Rocky has 31 days to prepare for his Discrete Math final. He has decided to do at least one tough problem each day, but no more than 50 problems total. Show there must be a sequence of consecutive days in which he does exactly 11 problems.
From  by Richard Brualdi  We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. To make sure that he's able to take a couple of breaks, he also decides that he wont play more than 12 games per week.  My question is, how do you prove/dis-prove that there will be a succession of (consecutive) days in which he will play k games, $k \geq  22$?  Specifically, I don't quite understand the use of the pigeonhole principle here.  Thanks in advance! :D  EDIT  I'll try to clarify what I'm asking:  The goal is to prove that for every possible practice schedule that the chess master makes, there will always be a consecutive sequence of days such that he plays exactly k games.  So for example, if k = 22, how do you prove that no matter what his schedule is like, he will always play exactly k games in r days, $1 \leq r\leq  77$.  My main question is: what is the general method for proving different values of k? Are there values of k (from 1 to 132 inclusive) that are impossible to obtain?  Note that he does not necessarily have to play 132 games within 77 days! Also, keep in mind the bounds of k.
Why is it true that every finite-dimensional inner product space is a Hilbert space? A Hilbert space is, by definition, a complete inner product space. If $(V,|.|)$ is finite dimensional inner product space of dimension $n$ then it is (topologically) isomorphic to $\mathbb{R}^n$ which is of course complete. My instinct here is to say "and therefore, $V$ is also a Hilbert space". I'm not sure about this step however since completeness depends on the norm and the norm depends on the selected inner product (assuming the induced norm $||x|| = (x | x)^{1/2}$ is used). So, must we place some condition on the inner product or is the topological isomorphism between $\mathbb{R}^n$ enough to guarantee that V is Hilbert? Terminal not opening up after upgrading python to 3.7 I have upgraded python to version 3.7 and now my terminal is not opening up.  I followed this post   Commands I ran are as follows,  sudo apt-get install python3.7 sudo update-alternatives --install /usr/bin/python3 python3 /usr/bin/python3.6 1 sudo update-alternatives --install /usr/bin/python3 python3 /usr/bin/python3.7 2 sudo update-alternatives --config python3   Entered 2 for python3.7  I can't open terminal and run commands.  My ubuntu version is, Ubuntu 18.04.2 LTS Identify a catastrophic sci fi story written before 1980: ruptured spleen, alien moon base, time and space travel, antimatter tool I read a sci fi short story sometime between 1974-1980 but it may have been written long before that.  I believe it was in a paper back anthology of sci fi stories, but I'm not certain.  I think the protagonist is facing the final climactic conflict when we meet him.  Most of the story is memories and flashbacks.  The first flashback is to his childhood.  He and his younger brother are sled riding.  His brother crashes, ruptures his spleen and dies.  He feels responsible and is wracked with guilt.  Much later, as an adult, he is exploring a long-deserted alien base on the moon.  There are unfamiliar artifacts on the base, including a time-travel device/vehicle, and a tool with an anti-matter tip.  It carves neat holes and eliminates matter simply by moving it through matter.  He accidentally drops the tool, which plummets unimpeded to the center of the moon.  He realizes that over time, it will consume and destroy the moon, which will be catastrophic for Earth.  He uses the time-travel device to travel to the moon at an earlier time so that he can prevent himself from dropping the tool.  He makes bigger problems for himself, forcing him to travel back further and further in time.  He travels so far back that he encounters the original alien inhabitants of the base.  Now we are at the present time in the story.  This is his final conflict, he has to escape, travel forward in time and resolve all of the problems he created.  Unfortunately, the alien inhabitants capture him, and the story ends.  Can anyone identify the story and author? what is/are the purpose(s) of inline? Durrett Example 1.9 - Pairwise independence does not imply mutual independence? The example in question is from Rick Durrett's "Elementrary Probability for Applications", and the setup is something like this:     Let $A$ be the event "Alice and Betty have the same birthday", $B$ be the event "Betty and Carol have the same birthday", and $C$ be the event "Carol and Alice have the same birthday".   Durrett goes on to demonstrate that each pair is independent, since for example,   $P(A \cap B) = P(A)P(B)$.   However, he concludes that $A, B$, and $C$ are not independent, since   $P(A \cap B \cap C) = \frac{1}{365^2} \neq \frac{1}{365^3} = P(A)P(B)P(C)$.  I understand the reasoning here, and that one can generally show that arbitrary events $X$ and $Y$ are not independent by showing that $P(X\cap Y) \neq P(X)P(Y)$.   I am a little new to probability, though, and don't understand why exactly $P(A \cap B \cap C) = \frac{1}{365^2}$.   My progress so far:   I do see why $P(A) = P(B) =P(C) = \frac{1}{365}$, and thus why $P(A)P(B)P(C) = \frac{1}{365^3}$.  It seems like the sample space $\Omega = \{ (a, b, c) \mid a,b,c \in [365] \}$ -- i.e., all of the possible triples of numbers from 1 to 365, where 1 denotes January 1st, 2 denotes January 2nd, etc. From that, I can conclude $|\Omega| = 365^3$, but I'm not sure where to go from here.   It seems like once a single birthday is chosen, the rest are completely determined if they're all equal to each other - is this a good direction to go in? How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ? I know that one can use "they" / "them" / "their" in place of "he" / "him" / "his" or "she" / "her" / "hers" when the subject's / direct object's gender is not known; for instance, just looking at the default user's about me section on Stack Exchange sites:     Apparently, this user prefers to keep an air of mystery about them   My question is what happens when someone wants to reference the subject using a pronoun instead.  Should the verb be conjugated accordingly to the pronoun? E.g.:     Apparently, they prefer to keep an air of mystery about them   Or should the conjugation be retained? E.g.:     Apparently, they prefers to keep an air of mystery about them   The former looks unclear because it wouldn't allow the reader / listener to understand if the writer / speaker is talking about a single person or about a group of people, however the latter sounds very weird, at least just to the non-native speaker which I am. Setting up PyCharm to reference ArcPy? How would I setup  to recognize import arcpy?   I'd like to use PyCharm to develop and debug geoprocessing tool. This question is a  of the common question asking for calculus references. It is here to  away the repetition, and give a canonical resource for calculus references.      I'm looking for a resources to learn single-variable calculus. How can multithreading speed up an application (when threads can't run concurrently)? Comment font sizes are inconsistent on Android Terminal returns not found for most commands Mac OSX Variation of Pythagorean triplets: $x^2+y^2 = z^3$ I need to prove that the equation $x^2 + y^2  = z^3$ has infinitely many solutions for positive $x, y$ and $z$. I got to as far as $4^3 = 8^2$ but that seems to be of no help.  Can some one help me with it? At what speed does our universe expand? Lie bracket for Lie algebra of $SO(n,m)$ Salesforce, Oracle and the Seven Dwarfs Over my time as a Salesforce developer, I've written some code which on occasion managed to surface Oracle plsql exceptions and curiously the error text contains the name of one of Disney's .  I've teased out Doc, Grumpy, Happy, Bashful and Sneezy. I'm sure the other two are not far away.   What is the underlying "thing" on the database platform that is named after a Dwarf? Why were these named after Dwarves?      java.sql.SQLException: ORA-30006: resource busy; acquire with WAIT   timeout expiredORA-06512: at "DOC.CAPEX", line 2200ORA-06512: at line   1: SQLException while executing plsql statement: {call   cApex.update_sysmodstamp(?)}([       java.sql.SQLException: ORA-30006: resource busy; acquire with WAIT   timeout expiredORA-06512: at "GRUMPY.CAPEX", line 2563ORA-06512: at   line 1: SQLException while executing plsql statement: {call   cApex.update_sysmodstamp(?)}([       java.sql.SQLException: ORA-30006: resource busy; acquire with WAIT   timeout expiredORA-06512: at "HAPPY.CAPEX", line 2715ORA-06512: at   line 1: SQLException while executing plsql statement: {call   cApex.update_sysmodstamp(?)}([       java.sql.SQLException: ORA-30006: resource busy; acquire with WAIT   timeout expiredORA-06512: at "BASHFUL.CAPEX", line 2715ORA-06512: at   line 1: SQLException while executing plsql statement: {call   cApex.update_sysmodstamp(?)}([       java.sql.SQLException: ORA-30006: resource busy; acquire with WAIT   timeout expiredORA-06512: at "SNEEZY.CAPEX", line 2718ORA-06512: at   line 1: SQLException while executing plsql statement: {call   cApex.invalidate_agc(?,?)}([      Error: ORA-20000: ORA-06512: at "SNEEZY.SIMPORTLOOKUP", line 850   ORA-06512: at "SNEEZY.SIMPORTLOOKUP", line 823 ORA-06512: at line 1 :   SQLException while executing plsql statement: {call   sImportLookup.get_by_ext_id_text(?,?,?,?,?,?,?)}([ What's the difference between the 'ref' and 'out' keywords? I'm creating a function where I need to pass an object so that it can be modified by the function. What is the difference between:  public void myFunction(ref MyClass someClass)   and  public void myFunction(out MyClass someClass)   Which should I use and why? Evaluating $\int \frac{\sin\left(x\right)}{1+x^2}dx$ $$\int \frac{\sin\left(x\right)}{1+x^2}dx$$  I have tried to integrate by parts but it doesn't work. How do I evaluate it? Any advice, hint or well-thought solution will be appreciated. Aliens abduct mathematician, astronomer, woman Bits I remember about this SF movie which probably originated in late 50s / early 60s:   Humans are flying in a light, general aviation, high winged, single engine aircraft. An alien in a flying saucer abducts the whole kit &amp; kaboodle (plane &amp; passengers). Takes them to a secret terrestrial base Eventually tells them that his computer is damaged and he needs their help to get back to his planet. He needs an astronomer for planetary masses &amp; other information. He needs a mathematician to calculate trajectories and perform other calculations. He needs the woman as eye candy, lol They eventually make it back to the alien home planet which is under attack by its enemy. The attacking aliens are bombarding the original alien's home planet with meteors / asteroids.   That's about all I remember. I'm looking to view and archive all my old DMs in Twitter, since I would like to revive conversations that took place 1 year ago.  Acording to Twitter they don't delete old messages, they simply archive them in their database.  Is there a way to view and download all of these messages, just like you can view all your old public tweets with third party apps?  UPDATE: I found  but apparently it's not working for DMs now