List all the possible outcomes of tossing 4 coins, what is the sample space?
Each one has two possibilites, so the total is 2*2*2*2 = 16 TTTT TTTH TTHT TTHH THTT THTH THHT THHH HTTT HTTH HTHT HTHH HHTT HHTH HHHT HHHH If you eliminate the duplicates, ie, the coins are identical, you get 4 TTTT TTTH TTHH THHH HHHH .
How do math geniuses understand extremely hard math concepts so quickly?
Some years ago I was doing a penetration test for a large mobile operator's voicemail infrastructure. The test in itself took about 5 minutes from the moment I started till the moment I was able to get into the system and change the default welcome message to something of my own. I presented my findings to the local manager in charge and, surprisingly, her reaction was - "so you expect us to pay you 2000€ for 5 minutes of work?" My answer, although a bit of a cliche, was this: "You're not paying for those 5 minutes. You're paying for the amount of knowledge gathered over the years that allowed me to figure out in less than a minute what's the problem with your voicemail." As Satvik Beri perfectly described in his answer, when you spend most of your time focusing on a specific issue, whether that's math, information security, programming or anything else - you gather enough experience to start figuring out things in a snap. That doesn't make you special, nor innately talented - just a really hard worker and extremely passionate about what you do. Think about that the next time you feel cheated by a consultant who solves your problem in 5 minutes.
Calculus and vectors help needed?
As simple as possible ... 2x+y+z=4 … (i) x-y+z=p … (ii) 4x+qy+z=2 … (iii) Generally, 3 planes meet in a point. So for any old values of p & q we could usually solve these equations for unique values of x, y, z. However, if we want the 3 planes to meet in a line rather than a point, this cannot happen. So we try to solve the equations in the normal way and see what conditions we need to impose on p & q to prevent a unique solution. (i) − (ii) eliminates z to give x+2y=4−p … (iv) (iii) − (i) eliminates z to give 2x+(q−1)y=−2 … (v) (v) − 2(iv) eliminates x to give (q−5)y=2(p−5) Now if q≠5 we can solve this for a unique y which from (v) and (i) leads to a unique x & z. Hence to avoid a unique solution we must have q=5 leaving above as 0=2(p−5). For this to be non-contradictory we must also have p=5.
In the film “Arrival”, how is it possible that learning an alien language alters Louise's perception of time and she can see glimpses of the future?
The ultimate point is that the future really does already exist. The aliens perceive the future and perform their roles in a story they already know. Humans perceive only the past and present. See Jennifer Hu’s answer for the physics that is used to justify this.The aliens’ written language embodies their understanding. A single stroke is part of multiple “words” and this implies that the aliens know the entire story they are going to write, stroke for stroke, before making the first stroke. As Louise learns to read and especially to write the alien language, she starts to see past, present, future, subjects, objects, nouns, verbs, and modifiers as all simultaneous. When her understanding changes, her perception of time changes. Large blocks of memories of her future start to come to her.Do I believe this is possible? No. The fact that equations can be rewritten in forms that appear to be time-independent does not make the world time-independent. I don’t think the formatting of equations says any more about reality than our ability to conceive unicorns. But it makes for a fascinating story.Note: This whole answer is based on Ted Chiang’s novella, “The Story of Your Life,” rather than the movie.