a remainder here and a remainder here and remainder here. And now what I do is I keep multiplying 5 by 2. So 5, 10, 20, 40. And now what I do is I take all the numbers where we had a remainder. So we had a remainder at these three lines. And so I now adopt these three steps. So I now add these up and I get 65, which is indeed 5 times 13. The reason why this works is what you're really doing is converting the numbers into binary and using the usual algorithm for binary. So what this just says is that 13 is equal to 1, 1, 0, 1 in binary. So this 1 comes from here. And this 1 comes from here. And this 1 comes from here. And this repeated division by 2 is just converting a number to binary. And then this multiplication by 2 is just what you do for binary multiplication. And this is a really terrible method for multiplying numbers because it's like the usual multiplication algorithm in base 10, except it's much slower because you're first converting to binary and then multiplying them in binary, which takes ages.  But the idea behind it turns out to be really, really useful. So what we're going to do now is have the problem of how do you work at a to the b modulo m? So you remember I actually discussed this earlier. So I'm just going to kind of review it again.  And this takes an insanely long time. It's sort of O of x of n because the number of times you have to multiply a by itself is about equal to the number b. And you remember the size of b is exponential in the length of b. So this is an exponential time algorithm and is completely awful. And you remember I also said we should also reduce mod m each step. So as well as algorithms sometimes take a lot of time, they also sometimes take a lot of space. And the amount of space an algorithm takes is obviously at most the amount of time it takes because each time you use some space, you have to use some time to take that space. But if you want to be really careful, you occasionally get algorithms that use a lot of time, but a small amount of space. So we can reduce the amount of space this algorithm uses by reducing modulo m every time we multiply and that makes sure these these numbers don't take too long. So the third method improvement is to use Russian peasant exponentiation. And this is actually the algorithm I mentioned earlier. But let me show you how you do this. Suppose I want to work out two to the 13. What I do is I take the number 13 and I keep dividing it by two just as the Russian peasants did for multiplication. And I keep track of when I have a remainder leftover. Now I take this number two. And instead of multiplying it by two, I keep squaring it. So here I get four, which is equal to two squared. Here I get 16, which is equal to two to the four. And here I get 256, which is equal to two to the eight. So now what I do is I take these numbers and multiply together the corresponding numbers here. So I get two times 16 times 256, which is equal to 8192, which is two to the 13. So you see this is just like Russian peasant multiplication, except instead of doubling numbers here, we're squaring them, which means instead of multiplying two and 13, we end up taking two to the power of 13. And we should do a quick estimate of how long this actually takes. Well, the number of divisions here is going to be about o of n. And so the number of times you have to square is going to be about o of n, where n is the length of this. And that's going to each squaring is going to take, you know, maybe o of log n steps. So we're going to get sort of terms like o of n log n and so on. I'm not going to work at them exactly. And then we have to multiply these together and divide, reduce modulo m each step. And that's going to take o of n log n to the power of something. And as I said, I don't really care what this exponent is. So the conclusion we get is that exponentiation is actually a really fast algorithm. It's not much more than linear in the length of the input, especially if you use some sort of fast multiplication involving fast Fourier transforms, which is going to give you extra factors in the exponent of log n, which is why I'm being a bit vague about it. So the conclusion is that Russian peasant exponentiation is really fast. And this working out h to the b modulo m is one of the most commonly used operations in computational number theory because it's so fast. So we can ask, is this actually best possible? Can we get fewer multiplications for exponentiation than used by the Russian peasant method? And the answer is it's not best possible. We can actually do better than this. And this seems surprising because, you know, the Russian peasant exponentiation method seems so efficient. I mean, you certainly can't do, you can't get more than a factor of two better because just repeatedly squaring something is going to take it takes up at least half the multiplications. However, we can do better. Let's work at a to the power of 15.  What about the Russian peasant method? So what we do is we work at a, we work at a squared, we work at a to the four, we work at a to the eight, and then we have to multiply together a squared, a four, a to the eight. So we get, we then get a to the nine, a to the 11, a to the 15. So what we see is that each exponent here is the sum of two exponents that come before it. And if we work out here, we've got one, two, three, four, five, six multiplications.  But there's an even better method. What we do is we work at a, a squared, a cubed, a to the four, a to the five.  So we have one, two, three, four multiplications so far. But then what we do is we're trying to get a to the five. Sorry, we don't need a cubed. So we get a squared, a to the four, a to the five. And this takes one, two, three multiplications. And now what we do is we take a to the five, a to the 10, a to the 15.  And this takes another two multiplications. So altogether, we get five multiplications. So we've actually done better than the Russian peasant exponentiation method. Notice we can do something similar. Whenever you're working at a to the m times n, we can first of all work at a to the m and then raise it to the power of n. And this will sometimes be faster than the, than the sort of Russian peasant exponentiation method. Actually, the, the, I'll just say a little bit, people actually studied this problem. What is the smallest number of multiplications needed for working out a to the n. And this is sometimes denoted by L of n. And people looked at this function and you can get it within a factor of two. So L of n is going to be at most two times the logarithm to base two of n. And it's at least logged the base two of n. So it's known to within a factor of two, but trying to get it exactly is fairly tricky. And it has some very subtle behavior. For example, it seems obvious that L of two n is equal to L of n plus one, because it's obvious that the most efficient way of working out a to the power of two n is going to be to work out a to the power of n and then square it.  But this is actually false. The smallest counter example is L of one nine one is equal to L of two times one nine one, which is equal to 11. And the pointers, you can work out L of 382, which is two times one nine one by using some odd numbers in a slightly more efficient way. The exact sequence you can get is one, two, four, five, nine, 14, 23, 46, 92, 184, 198, 382. And you work at each number in the sequence is the sum of two numbers in the previous sequence. And there are only 11 steps in this, checking that L of one nine one is 11 is rather more tedious because you've got to check a lot of cases, even more bizarrely, it's possible for L of n to be greater than L to n, which is even more astonishing. An example of this was found by Neil Clift only a few years ago. And his value of n was three, seven, five, four, nine, four, seven, zero, three. So he used a computer to find this. And I really hope he used a computer. If he did this by hand, it's really rather scary. If you'd like an open problem about this number L of n, there's apparently an open problem to show that L of two to the n minus one is less than or equal to n minus one plus L of n. It's been checked by computer for about the first 40 or so values, but beyond that it's an open conjecture. So another basic operation is to calculate a polynomial. So we might have three x to the four plus two x cubed plus five x squared plus x plus seven. And I just want to comment on this because the obvious method of calculating polynomials is not very efficient. So the obvious method is we calculate one x, x squared, x cubed, x to the four, which takes one, two, three multiplications. And then we multiply all these by the numbers we've got above there. So we multiply this by three, and this by two, and this by five, and this by one, and this by seven. And this takes another five multiplications. However, this is not a very good way of multiplying polynomials. In the days when people had to do calculations by hand, they were taught something called Horner's method for evaluating polynomials, where you evaluate the polynomial like this, you take x times three, and then you add two, and then you multiply this by x, and then you add five, and then you multiply this by x, and then you add one, and then you multiply this by x, and then you add seven. And you notice if you arrange the polynomial like this, you've actually reduced the number of multiplications by a factor of about two. So even when you implement polynomial multiplication on a computer, you should remember to do it in this efficient way. So next I want to discuss the problem of finding a square root of minus one modulo p, but I think I'll put that in the next video.


Individualism, a Reader, edited by George H. Smith and Marilyn Moore, narrated by James Foster. 5. From What Is An American? J. Hector St. John de Crevacour Letters from an American farmer, New York, Duffield, and company, 1908. J. Hector St. John de Crevacour, 1735 to 1813, served as a surveyor with the French Army during the French and Indian War, 1754 to 1763. After the war was over, Crevacour purchased a farm in New York and prospered. The following excerpt is from the third of Crevacour's Letters from an American farmer, first published in 1782. It is a classic statement of American individualism. I wish I could be acquainted with the feelings and thoughts which must agitate the heart and present themselves to the mind of an enlightened Englishman when he first lands on this continent. He has arrived on a new continent, a modern society offers itself to his contemplation different from what he had hitherto seen. He is not composed, as in Europe, of great lords who possess everything and of a herd of people who have nothing. Here are no aristocratical families, no courts, no kings, no bishops, no ecclesiastical dominion, no invisible power giving to a few a very visible one. No great manufacturers employing thousands, no great refinements of luxury. The rich and the poor are not so far removed from each other as they are in Europe. Some few towns accepted we are all tillers of the earth, from Nova Scotia to West Florida. We are a people of cultivators, scattered over an immense territory, communicating with each other by means of good roads and navigable rivers, united by the silken bands of mild government, all respecting the laws without dreading their power because they are equitable. We are all animated with the spirit of an industry which is unfettered and unrestrained because each person works for himself. If he travels through our rural districts he views not the hostile castle and the haughty mansion, contrasted with the clay-built hut and miserable cabin where cattle and men help to keep each other warm and dwell in meanness, smoke and indigence. A pleasing uniformity of decent competence appears throughout our habitations. The meanness of our log houses is a dry and comfortable habitation. Lawyer or merchant are the fairest titles our towns afford, that of a farmer is the only appellation of the rural inhabitants of our country. It must take some time ere he can reconcile himself to our dictionary, which is but short in words of dignity and names of honour. There on a Sunday he sees a congregation of respectable farmers and their wives all clad in neat homespun well-mounted or riding in their own humble wagons. There is not among them an esquire saving the unlettered magistrate. There he sees a parson as simple as his flock, a farmer who does not riot on the labour of others. We have no princes for whom we will toil, starve and bleed. We are the most perfect society now existing in the world. Here man is free, as he ought to be. This has been Individualism a Reader, edited by George H. Smith and Marilyn Moore, narrated by James Foster. Copyright 2015 by the Cato Institute. Production Copyright 2015 by the Cato Institute.


What is going on guys welcome back to the channel and I appreciate you being here today.


Today, we're going to do relay programming. It's all about programming under pressure and debugging the mistakes that sometimes happen when you work really quickly or in teams. We'll use graph paper programming to simulate coding and use relay races to simulate having a deadline. Relay programming teams will be racing to finish a graph paper program. You have to check your teammates work or debug it. Fix it if there's a mistake, add your arrow and then run back and tag your teammate. Programmers use debugging a lot to find and fix problems in their algorithms or code. It has to be this one, because it has to be this one. There are lots of ways to debug problems. One of the easiest is to move step by step until you find where something goes wrong. Then fix it. Here, I'm trying to do a backhand spring but I keep falling off. I walked through each part step by step and realized where my mistake was. My coach showed me, I wasn't putting my hands on the beam correctly. So I tried another backhand spring on the beam with my new hand position and I stuck it. I was really glad I debugged my backhand spring. Did you get it out? Yeah. Debugging is finding and fixing problems.


and I'm straightening up, you clip in the back of Wozniak, Jenozki on the outside immediately putting the bike down, a nasty looking incident, they're both up.


To have the Mac Commonwealth have three teams, two at large bids is just just warms my heart that we are getting the respect that I think that we have deserved for a long time as being a high quality group of teams and this benefits everyone in the Mac Commonwealth. First of all, we're going to take it one game at a time. We are not looking to Sunday, we want to get through Saturday and we want to be playing our best lacrosse on Saturday. Having that dominating three quarters against Masaya was so great for our creativity on offense, our risk taking on defense, I thought that we really put together 45 minutes of some of the greatest lacrosse that this program has ever played. We have to come up with a game plan this week to make sure that we're not dropping off our play in the fourth quarter because our middies are tired and I can't say enough about Kaitlyn Kozlowski and Kari Russo who are two-way middies, 57 out of the 60 minutes a game and they're just playing great lacrosse the whole time. We are heading into these games knowing exactly what to do. So Saturday it's going to be very, very important that we keep those middies fresh and we're able to put together 60 minutes. I'm loving what our defenders have been doing the second time they're playing a team. The ways that York scored in game one, I can specifically in my mind, one person in particular, Taylor Wibbocken, just changed her play from game one of Masaya to game two of Masaya. And then York, same thing, changed her play in game one from game one in York to how she played York in game two. And if that's one person on a team of 29 that's doing that and we can get 16, 17, 18 to have that same approach, I think that we're going to be a force to be reckoned with. One of the things that has been a big difference maker in the last probably three, three to four weeks this season has been Sophie Shikofsky. There's a reason that we held number 14 Masaya to two goals over almost three quarters because we have faith in our goalie. And as long as we continue to have faith in our goalie, I think that we're going to be pretty difficult to beat. I think offensively we are going to need every single player to contribute. We've got Caroline Murphy and McKenna Blitz drawing double teams pretty much every time they have the ball. They're doing a fantastic job creating space and creating opportunity for other players to cut into the critical scoring area and finish with a great shot. The rest of us around that have to do our jobs and we need to take some of the attention off of Caroline and McKenna and we need to make sure if they're getting too much attention we are open and driving hard right after that. Meredith has a lot of very tall, very athletic players. They're not afraid to go to goal. They're not afraid to shoot under pressure. So that's not something we see typically that kind of size from person to person. We don't typically see teams that really go to goal whether they have two people on them or not. So we're really working hard to defensively prepare to try to get some charges called. You know we're going to make them move the ball. The other thing that they do very very well is they lead their passers from X to the critical scoring area. They lead their passers very well. So taking care of their primary feeder who I think has 67 assists, taking care of that and having multiple game plans if something's not working being ready to take that option away is going to be very key.