Hey folks, good morning. My name is Mithra Miskasky. I'm a co-founder and head product at Platform Mind Systems. And I'm joined here today by Naveen Bhairati. Naveen is VP of Platform Engineering at Mabinir. And together we are here today to talk about simplifying edge cloud computing operations using SAS management. So let's get started. So by way of introductions, I head product management at Platform Mind Systems. And Platform Mind, for those of you who may not be familiar, we are an enterprise Kubernetes as a SAS service vendor. And we specialize in providing an enterprise-grade Kubernetes platform that's deployed using a unique and powerful SAS-based management model that enables you to run your private or hybrid or edge Kubernetes cloud deployments through a SAS managed infrastructure model. Naveen, over to you. Sure. Naveen Bhairati, way of introduction. I'm working as VP of Platform Engineering at Mabinir. Mabinir is a company that's trying to change the game for 5G radio specifically by disaggregating the radio, as they're known for, and also producing other products like 5G Core, IMS, et cetera. So again, I'm here to help them along the journey from a platform Kubernetes perspective. Back to you, Mithra. Great. Thanks, Naveen. Here's a look at the quick agenda for today's session. We'll start by taking a brief look at the edge market opportunity. We'll then talk about the challenges of managing server edge micro data centers, and we'll contrast that with the management of a traditional data center. We'll then take a look at SAS-based management for server edge infrastructure as one possible solution to address some of the challenges around edge cloud infrastructure management. We'll then follow that with an introduction to Maveinir, and we'll do a pretty thorough deep dive into Maveinir's web-scale platform in the context of 5G rollout. And then we'll talk about how Maveinir and ProfMine are partnering together to utilize SAS-based management to simplify the rollout of 5G VRan and packet core software. So let's get started. So let's start with a brief look at the edge cloud market opportunity. So this is one of the latest research reports coming out of Gordner, and as Gordner sees, while the edge cloud workloads represented a very tiny subsection of all of the workloads run across the world, that percentage is expected to grow in a pretty significant way over the next five years or so. So according to Gordner, edge workloads only represent about 1% or so of all the world's workloads, but that percentage is expected to grow in a pretty massive way to about 30% or so, where 30% of all the workloads run across the world will be run in edge locations. And this massive gain is contributed to in part by the requirement to run a lot of intelligent devices, such as smartphones or smart cars and other smart devices that require, in a way, that computing resources are run at edge to support these devices that are spread across the globe. And so through this and many other digitization trends as the scale of edge workloads is increasing in a pretty dramatic way, Gordner also expects to spend for edge cloud computing to grow in a pretty significant way. So by around 2024 or so, edge cloud computing spend is expected to grow to somewhere around $80 to $100 billion or so. So to put it in simple terms, this is a pretty massive market opportunity and one that's expected to grow in an almost an exponential way within the next five years or so. So let's look at what does operating an effective edge cloud involve, right? And to be clear, when we talk about edge computing, there are typically two ways of doing edge cloud computing that are commonly referenced. One is server edge, and the second is device edge or IoT edge. For the purpose of this session, we're focusing purely on server edge. Some examples of server edge include retailers, such as, say, Macy's or Starbucks or McDonald's, where each of their store locations spread across the globe is an example of an edge data center location. Or telco that we'll talk about in this session in a pretty detailed way where an individual cell tower location for a telco provider represents an edge site location. And so these edge locations are typically referenced as micro data centers. And it's because they are a fairly compact version of a data center. They'll typically include one to about three physical servers or so, and they're spread across the globe. And at times, they're at locations with fairly limited access for a human being. And at times, these locations might be based on a fairly high latency loop bandwidth network, et cetera. Now, because these edge servers or these edge micro data centers are spread across the globe, it's not always possible to standardize on same type of hardware across them. So you will find the diversity of hardware as one of the characteristics of these edge locations. You'll also find a mix of legacy workloads as well as modern apps. Legacy workloads, because a lot of apps run at these edge locations for, say, a retailer or even telco edge, tend to be built in a way that cannot really be modernized. So they might require a Windows operating system to run them. But some portion of the apps, for example, the 5G VLAN software that we'll talk about in a bit, are following modernization and are getting built using containers. So depending on the nature of workloads, you might need to run a mix of legacy as well as modern apps at these edge locations. And then finally, the biggest characteristic for the edge micro data centers is the scale, right, the massive scale at which these data centers need to be operated. As an example, a traditional enterprise customer, even a large enterprise, when running at scale will typically have to manage tens of data centers at the most hundreds of data centers across the globe. In contrast, a typical internet service provider, a telco provider, will have to manage probably tens of thousands, anywhere between 20,000 to 30,000 or so of just radio cell towers in order to manage or support all of their subscribers or customers located in a particular geo, right? So we're talking of a significant order of magnitude difference when it comes to scale for these edge data centers when compared to the traditional data centers. And that scale in part adds to a number of complexities involved in management of the edge data centers, where traditional data center management paradigms just don't work well. So understand that better. Let's take a microscopic look at what a traditional data center management entails. So in a typical data center, you will find one or more of all of these. There is a large inventory of servers provided by any of the popular server vendors like Dell, HP, or Supermicro, Huawei, or many others. These servers are typically configured with Windows or Linux operating system. And then they're coupled with top of rack network switches or gateways and block storage devices to together provide the resourcing needed to run the apps. And a portion of the server capacity will be dedicated towards management software, right? The most common management software is VMware vSphere. For example, that's used to deploy virtualization on a lot of traditional data centers. There will be a pool of databases because a lot of management software relies on external databases to store their persistent state. And then on top, you will have a massive pool of applications, including internal apps built by the ops team, as well as dev test workflows, as well as production apps that are consumed by that vendor's customers. And so depending on how these data centers are managed, typically, there tend to be a lot of manual operations or extremely time-sensitive, non-automated operations that are involved in upkeep and caretaking of these data centers. For example, installation of operating system across the physical server or hardware tends to be automated for a lot of enterprises. However, configuration of these operating systems or security patching or some other operations tend to at times be handled manually or might require someone to be sent to the data center to do some configurations, et cetera. The install and config of a management software is almost always manual, and it's extremely intensive and time-consuming, especially troubleshooting issues with the management software or having to scale it out or performing upgrades to newer versions of vSphere, for example, tend to be very intensive operations and almost always require someone to be there in person to babysit them. Database operations are similar where dedicated database administrators need to be available for maintenance and upgrades. And finally, the app layer will be owned by the ops team or the end users or developers building it. So in a nutshell, the management of a traditional data center almost assumes that you will have an army of resources available at times to be able to go physically to those data center locations to address and triage issues. But this paradigm finds it very difficult to scale when you're talking about a massive inventory of micro data centers that might be located in absolute remote locations across the globe. So if you take an example of a retailer, each retail store, as Vista said, represents an edge location. And it's almost impossible for a retailer to physically dispatch people at each of these locations to deploy complex software frameworks like virtualization or container orchestration and then to babysit them or handle the upkeep on an ongoing basis. So as a result, if you look at retail edge or possibly retail co-age, what you will find in a traditional environment is they are kept relatively simple. There's an operating system installed on top of one to two to three physical servers. And then there will be likely some windows-based apps that are directly installed on that operating system. And so at a high level, the installation of apps and OS tends to be very manual, and hence they're infrequently updated because there isn't automation available to update them regularly. Troubleshooting requires a human and hence is time consuming. And finally, you don't have a choice of deploying complex technologies such as virtualization or container orchestration. And so this creates a number of challenges for operating edge cloud environments, especially at scale because it will take multiple days to troubleshoot problems. It will likely take more than months to update the application or to update a security patch operating system. And you'll find that there's a proliferation of control plans because there isn't a single vSphere, for example, that could manage all of your edge site locations. So there will be a large pool of control plans which creates another problem of how to manage these different control plans. So there's lack of centralization. And then all of this adds to extreme high costs, but also equally importantly, it adds a ton of time delay in terms of time to market for that vendor when they're trying to roll out any new or critical initiative to the market. So as one example of this, when we just referenced is retail edge where in terms of the topological configuration, you'll typically find that there is one more corporate locations for that retail vendor that are then connected to a few centrally located data centers, probably tends to add the most hundreds of data centers. And then these data centers are in turn connected to a large inventory of stores, tens of thousands of them spread across the world. These stores will typically need to run anywhere between one to three to five applications locally. So those apps cannot really be outsourced to public cloud. These are your points of sale applications or your security surveillance, radio monitoring apps or media apps, et cetera. And so that's the nature of server edge deployment which incorporates some of these complexities in managing this infrastructure. Another example is telco slash 5G edge, which also includes a lot of very interesting nuances in managing different parts of the 4G or 5G infrastructure. Naveen, would you like to quickly talk about the deployment and conflicts of these telco slash 5G edge locations? Yeah, thanks, mother. I think one of the things that came about when I came to Mevenir is just the amount of latency that is tolerable is so different than the enterprise workloads, typically speaking. On the access side, the requirement is 200 microseconds from the time they call it UE user equipment, which is a cell phone or a car, which is IoT sending data using 5G signal. And the tolerance for that is 200 microseconds from the time the signal comes in to the time it gets processed and then response since it gets back to the actual device. So I think it's very different from a latency perspective. And also as you go into the core, it's a little bit more tolerable. So at the edge, it's more like two to 20, depending on the application and what's going on. And if the customer is going to video application or ARVR, any other newer stuff. And then at the core, it's more of the 3200 milliseconds. So very different in a telco world.  Got it. Thanks, Naveen. And so those are kind of at a high level, some of the challenges of having to manage it's server deployment set scale. And there isn't really good centralized management software that's designed to manage these fleet of micro data centers to scale. So as a result, Gartner products that about 50% of edge deployments will actually fail over the next few years. So to be specific, they predict that through 2022 about 50% of edge cloud computing solutions while they worked as proof of concept will actually fail to scale in production deployments. And we can see why for a lot of these reasons. And so as a solution, one possible solution that we would like to present in this session is using SAS managed infrastructure as a way to fundamentally address some of the complexity and challenges that are involved in managing edge data center locations and doing that at scale. So when we talk about SAS management, it typically entails from our perspective, a cloud hosted management plane or sometimes you call it the control plane. But the unique characteristic of this management plane is that it is detached from the underlying infrastructure that it is designed to manage. So while the underlying infrastructure might live in a customer's data center, central data center or at one or more of these edge locations, the management plane itself is outsourced and is running at a different location, right? And so this kind of architecture has several benefits. One of the key benefits is that it can offer centralized management of all of your inventory of not just the core data centers but also the edge locations together because of this kind of architectural paradigm. Because the SAS based management plane can scale out horizontally and outwardly independent of the data plane that it's managing. And so as you add more sites to your edge deployment, the management plane can scale out dynamically to address these additional site locations without those sites ever being impacted or even being aware that that management plane is actually going through a scale out. The management plane can handle a lot of heavy lifting of operations within itself. So instead of offloading all of the work or a lot of work around say installation and monitoring and management of your edge locations, all of that can be offloaded to that cloud hosted management plane which in turn enables you to deploy some pretty complicated software at these edge site locations and still keep up with the SLA and the high availability and uptime requirements that you have. So the SAS management plane in other words can guarantee that you're able to deliver your end apps with a very high SLA to your end users. For example, your radio cell towers as Naveen will talk about in a minute can run with a very high SLA of uptime while still being able to run complex technologies such as Kubernetes at each of those cell tower locations. So that's one of the beauties of this model. One of its characteristics is it can let you provision all of your software resources completely automatically. So for example, again, in the case of a radio cell tower or a retail store location, once your physical servers are wrapped in stack and have network connectivity at that edge location, then your remote hosted management plane can take over and it can boot up the servers, get the operating system installed from ground up. So in other words, remotely provision the bare metal and install the operating system on it and then deploy some complex technologies like virtualization or Kubernetes all remotely again and then allow you to deploy your legacy apps as VMs or straight on the bare metal or the containerized apps on Kubernetes. So this kind of model lets you as an end user build out your CI CD automation that will deploy one or many edge site locations through a single click operation, right? So it's complete software based automation using APIs that are exposed by this remote hosted management plan. And so in summary, the SAS based management model can fundamentally attack some of the core challenges that edge server infrastructure management faces. It can provide a very high SLA. It can remotely auto provision bare metal which can be a huge differentiator especially when you're talking about managing sites that don't have access. So you cannot send personnel at these inside locations. And finally, it gives you ability to deploy fairly complex technologies, virtualization or container orchestration or et cetera. So with that, let's take a deep dive into Mavenir and their use case to understand this better. Naveen, over to you. Thanks, mother.  I think the edge is going to be big. The question is how big is it? So, you know, I want to say at least there's going to be a million pods at the edge for each network. And the reason I say that is because when you think of the number of towers that are out there, when you think of how routing is done from a traffic perspective today, traffic's still going all the way into the data centers and then going out from a core perspective. So I think they're trying to, you know, move that as close to the tower as possible. I think there's added benefits like, you know, cost of the infrastructure is going to go down, you know, everything is containerized. That means the infrastructure is capable of coming up whether it's the RAN or the packet core or the IMS on the same infrastructure that, you know, Telcos never had the chance. So I think if you look into 5G and the aspect of 5G that's talked about a lot is radio, the purpose of that is the carriers have felt long that the incumbent vendors have always been, you know, more guarded about the radio hardware and software. It was shipped as a black box. It was not actually meant to be opened up and looked inside, right? So with the 5G spec, it's basically been disaggregated where you have the tower side and then there's everything else, which is running on x86. So let's go to the next slide and I'll jump into some of the details of this. So I think we at Mavinier looked at this and said, how do we address for Telcos a common infrastructure platform to run not only their radio, but also radio core IMS, any other workloads and also be able to move this to the public cloud. When we say this, it's the ability to move infrastructure using the smallest components. And like Madura said, how do we automate this? How are you able to scale up, scale down? Back in the day, people would ship software after they shipped hardware because you had to ship the hardware to go, hey, this region is at 90%, we need to ship more hardware, get that rack stacked and then somebody had to go install all of this. So the whole idea is it's plug and play. When you say plug and play, you could move some of this extra workload to public cloud and it should be seamless from an operator perspective. And the way the stack is structured is so the cloud itself becomes a component that the CAS and the past platform utilizes. So that becomes the fundamental component that we rely on. And this partnership with platform nine has been very beneficial because we're able to lean on them from a SAS model of the CAS and help scale our services faster. Now, on top of it, we also have the application layer. So because of the application nature, there are some common services provided as well. And then on top of it is some of the management layer which is where do you want this? How do you wanna orchestrate some of this network functions? Again, the entire idea is provide a platform that can run all of the services and network needs and also if needed, some of the other services like IT functions as an example and be able to move from public cloud into private cloud and private cloud back into public cloud as needed. So it's load driven and not constrained by hardware. And I think that's really the goal we're trying to accomplish here. It's just in the beginning stages, I think it's got a long ways to go. Let's go to the next slide. So this is a little bit more of a deeper dive into the aspects of some of the frameworks. So what does that mean? So today when we talk about private cloud, traditionally telecos have run on some version of physical hardware like Dell or Supermicro, they have switching layer and then they have the concept of let me go build you an operating system, let's then put something on top like open stack. And then on top of that, we can provision some services. What we're trying to talk about is, it could either be that or it could completely be an AWS Google. And then on top, we bring in a CAS platform which means Kubernetes fundamentally goes on bare metal and Kubernetes is gonna take care of everything whether it's scale up and scale down. Yes, there is some orchestrator that's orchestrating this resources from an application viewpoint, but application is seamlessly able to scale up and down as needed. And also it can be deployed as far as you want, which is really under the cell tower if needed in a single server, still part of a Kubernetes cluster because maybe that runs as a stretch cluster. So the idea of a stretch cluster is, if you have multiple availability zones, you consider the cell tower and availability zone and you kind of load balance it out with some other place. So if you have an outage, you can deal with some of these things. Now on the past side, we have a pretty standard infrastructure setup, which is how do we set up somebody with the traditional stuff, which is an ingress controller, a service mesh. And you need things like Prometheus Grafana and EFK to look at all the logging that's coming in, look at all the KPIs coming in and also look at some of the other stuff, which is can you really make this an observability driven platform? So there's an analytics engine on top that would power this to make some real-time decisions. And on top of that, we have the application now, which could be any of the network functions starting with the radio, which is split into DU, a C-U-C-P, a C-U-U-P, or it could be 5G core and there's a whole suite of services along with Rick, which is one of the newer, things coming to 5G, which would be radio controller driven in an interactive manner based on the KPIs. So this paradigm is completely different from the previous world where everything was a black box and radio and everything on the infrastructure was very manual, very human-centric is what I would say. And now this is going towards that automation and it's all going to be driven through Kubernetes because fundamentally the industry, and I believe this is really true, which is Kubernetes has been built for enterprise but will help Telco stretch to the limits of the edge in a very easy and hopefully in an automated manner. So let's go to the next slide here. So some of the detail that we showed earlier, this is a slightly different view. When we talk about a service and it's being offered to some of our applications internally, what does that mean? We talk about how do we make the application developer life easy so they can go, hey, I wanna move or I wanna deploy a service in a location and they really should not be aware of some of the constraints of building infrastructure and it should be coming up in as fast as possible manner. So primarily what we provide is on the bottom side, it's Kubernetes heavily driving the application deployment, driving the application placement and exposing the applications to be able to be scaled up, down, being able to manage the life cycle, maybe you wanna upgrade the application.  All of that is fundamentally dependent on this CAS layer, which we truly believe should be managed from a central location because some of these aspects are not something you wanna manage in hundreds of thousands of locations with millions of pods. I think what you wanna do is you wanna be able to bring that view back in and aggregate that view and be able to show to the operator what is it that you wanna do next or if you're even a little bit more aggressive, give that to an analytics driven machine learning model where you go, what would you like to push next? So I think that's really where the industry is going. And on top of that, in the MTCI layer which we're working with the Linux foundation to open source this and the XGVela group there's a set of application services that are very telco-centric. So think of them as paths, but telco paths. So these are services that are being open sourced by Mavenir and these provide anybody to come and make it a plug and play environment.  And I truly believe these are gonna be a million pods in each network at the edge. So it's almost running like many EKS's and that's where the real scale kicks in. I think that's where platform nine offers are pretty good, supreme solution where it's literally for us, it's plug and play. So let's go to the next slide, mother. This is more of a physical representation just to illustrate what the setup looks like. You have hundreds of thousands of towers on the left side. So you may have about four to five towers or up to 10, 20 towers, these get aggregated and there is a 5G term called it's a front hall. It's the data flowing from the tower back into the VDU. The VDU is the first component that's running on an X86 server. Now the nice thing about this is it could run in the LDC if there was a local data center concept or it could run under the tower itself in an X86 server. Now all of this gets aggregated and it also talks to the CUCP and CUB, there's a couple components that are associated. There are a little bit more upstream that are more on the regional data center side. And you'll notice that there's a management system that is more oriented towards the application and the application function orchestration but that drives the infrastructure as well underneath. And on the right side, you have the core which is more of the national data center. So all of this traffic goes back into CUCP and back into the core and that's where some of the key decision making happens. But on the left side, you have the radio and the actual physical radio, what I would call the digital signal being processed by the VDU. And then some of the brain or some of the components on the right side which take care of more of the, is it an SMS? Where does it go? Is it a voice call? Where does it get routed? So some of the other components are on the right side. It's not a, it's a very small picture just to show the complexity of the network in a network operators environment. And truly putting Kubernetes everywhere and stretching the edge which was very much a monolith running in a black box and replacing that with X86 hardware and running at scale at the edge. I think truly is the next steps in the evolution of this, what I truly call the edge. So I think this is the, let's go to the next slide, Naveen. Naveen, when you speak of this, when you speak about stretching Kubernetes, you're actually thinking of putting Kubernetes nodes also in those radio cell towers, right? Absolutely. So when the radio cell towers need a box, the X86 server sitting underneath is really a worker node running on bare metal with Kubernetes on top of it. It doesn't get any closer to edge, I would say. That's right. Great. Naveen, thank you so much for walking us through that. So in summary folks, what we believe is SaaS management can in a pretty significant way simplify your operations at scale at various layers of management across your edge deployment. As Naveen said, in a telco example, for example, it's your inventory of radio cell towers but also all of the underlying regional data centers or local data centers or the core software that all needs to be managed at scale and across the globe to truly run an effective 5G network. And so incorporating SaaS management can really be a game changer in this kind of scenario in the level of simplicity that it can bring to your management and hence the kind of SLAs that you were able to commit to your end customers. So those are some of the key takeaways that we hope that you take from this session. Again, thank you so much for joining and Naveen and I would be happy to take any questions if you might have them at this time. Hey guys, so there's no questions. This concludes our session and thanks everybody.


So, as I was saying, I'd like to thank Terry, Harrod, Rex, and anyone else who's willing to accept the blame for inviting me to speak. And the topic I was given was integrating model organism data around clinical genomics. I'm going to try and take that fairly literally and give a talk that I actually think Rex had previously heard, showing how we believe that fundamentally just rolling up our sleeves and getting deeply immersed in the middle of this is the only way forward. So this is a fairly typical problem, and in fact, as you've heard earlier in the day, sudden death is perhaps one of the areas where the rubber meets the road in clinical genomics. This is the fundamental genotype-phenotype problem, writ large. So this is one sub-ship of a pedigree that has about 400 affected individuals. I worked on it when I was a post-doc in Cricket's lab almost 25 years ago. So, single gene, sorry, I think you were just a first-year faculty member of that sex. It's a laminopathy. There's perfect segregation for one of two of the phenotypes in this family, and there's a large effect size. If you belong to this kindred and a genotype positive, you have about somewhere between a 510,000 X increased risk of sudden death before the age of 50. But unfortunately, there are multiple phenotypes in this family. Many of them have just asymptomatic phenotypes like this, a first prolongation of the distance between these complexes. Some people will have heart failure as early as the age of 10, and others will die suddenly and unexpectedly with no prior warning. And this is fundamentally the issue is that despite knowing perfectly what the cause is of the underlying disorder, we're unable to actually make any clinically useful predictions. It's worth pointing out that there are 12 different lamin syndromes reported, and actually all but 11 of them are found in this single family when you look deeply enough, although I don't believe that we've ever actually systematically addressed every single one. And the only one that isn't would be the Hutchison-Gilford Progeria, which as Mark mentioned earlier is an example of a single residue. There are multiple modifiers. That's what we would usually say in this setting, either genetic, epigenetic or environmental, but none of these are measured for almost any condition that we would hesitate to tackle. And there's no real empiric support for the strategy to move forward. And importantly, as I mentioned, no real additional information here to change clinical care. And as far as the patients are concerned, that would be improving their symptoms or improving their outcomes. Just as an aside, and we'll come back to this later, these are some of the phenotypes that you might expect outside the heart in a cardiomyopathy kindred, so everything from a Bifid uvula, which actually turns out to be a reasonable index of fusion defects mediated by TGF beta, through palmar plantar carotiderma, abnormalities of the petal arches or the axial skeleton that are based on integrals of long-term muscle tone during development and beyond, and then structural defects such as quartations. Those are all seen in dilated cardiomyopathy kindred. It's not all seen in this particular kindred. Clinical genomics is perhaps the other extreme where we start with a genomic sequence that we're trying then to interpret it. This is an example from the MedSeq study run by Robert Green and Cricket and I are both members of this. It's one of the CSER consortium studies, a randomized controlled trial of whole genome sequencing in a healthy primary care cohort and in a cardiomyopathy cohort. And this was one of the variants that was identified, this upper one here, identified in a primary care patient. It's a likely pathogenic variant in KCNQ1, one of the long QT genes, a potassium channel gene. During the initial patient primary care disclosure, the patient herself had an anxiety attack as a result of immediate concern regarding sudden death risk. And as you might imagine, throughout the rest of her evaluation, the concept of making her feel better or live longer was at the forefront. We could have thought then about what we would do to assess the pathogenicity of this variant. There are numerous assays in in vitro, both in hydrologous and homologous expression systems. There's an example from IPS cells from one of Cricket's papers. But they're also in vivo assays, including measuring along QT, looking at the QT morphology in lots of different ways. Worth pointing out, for example, here that the QT is really a surrogate for the underlying phenotype and it's one that we use simply because you can teach the average person to measure it. And this oversimplification of QT is worth pointing out. If I was to show the ECG to Dan Rodin, for example, he would have a binary yes or no repolarization disorder output, but the average person is left trying to work out where the end of the T-wave finishes or starts. And in fact, one of the most interesting things about this is as we were talking to this, talking about this problem with the patient in the cardiomyopathy clinic, the patient said, but my QT was normal. And so the question, I suppose, we have to ask ourselves at this point, is it always the phenotype? Is segregation, is penetrance, is pleiotropy, always a function of the underlying phenotype? Are there truly biological complexities that move beyond that? And that's worth, I think, thinking about as we go into the discussion. It's worth pointing out that the relationship between all of these metrics and actual objective risk to the patient is obscure. And what we really need are quantitative assays mapped onto people. The one thing I will put in here as a provisor, this is from Cricket's work, nothing to do with me, not that I don't think it's great work, actually quite the opposite. But it's a salutary example of how mutations that are known to cause autosomal dominant, highly penetrant, sarcomeric, hypertrophic cardiomyopathy in individual families in the general population may have no phenotype whatsoever work from the Framingham cohort. We go to the potential clinical studies that we could have done in this individual. We could have looked at her QT. We'd have looked at her corrected QT at the morphology of her electrocardiogram or at subclinical extracardiac phenotypes. None of these actually were particularly useful in this individual. And you can argue that they may not be useful in any individual. It's worth pointing out it was 95 years after the electrocardiogram was first introduced that we recognized the morphologic differences between the different forms of the long QT syndrome, required the genome to be cloned and the genes in each family to be cloned before we recognized it. Again, just an index of how poorly most phenotypes are actually resolved, even the ones that we think are incredibly precise and highly useful. There are a number of provoked phenotypes in the long QT syndrome, including posture, exercise, and recovery. And in fact, actually thanks to Andy Cron, there's actually a test of what the best discriminant might be in somebody who is genotype positive but not known to be phenotype positive. And they are actually the most useful thing, turns out to be the QT, corrected QT at 400, at four minutes in recovery after a standard exercise test. Unfortunately, that also doesn't perfectly apply to this situation. And again, throughout the entire evaluation, we did all of these things throughout the entire evaluation. The only thing the patient was interested in was am I actually at risk of sudden death rather than can you give me a label? And I suppose the other question we have to ask ourselves is, is the risk associated with the genotype or with the phenotype? A detailed family study just shows you how difficult this will be, so this is the actual pedigree when we did a more extended family study. We took a condition-specific family history, so there was actually a sudden death that had been previously labeled as an MI. There was also a congestive heart failure case in a grandparent, but it was complicated by the fact that this individual also had been treated for CLL. When we actually examined this woman, she had an ejection systolic murmur. Again, are you really interested in any of the subjective findings that a physician might record in the electronic health record? I would wager probably not. Certainly not something you'd be willing to spend money on. On the other hand, it is the basis of almost all diagnoses that we use in clinical care today. The QT itself was uncorrected at 466 milliseconds corrected at 461, so right on the margins for a female, but her EKG morphology was normal. But interestingly, our echo revealed focal left ventricular hypertrophy and mitral valve thickening not reported in the long QT syndrome. On the other hand, not systematically studied in any of the large cohorts with long QT. Our MRI was normal, and our provoked phenotype actually was, again, perfectly. This is almost too good to be true, but it was exactly 400 milliseconds of four minutes in recovery. So we have definitive abnormalities observed in the context of a genotype that we're not certain has correlated with this. And with all of these elements, I suppose the most important thing is that we have nothing really useful to show the patient. We've added about $8,000 to the evaluation. And remember, this is a known gene in the typical family. So phenotype, I think, is now limiting in multiple areas, not just in genomics, but also it turns out in clinical care. I think this is the momentum behind the precision medicine initiative, as much as genomics itself. There are fundamental issues with almost every phenotype. They're largely dominated by morphology. They're semi-subjective at best. They're late or even end stage. They are aggregated and have been so for decades for statistical power for clinical trials. We know coronary disease is probably 30 or 40 different disorders, yet we lump everybody together for most of the large trials, which have maybe 10, 20,000 people in them. They're largely legacy phenotypes. We never really thought about them in the biological context, for example, that Mark and Les were talking about this morning. They're often binary. They're cross-sectional, and there are almost no systematic perturbations, with the exception of a few stress tests in cardiology. So where's all the information? Why are these, why might alleles be silent? Well, it's pretty obvious from just standard genetic paradigms. We've looked, we've spent maybe many dollars on looking at Mendelian disorders and at GWAS. We're now in a resequencing phase, but the majority of phenotypes are likely in this middle ground, where there's actually an unmeasured conditioning variable, or there's now a phenotype that we don't have an assay for. And so that when you think about these things, the genetic architecture, I think, is an important piece of how we move forward. We don't know what the genetic architecture is for almost any trait that we think is important in disease. There are huge limitations to the way that we've thought about genetics to date. We've focused on extreme phenotypes out of necessity. There are a few prospective cohorts. There are very few modeling assessments of how many genes might be involved, if, for example, familiarity is detectable. And then the other thing to remember, I suppose, in all of this, is that heterogeneity also scales. So investing many billions of dollars in many much larger GWAS studies, for example, is unlikely to change the underlying thesis. And what we really need is to move away, I think, from the perfection that precision medicine implies to just thinking, perhaps, about more precise medicine as an intermediate step. So how might model organisms actually help here? Well, I think there are a lot of different strategies. And in fact, this is a diagram from Zach Kahane, just thinking about all of the different data sets that you might need in order to build a systems-level understanding of the relationship between genotype and phenotype. Model organisms, if properly tailored and the phenotypes are properly chosen, I think, can infiltrate this space very, very effectively through saturation screens to identify all the known genes for a particular trait. Although, obviously, it's important that we look at all the alleles, not just the F3 leafels. Reverse genetics to take individual alleles and then model what the phenotypic spectrum is. Again, we've tended to focus on areas that have been predetermined by legacy rather than on a universal phenotyping strategy. We need to begin to model environment. We need to think about where the gaps are that we could plug with model organisms. And importantly, I think the final goals will be iterative systems-level modeling and mapping to human genotype and human phenotype. So model organisms, I think, give us really some very important insights. One of the things that we found using the zebrafish, and I'm going to talk simply about the zebrafish simply because that's one of the areas that I've worked in, almost everything we do in the fish is done in 384 or 96 well plates. And I suppose the thing that we've learned is that comprehensive phenotyping is actually a much better way of going about this than taking one gene and a hypothetical phenotype. That you can begin to collect huge amounts of information if you begin to scale the phenotype and make it systematic and orthogonal. So these are just some of the traits that you can measure in fish. I don't have time to go into them in any detail, but it's worth pointing out that simple morphology clusters ion channel blockers as well as patch clamp during the first two or three days of development. We've undertaken strategies to do this in a variety of different disorders. This is a shelf screen for QT that we did in collaboration with Dan many years ago. And it's worth pointing out that in under extreme circumstances mapping exactly the phenotype in fish with a heart that's 100 microns long that we would map in humans, i.e. action potential duration, repolarization abnormalities. We identified this network of 15 genes, three of which have subsequently been found to represent some of the deviations in this QQ plot. We've turned the problem on its head. This is actually something I did almost 20 years ago, which was to look at the genetic architecture of atrial fibrillation, paroxysmal arrhythmia. So when you don't have the symptoms and you don't have any underlying phenotypes, it turns out there's no way of identifying unaffected individuals. And I would hesitate to suggest that that's actually one of the systematic themes in most of genomic medicine is the fact that we don't know who's unaffected because the resolution of our phenotyping is so low. We were able to show, for example, that this has a large Mendelian effect that only about 10% of the heritabilities explained by GWAS loci, even although these are the largest effect GWAS loci bar one in the entire genome. There are large missing intermediate effect sizes, and it's proven incredibly difficult to clone the genes simply because we do not know who's unaffected. So we're going to have to move forward with pedigrees where we basically have to wait for people to declare themselves or we need to identify new phenotypes. We've actually taken an approach in model organisms to try and identify what those phenotypes might be. We used a couple of machine learning strategies to try and identify genes that were at loci that were identified by GWAS that might be shared amongst all these. We found a unifying network that's about an order of magnitude more likely than any other network that is shared by all the 12 loci. And importantly, every gene in this network and either overexpression or loss of function perturbs the cell-cell coupling in the heart. But then when you try to move this to a clinical population, there are no metrics by which you could do that in a clinical population. So the other strategy that we've used is to try and think about what the perturbations might be. And these can be thought of in terms of physiologic tests or in terms of therapies. And as I argued earlier this morning, I think thinking about it in terms of therapies is a quick way to early low-hanging fruit. We modeled this disorder, which is actually a right ventricular disorder in an organism that does not have a right ventricle. It was chosen without a hint of irony. The disorder is characterized by arrhythmia, sudden death, and continuous abnormalities, and has traditional heart failure metrics. We did all the genotype anchoring that you heard about this morning so eloquently from hard. We built a conditional expression model. It turns out this is actually a relevant model of the allylic architecture of the disease. There's codome, co-expression of both the mutant and the wild type. You can see here the fish heart ends up, in fact, you can actually visualize the phenotype of the naked eye. These are siblings. We were able to identify phenotypes that were actually novel for this disorder, showing a dramatic loss of sodium current density at the membrane. There's tremendous heterogeneity of the cell biology in the heart, even in the single chamber fish. We were unable, obviously, to map any of that onto the human, but we were able to identify naturalistic peptide abnormalities. This is a reporter system that we built, and then went on to do a chemical screen for suppressors of this in the genetic mutants, just simply plating the mutants and then looking for suppressors. We found several compounds, including one that was effective at nanomolar potency, which subsequently has been used, and this is published, so I won't go into it, to rescue the phenotype, including the sodium channel defect of a variety of other phenotypes and even ventricular activity in the mouse, now actually also in human IPS cells. So I would hesitate to suggest that actually the fundamental problem with the entire field of genomics is actually this phenotype gap. This was work that we did actually, Teri had asked us to do it as part of the UDN, trying to cost out the scope of phenotyping that might occur in the Undiagnosed Diseases Network. And if you look at this, it's not surprising why we're unable to deconvolute a genome when we have only 10 to the four total phenotypes in the whole spectrum of clinical medicine with 10 to the nine genotypes, 16 plus transcription variants, 10 to the 20 proteomic lipidomic variants. We haven't even gotten to cell-cell connections or exposures. This was chosen by my wife. It's wind as in bag of wind. We need new translatable human phenotypes. The things we've picked are actually almost hysterical when you think about it. Glucose in the urine by taste, 20 year gap between when you develop the metabolic phenotypes of diabetes and when the glucose ends up in your urine. Cholesterol, there are 1,000 things abnormal in the LDL receptor families. We only measure cholesterol because it's visible to the naked eye. There's been deliberate reduction in complexity, limited dimensionality, and no clear organizing stimuli for any of the phenotypes that we look at. I'm gonna have two more slides. We need to reappraise all of our existing phenotypes focusing on resolution and computability. I would wager that there's not a single piece of information in most electronic health records that I care about because it's being collected in such a non-structured and random way. Every MRI done at Brigham and Women's is done using a different protocol. Just even unifying the structure in these would be important. I mentioned the ECG, which is probably the only phenotype for which there are standardizing data. We haven't pushed functional genomics into the clinic, although as you heard from Cricket, that type of pathway analysis could be quite powerful. I think there's a real opportunity to do next generation phenotype. You can imagine almost anything in this space from credit card reports through to facial recognition. I'm not gonna dwell on that. And the last slide, really, we need rigorous probability estimates for every step in this pipeline. I would point out the fact that there's a ton of extant data that we haven't organized in a way that can impact the clinical arena. We need quantitative family histories, population, lifetime risk studies, network structures and their responses, measured exposures, shared phenotypic lexicon that goes all the way from yeast to patients with much more mechanistic and proximate phenotypes is less outlined. Learning information systems and co-clinical modeling I think is an important part of that, all at population scale. And if we've learned anything from the genome, it's the fact that we need comprehensive, multi-scale dynamic phenotyping. This should all be integrated into clinical care with the goal of an next generation computable phenotype. And so in summary, I think it really requires deep knowledge of the conditioning variables for us to be able to interpret even a single genome. And many of the things we talk about here are just not measured any longer. The electronic health record is almost eliminated family history. Exposures, we measure the exposures to five or six things at max. Scalable anal modeling is emerging as a partner for clinical genomics, but I think it has to be embedded at very key points in order to be effective. It's worth pointing out, and I put this paragraph in after this morning's discussion, that at the same time as this is happening in the genomics community, there's a huge effort in the clinical community to redesign clinical care. And I believe that aligning discovery, genomic discovery, and beyond clinical care redesign and cost will actually mean that we have to integrate this stuff immediately to avoid unaffordable duplication. And perhaps one of the most important things we need is, again, just to emphasize this shared lexicon. And then finally, we need a minimal clinical data set, as Dan outlined, some degree of universality, which would have to be cellular, as well as quantitative linear stimulus response pairs, and all embedded in a way that complements existing care and is fundamentally part of how we deliver and educate the information to our providers. Thank you for your attention, and sorry I went a little bit long. Thank you, Callum. So we'll have time for a quick question. Does anybody have a quick question before we move on to Cecilia? Okay, so we'll move on to Cecilia Lowe, and then we'll have follow-up questions during the discussion.


As you saw at the last Democratic primary debate, Pete Buttigieg didn't just vociferously argue against Medicare for All, but what was interesting is he used the exact same talking points that materialized from within the health insurance industry. Now, not too long ago, he sounded really different when it comes to the issue of Medicare for All. In fact, he was a supporter of Medicare for All. So back in February of 2018, the People's Summit had called him out for not supporting Medicare for All, but he pushed back on Twitter saying, when slash where have you ever heard me oppose Medicare for All? Now, understand that the reason why they called him out was because he didn't explicitly vocalize support for it. But he pushed back right there and responded saying, I, Pete Buttigieg, politician, do henceforth and forthwith declare most affirmatively and indibitably unto the ages that I do favor Medicare for All as I do favor any measure that would help get all Americans covered. So that's what he said back in February of 2018, but you actually don't have to go back that far to see Pete Buttigieg supporting Medicare for All because this year he posted a picture of a receipt that said Medicare for All and he responded saying, I'm on it. And on top of that, in an interview with Morning Joe on MSNBC, he described Medicare for All in a way that I thought was brilliant. He said, you know, this isn't really a radical idea. This is the real compromise, if you consider the ACA being right wing reform and, you know, a national health system like we see in the UK being the true left wing reform. But really, Medicare for All, it's not as extreme as Republicans want us to believe. This is what he said specifically. The ACA, which was a conservative proposal, came to be caricatured as left wing by a very disciplined right wing message machine, right? What is Medicare for All? It's a compromise. I mean, in the UK, you got national health care. That would be the left wing, the true left wing position. The true right wing position is free for all, all corporate. And the compromise position is a single payer system where you have private doctors but a public payer. We've got to stop allowing the right to move the goalposts. So that's what he was saying about Medicare for All as recently as February. He supported it. But fast forward a couple of months later and he's putting out attack ads against Medicare for All. He actually has run to date on top of that. At the debate, we saw him take on Elizabeth Warren for her position of Medicare for All, accusing her of wanting to eliminate choice and blowing up the private insurance market. But it wasn't just at the debate where he attacked Medicare for All and was relentless because the day after the debate, he wasn't finished attacking Medicare for All and spreading misinformation about Medicare for All. Because this is what he said. Here's some new criticisms that he raised about Medicare for All. It sounds very familiar if you listen closely. Talk about that moment that we just played in our intro where you went at it with Senator Elizabeth Warren about her Medicare for All plan and you wanted a simple yes or no answer of will she raise taxes. And what she ended up saying time and again was Americans household costs will come down with her plan. What does it matter to you? What bucket it comes out of? Well not only is it important to have yes or no answers to yes or no questions at a time when people are so frustrated with Washington speak, but also there's still been no explanation for a multi-trillion dollar hole in this plan. I have a lot of respect for Senator Warren, but last night she was more specific and forthcoming about the number of selfies she's taken than about how this plan is going to be funded. And that's a real problem, especially when there's a better way to deliver healthcare coverage to everybody. Look what most Americans want is to have an opportunity to walk away from their private plan to have that what I call Medicare for All Who Wanted alternative, but also to be able to keep their private plan if they would prefer. And I think that's the right answer, especially when you do the math and realize it's also an answer that is paid for unlike the Medicare for All whether you wanted or not plans that still have this giant question mark over how it's supposed to work. So now he's asking how are we going to pay for Medicare for All? On top of that, he is implying that we're not really even sure how Medicare for All will work in practice. And let me explain something to you. These aren't just Republican talking points. These are health industry talking points. Even industry insiders see that that's the case. Because as Wendell Potter tweeted out, I can't tell if Pete Buttigieg is campaigning for president or my old job at Cigna. Mayor Pete DM me if you want to know how insurance companies really operate. So the question is what changed? What got Pete Buttigieg to change his position on Medicare for All that quickly? Because just months after proudly and boldly declaring support for it time and again, all of a sudden now he seems like he's using the same talking points that the industry uses. He sounds like a Republican, quite frankly. So what changed? Well, if you follow the money, you'll know exactly what has changed. He started taking money from the health insurance industry. And no, he's not just taking, you know, a few dollars here and there. When it comes to total donations, total health industry campaign contributions out of the entire Democratic Party 2020 primary field, do you want to know who's number one? It's Pete Buttigieg. Pete Buttigieg has gotten more money from the health insurance industry than every other Democratic candidate. In fact, the only individual running in 2020 to outspend Mayor Pete is Donald Trump, the sitting president. So looking at this chart here, as you can see, he took $548,000 in total from the health insurance industry. Now to be clear, this does include small and large campaign contributions. So you know, there is a percentage of Mayor Pete's health sector donations that probably came from people who just, you know, work at hospitals or who are nurses. And that's why you see Bernie Sanders and Elizabeth weren't on this list as well. But what's different about Pete Buttigieg is he hasn't sworn off fundraisers. He hasn't sworn off big money. In fact, he's actually taking quite a lot of big money and more than 50% of his total individual contributions came from large campaign donations likely bundled by rich donors in fundraisers that he held in the Hamptons. So if you're wondering why he sounds so different, you know, when it comes to Medicare for All, there's a clear reason why he sounds like a health industry stooge. It's because he is one. He sold out. He took their money and it corrupted him that quickly in a matter of months going from supporting Medicare for All to using the same lines of attack as Trump uses against Medicare for All. How sad is that?   What matters to him is these for profit insurance vultures are helping him get elected. So he's willing to turn a blind eye and sell the American people out. If it means advancing his own career and he says all of this about Medicare for All while denouncing the duplicity and doublespeak that politicians use in Washington, D.C. No, Pete, you're no different than any of these insiders in D.C. You're part of the problem. And in fact, you're one of the worst offenders because more people in D.C. when they take money, when they get bought off, they try to hide it better. But you're not even trying to hide it.  You're talking about why we should do Medicare for All and all of a sudden you sound like a health industry executive. That's disgusting. That's embarrassing. It doesn't matter to you that passing Medicare for All would literally save lives. You are perfectly satisfied with the pro-death status quo if it means that these for profit insurance industry vultures will help you get elected. Well, that's disgusting.  You shouldn't just lose this race. But your career and politics should be finished permanently because you showed all of us your true colors that you're willing to sell out if the price is right. And that's incredibly disgusting. It says a lot about you. It says that you're not in this to help normal Americans. You're in this for self-aggrandizement.   I can't wait until he drops out. And I would tell him to stay in South Bend. But it doesn't really seem like his constituents. They are very happy with him either. So maybe you should just do what we all know you're going to do and become a lobbyist for the health industry or some lobbying firm in D.C. Because that's what you're going to do. You were a consultant for McKinsey. And we all know at the end of the day, you're going to be another one of these multimillionaires who makes a ton of money getting the government to not do policies that the American people want, that a majority of Democratic Party primary voters support.  That's just, that's disgusting. That is, you know, there's a different kind of evil to that, right? Just not supporting a policy in and of itself, being apathetic. That's a problem. But to actively try to drive down support for a policy that the grassroots has been fighting for, that shows how loathsome you are. Not just as a politician, Pete, but as a human being.


Hi, I'm Zor. Welcome to Unisor Education. Today we will solve a couple of problems which I will qualify as arithmetic problems, but because they are about numbers. Now, this lecture is part of the course called MESS and MESS Plus and Problems presented on Unisor.com. By the way, if you found this lecture somewhere else, like on YouTube, etc., obviously you are welcome to watch whatever you want, but the website actually gives you a little bit more because together with every lecture on the website there is a detailed description of the same material, which basically is like a textbook. So you have both the lecture, visual kind of presentation and textbook style presentation on the same page basically. The website is totally free, no advertisement, no strings attached, so you can use it as much as you can. Now, this course is a subsequent to a prerequisite course called MESS for Teens on the same website, and I do suggest you basically to familiarize yourself with whatever is in that course before you are approaching all these MESS Plus and Problems. Well, primarily because all the problems are related to some theoretical material presented in the MESS for Teens course, the prerequisite course. Also, there is a physics for Teens course on the same website. There is a relativity for all kind of introduction to special relativity course. Well, use it as well. So, a couple of problems today. The first problem is the following. Consider you have a prime number P, which is greater than 24, and I'm dividing this prime number 4 by 24, which means it's represented as this, where is a remainder. Now, what I would like to prove is that R is prime as well. So, P is prime, I divided this by 24 with a remainder, and I'm staging that remainder is supposed to be a prime number. Okay. Now, as usually before I proceed, it makes sense for you to basically pause the video and try to solve this problem yourself. Now, the textual part, the notes for this lecture on the Unisor.com website does not contain the solution, but it contains a hint. So, maybe if you can think about this problem and you don't have a good solution, take a look at the notes for this lecture, it gives you a hint, some kind of a beginning of logical proof. Okay. Now, that might help you. It's definitely much more beneficial if you solve the problem yourself. Now, here is what I suggest as a solution. Well, let's think about this. Let's assume R is not prime, which means it's supposed to be represented as two different numbers. Both of them should be greater than one. Yeah, greater than one. Okay. Now, let's think about what kind of numbers that can be. Now, don't forget that P is supposed to be greater than one. Okay. Now, let's say P is greater than 24 and R is supposed to be less than 24. So, R is a number which is less than 24, represented as a product of something. Now, what are the numbers? Obviously, we need some kind of prime numbers into which R is supposed to be represented. Well, since 24 is 2 times 2 times 2 times 3, none of these numbers can be 2 or 3, because if this is 2 or 3, then P would not be a prime number because 24 is divisible by 2 and 3 and R is divisible by 2 or 3. So, P will not be a prime number. It will be divisible by 2 or 3. Okay. So, it cannot be. So, what's the next prime number? The next prime number is 5. Now, R is not supposed to be just 5 because 5 is a prime number and there is nothing to prove. We're trying to prove that R is a prime number. So, it's supposed to be 5 times something. Well, and that something also should be not equal to one of these, not equal to 2, not equal to 3. Obviously, not equal to 4. So, the minimum will be 5. But 5 times 5 times 5 is 25. R is supposed to be, it's a remainder. So, it's supposed to be less than 24, right? R is supposed to be from 0 to 23. That's the remainder. So, there is no number less than 24 that can be represented as a multiple of at least 2 5s or more because 5 times 5 is already 25. So, it's impossible to represent it as the product of 2 numbers because each one of those numbers supposed to be at least 5. 2 and 3 cannot be because it will be and 4 as well because 4 is also divisible by 2. So, 2, 3 and 4 are not supposed to be among these R's. So, it's 5 and that's why you cannot have non-prime number as a remainder. Okay, that's my first problem. I'd like to mention that the problems which I present here are not typical problems which you might be presented in school because those problems are more related to just to check how well you know the theoretical material. These problems are supposed to force you to think about something which you don't know how to solve.  No, there is no recipe here. You have to invent it and that's the most important part I think in studying mathematics.  So, that's the whole purpose I think of studying mathematics. It doesn't have much utility nowadays. I mean you might actually need to add numbers. I mean that's as much as much math as you might actually need. But more than that really requires certain really good thinking. That's why I would like to present this course called Maths Plus and Problems for you to develop this particular kind of non-arthodox thinking. Okay, next problem. Next problem is considering number 10 to the power of n plus 18n minus 1. And it should be, I would like to prove that it's divisible by 27 regardless of n. Well, just let's check if it's divisible if n is equal to 1, that's 28 minus 1, that's 27. If n is equal to 2, it's 136 minus 1, so it's 135, which is 5 times 27. So, for any n, this is just example, but any n is supposed to be divisible by 27. Okay, and again as usually, pause this lecture and try to prove it yourself. Okay, now, how can we prove it? Well, first of all let me start from the easy thing. Divisible by 27 means it's a 3 to the power of 3, which means we have to divide it by 3 two times. So, what I can immediately see, that I can divide it by 9, which is 2 3s. Why? Because 10 to the power of n is 1 and n 0s, right?  Obviously, it's divisible by 9, it's 9 times 1, 1, 1, 1, n digits and 18 is 2 times 9 times n. So, this is divisible 10 to the power of n minus 1 and 18 divisible by 9. So, divisible by 9 I have 3 and 3. What remains is that if I will divide it by 9 and what will be as a result, I will have this number plus 2 n. I would like to prove that this is also divisible by 3. That would be my third times. So, by 9 I have already divided by 9. I have remaining 2 n from 18 and this number, which contains n units, that's the second one. So, the sum is supposed to be divisible by 3. How can I prove it? Now, it's a known fact that the number of n minus 1, n minus 1, n minus 1, n minus 2, any number, any number, gives by division of, division by 3, gives a remainder the same as sum of its digits. It's a known fact, but I'll just very briefly prove it. So, this number is supposed to be written as 10 to the power of k times a k plus 10 to the power of k minus 1 times a k minus 1 plus etcetera plus 10 to the power of 1 a 1 plus a 0. So, if you have for instance 3571, it's 10 to the power of 3 plus 10 to the power of 2 5 plus 10 to the power of 1 7 plus 10 to the power of 0, which is 1 times 1. So, 3 5 7 1. This is a decimal representation, right? Okay. Now, what can we do about this?   Now, these are divisible by 3, actually by 9 as well. Why? Because 10 to the power of k is 1 and k 0 minus 1 would be 9 digits plus 10 to the power of k times 9 digits repeated, which is obviously divisible by 9 and by 3. So, which means that the visibility of this and the visibility of this, this is number n. Number n gives exactly the same remainder by division by 9 and by 3 as some of its digits. This is sum of its digits, right? So, it's a digital decimal representation with digits. So, some of the digits must have exactly the same remainder as n if you divide it by 3 or 9 because all these are divisible by 3 or 9, okay? So, their difference must be divisible by 9 as well because this is equal to n minus sum of its digits. So, n and sum of its digits must have exactly the same remainder if you divide it by 9 or by 3. Well, in our case, we need right now to prove that this is divisible by 3. All right, let's think about this way. n is supposed to give exactly the same remainder dividing by 3, 3 times let's say m and some kind of a remainder r as sum of its digits, okay? Now, this, well, this is a number, awesome number and it's supposed to give the same remainder dividing it by 3 as sum of its digits. But in this case, we know sum of its digits. Sum of its digits is n because it's n times 1 repeated, repeated one after another. So, sum of its digits is n. So, if you divide this number by 3, you will have exactly the same remainder as number n. So, 1, 1, 1, blah, blah, blah is equal to 3 times let's say n, but remainder will be the same, okay? Now, if I will put 2 here, that would be 2m and 2r, right? It's 2n. So, sum of this would be equal to 3 times n minus sum of its digits. So, sum of its whatever 2m plus n plus 3r and it's divisible by 3. So, sum of these is divisible by 3. Now, this is a very, I would say, delicate moment in logic. So, first, you basically prove the theorem that the number, any number, by dividing it by 3 gives the same remainder as sum of its digits. And then, this number has, in dividing it by 3, gives some kind of a remainder as the sum of its digits. Sum of its digits is n. Now, this is just the number n and we also know that number n is supposed to give the same remainder r. And now, all you do is just multiply it by 2 and add them together and you will get the visibility by 3. So, that's the third 3. So, 9 was divided before and now we have proven that this is the result after you divided by 9. This is the result of dividing by 9. You still can divide it by 3 and that's the 27. Okay, the last problem. Now, let's assume that we have 100 different numbers. The problem is, out of this set of 100 different numbers, different natural numbers, I can choose a subset, subgroup, whatever, a few numbers, sum of which is divisible by 100. This is supposed to be proven. So, again, we have to prove that out of any group of 100 numbers, I can choose a subgroup, subgroup, subgroup, subgroup, sum of which is divisible by 100. Okay, now, obviously, you can pause the video and think about this process. Again, there are hints in the textual part of this lecture. So, go on the website and you will have notes and notes contain the hint. Okay, so, here is how I suggest to solve it. Let's form partial sums. So, S k is equal to sum of n i from 1 to k. So, S 1 is n 1. S 2 is n 1 plus n 2. S 3 is n 1 plus n 2 plus n 3, etc. Up to S 100s, which is sum of all of these numbers. So, basically, S k represents sum of certain numbers. Okay, now, there are two different cases to consider. Number one, there is one particular sum which is divisible by 100. Well, fine, that's exactly what we wanted to prove. So, if there is such a sum, then whatever constitutes its components is the subgroup sum of which gives the result, which is divisible by 100. So, in this case, everything is just trivial. So, now, assume that non of S k is divisible by 100. Okay, so, non of S k is divisible by 100. Now, this is where the real thing starts. Well, if it's not divisible by 100, it should give some kind of a remainder when you divide it by 100, right? Okay, so, we have 100 different S ks as 1, as 2, etc., as 100. But how many remainders are available if you divide it by 100 and if it's not divisible by 100? Well, the remainder can be either 1 or 2 or 3 or 99. So, there are only 99 different remainders. If you divide it by 100, if you divide it by 100, number which is not divisible by 100, divisible gives you 0, obviously. So, if it's not divisible, it gives you 1, 2, 3 or 99. So, there are 99 different remainders. Now, how many sums do we have? We have 100 sums, as 1, as 2, etc., as 100. Now, there is a principle. If you have, well, I think originally it was about rabbits. If you have 100 rabbits and 99 cages, you have to put at least two rabbits into some cage. Because there are no places to put the hundreds. Only 99 places, but you have 100 rabbits. So, if you would like to put them into cages, then obviously at least two of them must go into the same cage. Which means two different, at least two different sums, let's say, S m and S n, should give exactly the same remainder if you divide it by 100. Okay, then let's consider the following thing, S n minus S m. Well, if two numbers give the same remainder, then their difference would be divisible by 100, right? So, if you have, let's say, 127 and 3,527 remainder 27 in both cases, if you subtract one from another, you'll get 3,4,0,0, obviously. So, if you subtract two numbers which have the same remainder, if divided by 100, you will have the result, the difference would be divisible by 100. So, let's do the following. What constitutes S m? It's all numbers from 1 to n. What constitutes S m? It's not all numbers from 1 to n. So, let's take numbers S, sorry, n, m plus 1, n, m plus 2, etcetera, n, n. What are these numbers? Well, this is exactly what constitutes this sum, because S m is from the first to n's. S m is from the first to n m's. So, if I subtract them, then this group represents this particular sum. So, this is exactly the group of numbers. If I can choose these numbers, some of them would be actually the difference between these S m and S m, and obviously divisible by 100. That's the end of it. So, again, what I'm suggesting to you, go to the website, Unisor.com, to the course mass plus and problems, go to arithmetic part of it, and it's arithmetic 0,5. So, read the problems, read the hints, and try to recreate all the solutions yourself. That would be very, very helpful for you. Okay? That's it for today. Thank you very much, and good luck.


Project-based learning, which uses cloud-based services such as Yammer, provides a platform for students to collaborate and learn from each other. It allows them to communicate easily, share ideas, and work together on projects. Additionally, it encourages self-learning and self-improvement, as well as interdisciplinary connections between students of different levels. Furthermore, it helps students develop digital literacy skills and fosters their ability to think critically and solve problems. Finally, Yammer can be used to facilitate the small group method, which includes initiating, planning, implementing, presenting, and defending projects. This article was authored by Dmitro M. Bodnenko, Halina A. Kuchakowska, Alexandra V. Lokasiak and others.


Personal notice. Danger is my stock in trade. If the job's too tough for you to handle, you've got a job for me. George Valentine. Rightful details. Yes, it's Let George Do It. Brought to you by Prem. P-R-E-A-M. The new miracle way to cream your coffee. We'll begin tonight's transcribed adventure of George Valentine in a moment, but first... All the new way to cream it is to purr-ee-bit get Prem. That's Prem with a capital P. Overall, other products for creaming your coffee. You'll prefer Prem in instant powdered form. P-R-E-A-M. Here's why. First, Prem has a rich, delicious flavor because it's made entirely from fresh, sweet cream and other milk products. Yet when you use Prem in your coffee, every cup contains only about half as many calories as if you use coffee cream. And then Prem is wonderfully convenient. It never sours, never turns. Keeps indefinitely while sealed. Open for everyday use. Prem stays fresh tasting to the last spoonful on your kitchen shelf or in your refrigerator. And last but not least, this new dairy miracle actually saves you more than one-third the cost of coffee cream. Tomorrow starts serving Prem whenever you serve coffee. P-R-E-A-M. The new way to cream it is to Prem it. Prem gets Prem with a capital P. And now, tonight's adventure of George Valentine, the ghost of Ireland Betty. Mr. Valentine? Valentine? Huh. Dear Mr. Valentine, I live in an enormous, rambling house built like an Italian villa. Oh, that's funny. What else did she write? What was the rest of it? What? What? Crying. It's crying. That's what it is. It's crying. I know it is. It's crying. That's what... Where is it? It's crying. I know it's crying. I know it's... Oh, for heaven sakes. Where are you going in such a hurry? Miss Dowdy, I'm sorry, but I just... Listen, Miss... Yes, my dear. Christine, you've had such a long trip. I thought you were going to lie down and rest awhile. Did you hear something, my dear? No. No, I didn't hear anything. Mr. Valentine, who is he? Why were you writing him a letter, Miss Dowdy, about the house? Here, it's a letter you didn't finish. Oh, that. But I did finish another one. That one blotted. Yes, I've already written, Mr. Valentine. He's just a friend of mine. Oh, it's lonely here, and I thought perhaps he might like to call on me.  There's nothing to be afraid of in the old place. You may hear a shutter banging once in a while, but... That's all. That's absolutely all. All right, Miss Dowdy. Sorry, my imagination seems to... I'm sorry I bothered you. Miss Dowdy, I got the letter you wrote to me all right, but... Please, please don't talk so loud, Mr. Valentine. In here, Miss Bruce. Of course. Lovely old room. Christine is asleep. I don't want to disturb her. That's Christine O'Casey? Yes. She just got here, you know, from Canada.  Won't you sit down? Thanks, Miss Dowdy, but I'm not sure we're staying. Oh, but I wrote you this... Yes, you wrote in your letter that you wanted a man investigated, a Professor E. L. Gifford from Denver. Yes, that's right, Professor Gifford. You sounded as though you wanted us to, um... well, get something on him. Oh, now you must understand. And yet, as I gather, all you suspect about this Professor Gifford is that he's liable to buy this house. Yes, yes, that's just it. You see, Mr. Eustis, the real estate man, has received this offer from Denver. Yes, I know. I've already talked to Mr. Eustis. And I find he's also the executor of a state for one Miss Betty O'Casey, who used to own the house. Yes, for Betty, the one who died. That's Christine's aunt. And I find you don't even have anything to do with the sale of this house, Miss Dowdy. You don't even own it, any part of it. No, no, I don't. But I was Betty O'Casey's companion for more than 20 years, Mr. Valentine. Sure, I understand. She left you pretty well taken care of, right? Only the house went to the principal beneficiary, Christine, her relative. And it's Christine's desire to sell, according to Mr. Eustis. That's why she's here to speed up a sale, any sale. Okay, but Miss Dowdy, what do you have against this Professor? Why shouldn't he buy the house? Why should you want to hire me to interfere with any possible sale? The Professor, he's not the right type. Not the right type? Yes. Well, if the house is Christine's, what does she say? Haven't you talked to her about it? Christine isn't, well, she's not sympathetic to the house. What? She's not sympathetic. She doesn't understand. She's young and cynical. And this isn't just a usual house, Mr. Valentine. Well, I grant you it's pretty old, valuable, maybe, but not so easy to get rid of. We looked all over for this house years ago. It was just what Ireland Betty wanted. Oh, it made her so happy. Oh, wait a minute. Who? Betty O'Casey. Ireland Betty. Oh, I know. It does make her sound rather like an apple woman, but it was nickname. Many people called her that. Ireland Betty. Excuse me, but it sounds more to me like the name of a race horse. Oh, she would have loved to hear you say that. She owned some horses once. Oh. She was so enthusiastic, so interested in, in everything. And she was no more recently Irish than I am. But she called herself second cousin to a leopard carton. Sounds like quite a woman. See Miss Brooks, not a woman. Not at all like her niece, the little modern realist, the unbeliever. Unbeliever in what, Miss Doughty? Leprechaun? Well, unbeliever in this house for one thing. Well, you'll have to be more specific. What is there about this place, Miss Doughty, that you don't want it sold? Because that is the idea, isn't it? You don't want anyone else living here. Why? Mr. Valentine, Ireland Betty and I became great believers in... Well, she looked all over to find this house. Because this house was supposed to have... I'm sorry, Mr. Valentine, that I've troubled you. You're too young yourself. And a good deal more honest than I thought you would be. What? Now wait a minute. Oh, uh, uh, excuse me. Uh, sorry. Wait a minute. How did you get in here? With a key, naturally. Greetings, my dear Miss Doughty. You haven't changed a bit. Where did you get her? Let go of my hand. Oh, this it won't bite you. Tape measure, see if there's space for a piano in the parlor. Besides my recording instrument. Uh, excuse me, Miss Doughty, but is this, uh, is this the guy? The guy? Professor Gifford. Yes, um, Mr. Valentine, Mr. Brooks. How do you do? How do you do? Mr. Valentine was just leaving. No, no. What? No, I wasn't going anywhere. I was just going to say I'd like to, uh, stick around for a while. Oh, Mr. Valentine. But of course, you couldn't help me, uh, hold one end of the tape measure, Mr. Valentine. Come along. I am sorry I intruded, Miss Doughty. But I was to meet Eustace here. She gave me the key. Oh, Eustace. Mr. Eustace. You come along with Mr. Valentine. Right. Us and an almost empty piece of real estate, Professor Gifford. What's your business, Professor? My own happiness, Mr. Valentine. What's yours? Now what I meant was, what are you a Professor of? Nothing. Huh? The sort of nothing that is something. The sort of something that is nothing until it is seen or heard by man. Until man, like myself, is willing to admit that he sees what he can't understand. What no one understands. Huh? That makes as much sense as a scramble jigsaw puzzle. Oh, it's quite simple. I'm a student of psychic research. Supernatural being. Oh, brother. No, not that. Only fools laugh, Mr. Valentine. Okay. I'm not laughing. You, uh, believe in ghosts, huh? So does Miss Doughty. So did I and Betty. So does this house. Yes, most of all, this house. So that's why you're buying it. Place is infested with ghosts, eh? Not every house has a ghost, you know, any more than it has good plumbing. But that's why Betty bought it originally. Been a ghost reported off and on for past hundred years. Shall we rejoin the ladies? Yeah. Yeah. I'm getting scared. I almost had this house once before, Valentine. I, Ellen Betty, rather, had her cat set for me. She did not. Yes, I would have had it for nothing then, if it hadn't been for her bodyguard here. Would you throw him out of here, or do I have to? No sense of humor, Mr. Doughty. All right, Mr. Because it's true. Ellen Betty pursued me like the beautiful banshee that she was. She came galloping at me like... Yeah, I said never mind. Mr. Valentine. And that's for you, Miss Doughty. No, no, please. Be quiet. Listen. Oh, but, Miss Doughty, what... Don't move anyone. I never knew. Yes. That's what it is. I've never heard it before. Oh, come on. Cut it out, both of you. Sure. Sure, it's crying, all right, but that's not me. Go. Mr. Doughty, where are you? Mr. Euston. That's his voice. Get down here, somebody, will you? Come help me. It's Christine. No, Christine. I found her just lying there on the stone outside her aunt's window. There in the second garden, Mr. Euston? Yes, that's right. Wimpering, almost unconscious. She'd had a terrible fall. I was coming up by the side entrance, you see. What makes you think she just had a fall? Christine, what happened? Can you tell me? Did you fall? No. No, I didn't. Oh, then tell us. What was it? I don't know what happened, Miss Doughty. Oh, now look here. George is just woozy. Let her rest. Did you see the ghost, Christine? Oh, Professor, for the love of my... No, I didn't. I didn't. I tell you, there's no such thing as ghost. It's all your imagination. I saw the ghost once myself. The night your aunt died, perhaps even she saw it over the lawn by the summer house. Who saw what when? I wasn't here. You never told me? The ghost. Nothing to upset yourself over. And I'm sure Miss Doughty, unless she cares to lie, she'll admit that she too saw it. But what did I did? What if I did see it? It doesn't have anything to do with this. It didn't, Miss Doughty. All right.  And all knocked down by a ghost. All right, now come on. Come on, somebody. I think she just noticed where you were found, Christine. What? The stone floor outside your aunt's window, Christine. Exactly where you were. That's where your aunt fell, where she died. The night that we saw the ghost. Coincidence, I suppose. Boy, good coffee. You'll enjoy coffee creamed with cream, P-R-E-A-M. For this new 100% dairy product in convenient powdered form, has a truly delicious flavor. And cream is thrifty. Saves over one third on coffee cream caught.  You'll be delighted to find cream has many other uses, too. It's perfect for sauces, smooth, luscious gravies, and richer, creamier soup. If you'd like to discover how cream can help you serve tastier and more economical meals, you'll want to send for the wonderful new cream recipe booklet we're offering you absolutely free. This brand-new collection of cream recipes includes easier, quicker, sure ways to make such family favorites as souffles, dressing, shortcakes, plus tested time-saving recipes for delectable new desserts. For your free cream booklet, just drop a card or letter with your name and address to Cream Test Kitchen, Box 959G, Columbus 16, Ohio. Cream Test Kitchen, Box 959G, Columbus 16, Ohio. Don't miss this exciting offer. Send for your free copy of the new cream recipe collection right away. And now, back to George Valentine. Look out. Miss Brooks, you're here? Miss Brooks, where are you? Where did you go? I am here. I just went back to the kitchen to get some ice. What's the matter? Be quiet. There's my imagination at work. Oh, he and I lie down again. Put this on your forehead. My forehead's all right. The doctor said it's the outlater to take a look at. Yeah, yeah, sure, sure. You've been pretty sick up in Canada, haven't you? Yes, and I'm tired and I'm... My aunt had asthma, you know. Did she? But that wasn't what she died of, was it? Well, sort of. That's what the doctor wrote me. I wasn't here, you know. We've been having a fight and she had guests for the weekend, so I just got in my car and drove off. See that window there? French doors. There's no guardrail outside it. And he thought she must have opened the doors trying to get more air. She was having one of her attacks, and she fell to the stone below. They found in the next morning. Only now these people talk about a ghost. You don't suppose she opened the window and saw it out there itself? Stop it! Not you, too. How did you get hurt? Did you fall out of the window, too? No. I was out there and I tripped and fell. Really, that's all right. Why don't you want to talk about it to the others? What is it about this house that upsets you so?  I've always hated it. My aunt and I used to fight about it. She was so wonderful. But what she'd take up with those fads, those crazy people, those crazy superstitions, that awful Professor Gifford that she went chasing after trying to impress. All this talk about ectoplasm and manifestations and... You blame me? Christina, I don't believe in such things any more than you do. But I would blame you if you kept too much to yourself.  What do you mean? I mean what you heard when I was still in the kitchen. I mean why you've come back here and what you may have seen and what you suspect about these people, about your aunt's death. I don't really know. Is this the Valentine going to stay here? Yes, Christine. Where is he now? He'll be close. He'll be right here, won't he? Of course. Just...just chatting with Miss Dowdy. She's probably showing him around the place. Now, Miss Eustis, suppose you tell me the truth about what you saw that night. Well, I... Well, my room was third-story. I'd been reading in bed. There'd been a good deal of talk at dinner about the... the ghost. Being the proper time of year for appearance and all. Naturally, I tried to be tolerant, but... Well, perhaps I did watch a little. It was supposed to appear just momentarily out back by the summer house. Moving slowly and then sort of... floating away. And, uh, did it? Well, for heaven's sake, it didn't actually... I didn't hear anything and certainly whatever it was had no substance, no body. Naturally, the next morning I didn't say anything. Why should I? In effect of all the talk, a trick of the imagination. Of course. You'd have been laughed at. You don't believe in ghosts. I believe in the kind of ghost that can be tricked up by a human being. And maybe it's being tricked up again. I sleep down one of Christine's shoes. He is torn off. Yeah. Well, that's where she fell. Stone floor of the sunken garden beneath the overhang there. Dark. Loose shutter that's on. Stone steps here have mud on the side. Roses have been watered. And mud on Christine's shoes. But if she came down there, she would have been on the path. Hold it, hold it. Yeah, here we are. I feel the footprint where she stepped off the path by the top of the steps. The heel? It's the one. All right. And that must have been how she tripped and fell down there. Can't hear us, Angel, but they will have to go back. So, well, it's a lovely place, all right. They'll run down, maybe. But if the gardens out here were fixed up, I'm sure they... Brooksy. Brooksy, look out. Shutter. One foot. Shutter, I almost... Sure. I'm sure that's what it was we heard. I'm all right, George.  Shh, quiet. But George, from above it's so dark. I look just like Christine. That must have been... I never find anybody up there. There's nobody in this house but ghost, didn't you know? Well, Angel, we're going to catch one before the ghost succeeds at murder. Yes, Mr. Valentine, I guess that's where I lost my heel, where I tripped only out... Yeah, you were coming along the path, Christine. You stepped off it, tripped and rolled down the stone steps. Yes. You were out here walking with a summer house but you didn't want anybody to know. See, it was Eustace who found you. What happened? You see him coming and running for the house? Yes, that's right, I ran. Only why did you step off the path? Well, it was a shovel or a rake or something lying across the path. Oh, I get it. Only it's not there now. I didn't look. Mr. Valentine, I did suspect something for a long time. But I didn't know what I was looking for or why or... Or whether you could find anything back here that would show you how the ghost worked that night your aunt died. That's what I mean. A whimpering, crying ghost, the thing with no substance that's supposed to appear and then float away. Aunt Betty told me where she expected to see it. Mr. Valentine! Well, listen to that, rising to the bait already. Mr. Valentine, where have you gone? Here, here, here.  Oh, I got to see you alone. I've got to. Anything you have to say, you can say right in front of her only. Say it fast. It's about the ghost. Oh, it was only to impress Gifford. You've got to realize that. It wasn't to scare anybody, that's what you're trying to say. I'm smart enough to figure that Ireland Betty didn't fall from her window out of fright on seeing the ghost. She wasn't the type. So, if it was you who played the ghost that night, lady. No, no, no. That's not what I'm trying to say. It was Ireland Betty. It was she herself who did it. What? Miss Downey. Yes, she did. Oh, I know it sounds crazy, but she had to. She'd never been able to locate the ghost in the house. And she had bragged. And she didn't want Professor Gifford to be disappointed. That's why I didn't want him to get the house. He might have fallen down. Okay, now slow down. All you're claiming is there isn't any ghost, is that it? There never has been one in this house. Only why, then? Mr. Valentine, listen to me. If Betty was out here, instead of in her room, and you think she was murdered. Of course. I know she died before she ever got back to her room. And she was probably running just like Christine here was. Yes. Only if that's the case, then... It's okay, it's okay. I'll take care of everything. Mr. Valentine. Come on, Christine. Head back to the house. See you later, Miss Downey. Well, a whole thing's pretty simple, I guess, isn't it? I don't understand at all any of it. Don't you? You understand a good deal about the ghost? But Aunt Betty told me... Sure. I was supposed to look. But nobody says they've ever heard it, except you. The whimpering, the crying. Christine, were you here in the garden that night with your aunt? Is that one you thought you heard the crying? No, I've really heard it. Someone else... Were you hiding out here? Did you pretend to leave for Canada and then come back and hide and scare her? Oh, sure. You don't believe in ghosts. You knew the little old pixie would be playing the part. So she was scared and ran, just like you did down the path. Mr. Valentine, stop it. You don't understand. What's the matter? You're afraid to walk down this path? Are you afraid to go trip on something again? A rake that you might see when you're running?  No, it wasn't a rake. I don't know what it was. I know what it was, young lady. It wasn't anything. Yes, it was. Yeah, but I know there was. Christine, I said some crimes are simple. They are after they've happened because a guilty conscience does terrible, horrible things like stumbling over something that isn't even there, except a memory. But what was there when you made your old aunt fall? A rake, a string, a wire? No. No, I didn't do it. I didn't. Then come on, walk. Perfect accident. Not so perfect a few months after, is it? When you get so scared, you get sick.   I didn't. I didn't push that shutter. Sure, go on. Say it. Fill me in. I know it was you, all right, sister? You were the only one who thought Miss Dowdy was out walking with me.  Stop it! Stop it! Stop it! Listen. I never believed it. They all did, but I never believed it. Crying. What? I don't hear anything. Yes, you can. When bring a woman crying. If it's not a real person, one of those others. It has to be the ghost. The ghost of the house. Listen. Here it is. There. Yeah.  You're the only one who does. The unbeliever. And yet the only one who's heard anything all evening. Or ever. But hasn't it occurred to you, there never was a ghost in this house until you made one. The ghost of Ireland Betty. You know there's something about crisp, tangy, autumn weather that makes coffee time more enjoyable than ever. And what better time than now to discover cream? P-R-E-A-M. The delicious new way to cream coffee. In instant powdered form, cream is made from fresh, sweet cream, milk products, and nothing else. Naturally, it has a rich, delicious flavor. What's more, cream is so convenient, so economical, you'll prefer it to all other products for creaming your coffee. You see, cream never sours, never turns. Sealed, cream keeps indefinitely, open for regular use. Cream stays fresh tasting to the last spoonful on your kitchen shelf or in your refrigerator. Budget-wise, cream saves more than one-third the cost of coffee cream. And then, something every calorie counter should know. When you use cream in your coffee, every cup contains only about half as many calories as if you used coffee cream. So try cream in your coffee. P-R-E-A-M. The new way to cream it is to pre-mit. Well, Brooksy, that's what's known as getting a confession the hard way. Oh, the poor girl's an absolute wreck. Poor girl. She killed her aunt, didn't she? Ireland Betty. Second cousin to a leprechaun. There's the one I'd like to have known. Yeah, the Irish. Believe in anything. Oh, Brooksy, there is one little item. You know, when I told you to go back in the house there, when I asked you to make some noises to help me get the story out of the girl? What? Well, I didn't make any noises. Huh? Well, I mean, first there was a police on the phone and then Mr. Eustis kept following me around. I mean, you didn't think I did, did you? You didn't hear anything, did you? Whimpering or crying? No, angel. No, of course not. I didn't hear anything. It was just in the girl's imagination, that's all. Tonight's adventure of George Valentine has been brought to you transcribed by Prem, the new dairy miracle in instant powdered form. Try it. You'll prefer it to all other products for creaming your coffee. For Prem is the most delicious, convenient, thrifty way to cream your coffee ever discovered. Yes, the new way to cream it is to cream it. Let George Do It stars Olin Suley as George and tonight Lillian Bief appeared as Brooksy. Let George Do It is written by David Victor and Jackson Gillis and directed by J. C. Lewis. Also heard in tonight's cast were Jean Bates as Christine, Florence Ravenall as Doughty, Benny Rubin as Gifford and Howard Culver as Eustace. The music was composed and presented by George Wright. Tonight's performance concludes this current series. We hope you have enjoyed it. Your announcer has been George Crowell.