Actually, there is a common-sense way of interpreting this.
e is often introduced in terms of compound interest. If I put $1 into a bank, and the bank pays 100% interest a year, how much do i have after a year? Well, if the bank pays interest once a year, I have (1+1/1)1 = $2. If the bank pays twice a year, I have (1+1/2)2 = $2.25. Three times a year and it's (1+1/3)3 = $2.37... In the limit where the bank is paying me interest continuously, then after one year I have $2.71..., and we call this number e.
Now let's run this question in the other direction. What if the bank pays interest every two years? Now I only get (1+2)1/2 = $1.73..., and if it only pays every three years I get (1+3)1/3 = $1.59... What happens in the limit when the bank never pays interest? Well, without doing any maths at all, it should be pretty self-evident that I still have the dollar I started with.
There isn't, as far as I'm aware, a particularly efficient way to cut down the candidates for Mersenne primes. GIMPS, I think, tests candidates for relatively small prime factors (which can be optimised a little by the fact that any factor of 2^p-1 must be one more than a multiple of p). But the majority of the computational work is performing the Lucas-Lehmer test on very large numbers. In order to check if 2^p-1 is prime, you have to multiply together two numbers of the order of 2^p, and you have to do it p times. Overall that is of the order of p^3 bit operations of the sort a logic gate might perform (I think). So when you get up to p being 10^8, you have to do 10^24 calculations. Which is far, far less than the 2^(5*10^7) calculations you'd need for trial division, but still quite large.
When an encode is available, I will watch this. I'm sure the technical quality will be excellent - and I am always intrigued when a Pokemon you might not expect ends up being the fastest. I would never have thought of Kadabra. However...
As someone who has watched a lot of real-time Pokemon runs, this just seems like such an arbitrary category to me. To allow some applications of the tweaking glitch but ban others doesn't sit right with me. I would have preferred it if a run like this had been done on Pokemon Platinum, where the skip-to-the-end void tweak is impossible anyway. If it does end up not getting published, I'm sure it will be a great gruefood delight entry.
As for lower bounds, I wonder whether there is some clever lower bound you can get by ascribing values to each cell on the square, and demonstrating that no line can pass through more than a certain total value. You can get a very weak lower bound by giving each cell a value of one (ie "count the squares"), and no line can pass through more than 2n-1 squares which gives a weak lower bound of (n+1)/2. I am sure with a smarter weighting function this can be improved.
Lol. I was completely wrong as usual :) I had convinced myself it was totally not possible to cover all the squares in a 3x3 with two lines. And, amazingly, I had pretty much discovered your solution except one of my lines was not "optimal", to use JXQ's terminology.
JXQ: I haven't been able to find a way to cover every cell in an nxn grid with n-1 straight lines, but it's pretty easy to see that you can cover every cell but one. Just start with the configuration you mentioned, each line piercing a single column of squares. Now tilt the first line slightly so it pierces the top square in the second column, then tilt the second line a little more so it misses the top square in its own column, but pierces the top two squares of the third column, and tilt each line progressively more, until the (n-1)th line covers n-1 squares in the nth column - ie, all but the bottom corner.
Trying to pierce all the squares with diagonal lines (by which I mean, ones at a perfect 45-degree angle) also leaves one square. You can do this either with parallel diagonal lines or two sets of diagonals which are mutually perpendicular and are cleverly interwoven - but every time, one square is left behind.
I'm not sure it's possible. I'm pretty certain that if an example did exist, it would have lines which crossed each other - but of course, this introduces inefficiencies to the problem, as some squares would necessarily be pierced more than once.
For those who want to remain hopeful about time travel, it's worth remembering that when Karl Schwarzschild first did the maths that predicted the existence of black holes, he expected them to be solely a theoretical curiosity that would not actually exist in our universe. He was wrong about that, so maybe we're wrong about wormholes, too?
I guess the difference is that it's a lot harder to see a mechanism for how a wormhole might develop. A black hole is just large amount of matter which has collapsed under its own gravity - simple. But a wormhole is a far more complicated structure.
On another topic, it's estimated that the photons that arrive to Earth from the Sun are actually tens of thousands of years old, because it takes that much for those photons to reach the surface of the Sun from its core.
But this seems to imply that every single photon created in the Sun is created in its core. One would think that eg. the solar corona emits boatloads of photons itself. Is it really true that all photons from the sun are tens of thousands of years old, or would it be more accurate to say that some of them are?
Most of the photons are emitted deep within the core, simply because it's hotter there (over a thousand times hotter in the sun's core as compared to its still-very-hot-by-everyday-standards surface), and you need those incredibly high energies to power the nuclear fusion within the sun (which is what emits the photons).
There might be some emitted from nearer the surface, from things like the decay of free neutrons, but I would imagine the proportion would be insignificant.
Of course a much more mundane and closer-to-life example would be: A photon is emitted from the Sun. At that exact moment it has an unobstructed path towards the Earth. However, while it's on its way, something gets in its way, blocking it. Thus it hits the blocking object rather than arriving at the Earth.
But from the photon's own perspective this all happened in 0 seconds, which seems paradoxical. The moment that it left the Sun, and the moment it hits the obstruction, are the exact same moment. (The distance between the Sun and the object is zero, no matter how far apart they are from an sub-lightspeed point of view...)
The emission of the photon and the the imposition of the obstacle are separated by, to use a bit of jargon, a spacelike interval. (Spacelike means the distance between the two events is greater than the distance light could travel in the time between them). Spacelike intervals have two key properties: they are not related by causality, and the chronological order in which the events happens is frame-dependent. So, you are right that the light ray feels no time at all in its own frame, but, from the light ray's perspective, the barrier is put in before it sets off - which resolves the paradox.
Ah, I've spotted the mistake I made!
I set up the problem in terms of two variables, a and b. But when it came to the calculus, I failed to remember that a and b are not independent of each other, so I have to express a in terms of b before proceeding. Instead I just ignored a when differentiating with respect to b.
When I do that, I do indeed get to 1.635. Are we all agreed on this probably being the best number?
So what I did was broadly similar - I'm certainly assuming the optimal solution is a point somewhere in the middle with three straight lines coming out of it.
I also assumed that the correct solution has reflectional symmetry across one of the square's midlines. Using these constraints I was able to get the problem down to single degree of freedom which I could then optimise with calculus - but I must have got the calculus wrong as I could only get the number down to 1.641 (to three decimal places).
Optimality proofs will be through some calculus-of-variations methods similar to those that solved the isoperimetric problem - but I left university a while ago and can't really remember anything about the calculus of variations.
Is it the case, then, that the sine of any rational number (or, since we really ought to be talking in radians, a rational multiple of pi) is an algebraic number? And if so, is there an algorithm that can calculate it?
A few years ago, I would have definitely recommended TV Tropes, but unfortunately, the site fell victim to bowdlerisation. They started being very careful to not offend advertisers, making everything lighter and softer. This ruined many of the best pages. So I would say that the site jumped the shark.
When money is involved, using technological forms of cheating may be considered fraud.
But again, why is this considered "cheating"? Yes, the casinos won't want you to do it, are perfectly allowed to blacklist you, and may take you into the back rooms for an informal scrap, but I don't see what here is against the casino's rules. You aren't damaging or tampering with the machine. You're just playing the game, following an algorithm that gives you better than random chance. Unless "cheating" is defined to be anything that allows you to beat random chance, in which case the casino always wins by definition because any method of beating the casino is ipso facto cheating.
The Charles Ingram case is quite clearly different. It's clear that in Who Wants To Be A Millionaire? the contestant isn't allowed to be passed outside information. Someone watching the show, Googling answers and feeding them through an earpiece, for example, would clearly be wrong. But slot machines are different - there is no "outside information". The only "information" is the machine's PRNG, very clearly "inside information" - and even this was not stolen, it was merely deduced from careful observation.
Might be a controversial opinion, but whenever I hear a story like this I feel that the casino owners are getting what they deserve. Casinos make big profits from human vulnerabilities, so I'm not too bothered when a human finds an ingenious way of exploiting a casino's vulnerability. The rules are stacked so far in the casinos' favour that anyone who finds a way to beat them deserves an immense amount of respect.
Arctan is normally defined to take a value between -pi/2 and pi/2. It can also be defined in a number of other ways, such as the integral of 1/(1+x^2), which remove the ambiguity. In principle you could choose any other principal branch and you'd be fine as long as you applied it consistently, but it seems a lot more natural to take the branch covering x=0 as the principal branch.
0/0 is different. It really is undefined. There is no real concept of branches of that "function" because its solutions are infinitesimally close together.
I do wish more was taught about the history of communism at school, but the problem is since most of the people who set the syllabus in our schools are sympathetic to the ideals of communism.
You can be sympathetic to the ideals of communism while reviling what Stalin did - such a position is called socialism. In practice the socialist/communist divide is a pretty grey area, and the water is muddied further by the large number of people around these days calling themselves socialists who are really social democrats.
The problem with communism is that the enormous level of control a government requires to enact it takes away nearly all of its people's freedoms, including their right to resist. People in capitalist societies may sometimes find themselves oppressed by market forces, but at least they have some collective agency. They can join a union, they can campaign for a better society, they can use their ballots to vote for change; but people oppressed by their own governments have no such power. There is a great anti-authoritarian tradition in left-wing thinking, of which I guess George Orwell is the most recognisable figure. Sadly, the modern left is rather more sanguine about totalitarian regimes as long as they come from the left. Just the other day I read an article pinning all the blame for the current tragedy in Venezuela on the west, which had been supposedly working to undermine Venezuela's sovereignty and bring down the Chavista revolution - all the usual tropes of the far-left conspiracy theorist were there, at one point it even called one of the Venezuelan opposition leaders a Zionist. It was all a pretty disgusting denial of the inconvenient truth.
/rant over