We're playing a game. I flip a coin. If it lands on Tails, I flip it again. If it lands on Heads, the game ends.

You win if the game ends on an even turn, and lose otherwise.

Define the following events:

A: You win the game

B: The game goes on for at least 4 turns

C: The game goes on for at least 5 turns

What are P(A), P(B), and P(C)? Are A and B independent? How about A and C?

Consider the function defined by y = x^(sin(x)^sin(x)). Observe its graph. Find an increasing function which passes through each of its local maximums, and another increasing function which passes through each of its local minimums.

Extra credit: You'll notice the graph isn't drawn for x-values which make sin(x) negative. This is because most of those values make the function undefined - though it is defined for infinitely many points in those intervals, it just also has infinitely many holes. Since it lacks continuity here, it has no true local maxes or local mins, and doesn't impact the original problem. We can nonetheless cheat and fill in the holes by expanding the function to these regions with y = x^|sin(x)|^sin(x) (Using x^-|sin(x)|^sin(x) should also be technically valid, but is being ignored because it's discontinuous with the rest of the graph and not as pretty, but will be mentioned in my solution). Doing so adds more local maxes and local mins. The new local mins should line up with your function that finds the local maxes for the original function - but, find a new function which hits all of the new local maxes.

(x/5)^log_b(5) - (x/6)^log_b(6) = 0

Index of my unnamed series of posted problems

An 8x5 rectangle. If the bottom left corner is considered (0, 0), then two lines are drawn within the rectangle, from (0, 4) to (8, 1) and from (1, 5) to (7, 0). The smaller two regions of the four these lines cut the rectangle into are shaded. What is their combined area?

The image is of a large unit square with five smaller disjoint shaded squares contained entirely within it. The five smaller squares are congruent. Four of them are at each corner of the large square. The fifth is in the center, rotated diagonally, so the center of each of its sides is touched by the vertex from one of the other four squares. You are given that the common length for the five smaller congruent squares is (a-sqrt(2)) / b, where a and b are positive integers. What is the value of a + b?

I read this article and still walked away feeling like I didn't understand the situation that well.

Is it $56 billion that he's already been paid, and he needs to return it? $56 billion he's partially been paid, and he can keep what he has, but won't get the rest? Something more complicated?

For example - if a popular TV show is about to have its season or series finale, or a sport league is about to have its championship game. Are there any websites that track these, without all the noise of less important shows or games, to keep track of?

ESPN.com does seem to track upcoming sporting events pretty well, but it's not that easy to tell which upcoming games are "big" for the league in question or not.

It seems like every shower has its own unique way of controlling water temperature and pressure. Of all the showers I've ever used, no two of which have ever been alike, I like my controls the least. Plus the faucet has started dripping lately.

Is this likely to be something I can replace on my own, without a plumber? To me, that means: Can I likely do this without damaging the wall, without having to mess with pipes, and without needing to do anything involving words like "hacksaw", "weld", or "plumbing torch"?

Basically I believe in my ability to buy a faucet and control thingie from Home Depot; to use screwdrivers, allen wrenches, pliers, and regular wrenches; to use things like plumbing tape, lubricants, and caulk; and to remember to turn the water off to the house.

Would a project like this likely require anything more complex than that? I tend to prefer shower controls that have separate knobs for hot and cold, but I figure going from a one-knob setup to a two-knob setup is definitely going to require reconfiguring the plumbing. Should sticking with a one-knob solution be okay?

I don't know if it matters but I live in Florida in the US, and this place was built in the 1980s. I doubt this matters, but my current controls work by turning the larger knob left or right for temperature, and the smaller knob for pressure.

My place does have some annoyances - like the front door is an uncommon size that's difficult to find replacements for at places like Home Depot. Is there any chance of me running into issues like that when it comes to things like the size of the pipe openings?

Thanks for any insight.

Edit: Thanks for all the replies. It's pretty clear now that this is something that could very easily end up a lot more involved and time consuming and property damaging than I'm comfortable with.

I'm mostly thinking about insurance here. I've been told conflicting information. I live in Florida.

I live with someone who has a driver's license and a car, but I don't have either. I've avoided getting one because I have no interest in car ownership, and I feel like if I started driving regularly I'd probably die - I have driven before but I really don't think it's something I'd ever get good at.

It's undeniable that having one would be convenient though - for rare occasions like emergencies at a minimum but also other scenarios.

I know almost nothing about how this stuff works. If I get a license, am I required to acquire and pay for insurance, even if I don't own a car or regularly drive? Or will the person I live with have to pay more for their insurance? Are there any other costs or downsides associated with it that I might not be thinking of?

Thanks.

Hollow Knight is an incredibly competent game on pretty much all fronts. In my opinion, Hollow Knight is a masterpiece and we will discuss all the things that make it so great in detail in this Hollow Knight retrospective. However - the one thing that I find most fascinating about Hollow Knight is an aspect of the game that is seldom discussed. The most fascinating aspect of Hollow Knight is that it exists in the first place - because to put it mildly - Hollow Knight’s existence should not be possible. Hollow Knight was developed by only three people, in roughly three years. Three people managed to produce a game that looks this beautiful, features a combat system with a skill ceiling this high, crafted a beautiful world this big, wrote lore that deep, and crafted gameplay this fluid. Three people did a job that puts most teams of 60 people that work for half a decade on a game to shame. How did three pull this off?

In this Hollow Knight retrospective, we are going to investigate this question. We are going to find out how it is possible that Hollow Knight even exists!

In late July, a couple of startling papers appeared on the arXiv, a repository of pre-peer-review manuscripts on topics in physics and astronomy. The papers claim to describe the synthesis of a material that is not only able to superconduct above room temperature, but also above the boiling point of water. And it does so at normal atmospheric pressures.

Instead of having to build upon years of work with exotic materials that only work under extreme conditions, the papers seem to describe a material that could be made via some relatively straightforward chemistry and would work if you set it on your desk. It was like finding a shortcut to a material that would revolutionize society.

The perfect time to write an article on those results would be when they've been confirmed by multiple labs. But these are not perfect times. Instead, rumors seem to be flying daily about possible confirmation, confusing and contradictory results, and informed discussions of why this material either should or shouldn't work.

In this article, we'll explain where things stand and why getting to a place of clarity will be challenging, even if these claims are right.