Can you patent a math formula? Today I saw this video about this video game called No Man's Sky. They say Hello Games - the creater of No Man's Sky - is possibly using a math formula patented by another compony. This really surprised me, can you patent a math formula? And if so, how complex does it have to be? (Complex as in long or difficult, not the complex numbers.) Could someone for example patent the formula y = 3x - 5?
Math formulas are not patentable. Gottschalk v Benson held that mathematical formula and mere algorithms for computing numbers cannot be patented, nor can claims that are so broad as to preclude all possible uses of such a formula or algorithm. Alice v CLS bank is the Supreme Court's most recent restatement of this.
The relevant section of US patent law was already posted in an answer on one of the questions you linked, but I'll repeat it: 35 USC 271 (a): Except as otherwise provided in this title, whoever without authority makes, uses, offers to sell, or sells any patented invention, within the United States or imports into the United States any patented invention during the term of the patent therefor, infringes the patent. Merely making a patented invention is infringement; it isn't relevant whether you sell it. It is also infringement if you use it yourself, regardless of who made it or whether you paid them, or whether you would have otherwise bought it from the patent holder. There are some special exceptions for things like certain kinds of pharmaceuticals, but I browsed the rest of 35 USC Chapter 28, and didn't see anything that would appear to be relevant in this case.
Patenting does not require a working product in fields where the performance is predictable before hand. Chemistry and biology do have criteria regarding real results. The description in the patent application needs to be detailed enough that someone of ordinary skill in the field could make and use the invention. After company A gets a patent (might be 3 years or never), it can try to stop company B from making, selling, offering to sell, using, or importing whatever is claimed in the issued patent. You say "does essentially the same thing". If you accomplish the identical goal (speaker identification from noisy audio, for example) but accomplish it by following very different steps, then there may be no infringement at all.
There are no property rights on celestial bodies ... at present The Outer Space Treaty “explicitly forbids any government to claim a celestial resource such as the Moon or a planet.” And, since only governments can make such claims under international law, no one can. However, the treaty was drafted at a time when only governments could play in space and didn’t contemplate actions by non-governments. The rocket fuel will hit the fan when a corporation starts exploiting space minerals.
You need to check if the original game developers patented the mechanics/rules of the game. For your example, Monopoly was patented, but expired in the 50s and while it would not be a copyright violation to mimic the rules/mechanics, it would be a different intellectual property violation (patent infringement). Spry Fox vs LOLAPPS is also relevant as it extends copyright protection to the implementation of an idea, but ideas (like rules/mechanics) cannot, by themselves, be copyrighted. This means that there is an avenue for the original developer/company to sue even if the visual/audible aspects are different, but the overall gameplay is the same. Also see Tetris Holding LLC vs Xio Interactive Inc. as it relates to the visual aspect of the game.
If the app (and the service accessed from the app) truly doesn't have any EULA, ToS, or license agreement, to include restrictions on reverse engineering, you can probably create an alternate front end, so long as you aren't using their logos, etc. However, their data may be a different beast, depending on the nature of it. For example, extracting data from Twitter would potentially violate Twitter's license on the original text copyright held by the authors in question. Wikipedia explicitly includes redistribution in their license agreement with authors/content creators. If all you are extracting and storing is the temperature at a particular weather station, you might have less of a concern. Anything beyond merely factual runs the risk of a copyright infringement as you store the data in your own database.
No. The "pricing structure" here is essentially an algorithm. It can be patented but not copyrighted.
It is reasonable to interpret the statement in their Github repository README.md as a "public domain" license for anything contained there. However, their "usage guidelines" backpedals a bit ("generally are not copyrighted", the misleading implication that content used commercially is subject to restrictions that educational and personal uses are not subject to). Although it is true that works "created by the US government" are not protected by copyright, not everything associated with a government agency is created by the US government. An agency might have a policy that they will not post material that is not copyright-free, there is no practical means of knowing if an item is an actual government work, versus a government-supported or government-hosted work (where copyright is held by someone else). If you trust their implication that all of those items in the repository are indeed government works, then they are free of copyright. I don't know any reason to not believe them, although sometimes the government is wrong and they end up liable big-time for infringement. However... NASA Open Source Agreement Version 1.3 (another copy on a NASA web page) on first glance seems to contradict the "government work" theory. Here, they claim to grant certain rights to users and also impose impose restrictions (including obligatory registration). This does not make any sense for a work that is in the public domain. The license is legally defective in that it fails to fill in relevant blanks (agency name, title of work, URL for obligatory registration). Also notice that the license is only for software. The scope of that license therefore has to be something narrower – it applies only to software, and presumably software that is not "a government work". I have no idea what software NASA could legally give away and is not a government work which is therefore not protected by copyright.