This question has been always intrigued me. Why proof by Induction is admissible/acceptable ? We use this type of proof nearly every where and some times It's the easiest way of proof.
What makes induction (over natural numbers) a valid proof technique? Is $$ \dfrac{ P(0) \quad \forall i \in \mathbb{N}. P(i) \Rightarrow P(i+1) }{ \forall n \in \mathbb{N}.  P(n)} $$ just taken for granted as a proof rule, or can it be derived from more foundational axioms?  Similarly, can the principle of induction over well-founded sets be derived from something more foundational, or does it have to be assumed to be a valid inference rule?
Decorate Tables You have $r$ red, $g$ green and $b$ blue balloons. To decorate a single table for the banquet you need exactly three balloons. Three balloons attached to some table shouldn't have the same color. What maximum number $t$ of tables can be decorated if we know number of balloons of each color?  I know the answer will at least be $\min(r,g,b)$. Also we need to add tables with $2$ reds, $2$ blues or $2$ greens. I cannot think of the formula for it PHP error in .xml file? How do I run an executable file with a double click? I'm trying to run an executable (testdisk) but double click doesn't work. The only way is to run:  sudo ./testdisk   from terminal. But I'd like to open executables with double click in future. How can I do this? Finding centrelines from polygons in QGIS? I was wondering if there is a way to create centre lines from polygons in QGIS? I have a polygon layer, but ideally need the centrelines as a separate file. How to split the terminal into more than one "view"? I try to press the F6 key and in the YouTube tutorials those who make the video pressing them all go well, but not me :( Suppose I have 30 numbers that vary between 0 and 1.0 and which sum to 1.0.  The mean is obviously 0.033.  A client wants these scaled to lie between 0 and 1.0 but to have a mean of 0.5.  By the way, I'll probably have to do this for any set of numbers (positive, negative, outside the range 0 - 1) so that they lie in the range 0 - 1 and have a mean 0.5.  Any suggestions? How to make applications open on the correct monitor when using multiple monitors? I'm using Windows 8.1, which has much better multiple monitor support than Windows 7 (sort of). However when I'm in a full-screen app on one monitor, and I go to the Start menu on the other and open a desktop app, it opens on the opposite monitor because of it being on that one last time the app was used.  This suspends the full-screen app and switches to the desktop view on that monitor. I then have to drag the desktop app to the other monitor which I didn't have my mouse pointer near in the first place, then switch back to the full-screen app on that monitor by going to the upper-right corner.  This is so incredibly irritating, every time I plug in my laptop to my second monitor and try to use it, this happens. Not to mention whenever I have Netflix or any kind of video in the second monitor and it just randomly stops playing when I'm trying to work on the other screen.  Most likely there is going to have to be a third-party application to install to do this, right? I would really prefer if it was free, as this was less annoying in Windows 7 when the application would only open on top of the other that was on that monitor. This is one of the maybe two annoying things I don't like about Windows 8/8.1. Everything else is fantastic, to be honest. Has the Magical Community breached the Final Frontier? (aka Wizards in Space?) Have they? Given that Muggles put the first man in space in 1961, even by 1991 a lot of Muggleborns perfectly aware of the space-travel would have passed through Hogwarts and into the Ministry, even the Department of Mysteries (which seems to be a bit like their research division/Area 51 combo) And even leaving Muggle technology (which Wizards don't care for or about) aside, Humans have wondered about air travel to &quot;other worlds&quot; and meeting aliens from the time of the Ancient Greeks, surely some Witch or Wizard somewhere might have wondered if it really was possible to fly to the moon? So, is there any canon (aka Books + Pottermore + Rowling interviews) evidence of the Wizarding community having ever travelled or trying to travel to space / other planets or satellites? ETA: Note, I am also totally okay (and in fact would be eager for) historical &quot;Famous Wizard&quot; card-like tidbits from canon-sources.  AWESOME example (that I'm frankly not expecting): When the Muggles managed to land on the Moon, the Department of Mysteries had to scramble to hide the Silver-backed Moon Dragon conservatory they had been maintaining near the landing site. ALSO AWESOME example (that I am hoping for): Warlock Flibberti and Witch Tigget misused this earth-based spell and amusingly found themselves on the moon with their pet monkey instead! I wish to familiarize myself with the capabilities of QGIS up to and beyond QGIS version 3.0.   Can you recommend some good web resources and tutorials that would show me QGIS' capabilities?   I am interested in both using QGIS through its GUI and in developing applications using QGIS. Group of order $8p$ is solvable, for any prime $p$ Consider the following question:     Show that a group $G$ of order $8p$ is solvable, for any prime $p$.   I am kind of stuck, but here are my first attempts:  I chose the series of subroups $G&gt;H_8&gt;H_4&gt;H_2&gt;H_1=\{e\}$ where $H_k$ has order $k$. All of these subgroups exist due to Sylow's Theorems. We have the quotients $H_8/H_4\cong H_4/H_2 \cong H_2/H_1 \cong \mathbb Z/2\mathbb Z$, so $G&gt;H_8\rhd H_4\rhd H_2\rhd H_1=\{e\}$. Only the factor $G/H_8 \cong \mathbb Z/p\mathbb Z$ is causing me a headache, because although its of prime order (and hence abelian) I don't know whether $H_8$ is normal in $G$ (unless $p=2$).  If $p\ne 2$, then the number $k$ of Sylow $2$-subgroups is $1$ mod $2$. Since $k$ divides $|G|$ we might have $k=1$ or $k=p$. If we had $k=1$ then $H_8$ would be the only Sylow $2$-subgroup (and hence normal in $G$). But how could we show this?  Or is there even an easier way to approach this problem?  $\ $  edit: I am going to try a different approach: The case $p=2$ is clear. When $p=3$ or $p=7$ the group has order 24 or 56 and since the smallest simple non-abelian group has order 60, the group must also be solvable in these exceptional cases (as Mariano pointed out).  In any other case we get a Sylow $p$-subgroup $H_p$ of order $p$, which is is normal in $G$. The quotient $G/H_p$ has order $8=2^3$; and prime power order implies solvable. So $G/H_p$ is solvable and $H_p$ is also solvable. And since the composition factors of $G$ are those of $H_p$ together with those of $G/H_p$, we conclude that $G$ is solvable iff $H_p$ and $G/H_p$ are solvable, which is the case. Hence $G$ is solvable if $|G|=8p$. Do any queries exist that require RIGHT JOIN, or can they always be re-written with LEFT JOIN?  And more specifically, how do you re-write this one without the right join (and I guess implicitly without any subqueries or other fanciness):   SELECT * FROM t1 LEFT JOIN t2 ON t1.k2 = t2.k2 RIGHT JOIN t3 ON t3.k3 = t2.k3 Why are old initrd files of uninstalled kernels filling up /boot partition? Thunderbird - remove unused dictionaries from menu Convert .bbl file to .bib file How can I convert a .bbl file to a .bib file. I  lost my .bib file but I still have my .bbl file. Now , I need to add more references to my .bbl file according to a specific format which I need to produce using Bibtex, so how can I convert my .bbl file to a .bib file so I can add the new references according to the format.  Thanks for any insight. Where do you go for the 'nearest railroad' in Monopoly? In the American edition of Monopoly, there is a Chance card that says:  Advance token to nearest railroad Pay owner twice the rental to which he/she is entitled. If railroad is unowned, you may buy it from the bank.   However, is this the closest railroad, or the next one? If you are standing on the Chance spot in the light blues, do you go all around the board to Reading, which is the closest to you, or do you go to Pennsylvania, which is the first one you land on? Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. Sheet Specific Scripts I'm currently using the below script to timestamp when an edit is made to column 7 on a sheet named "Sheet 1".  function onEdit(e) {    if ([7].indexOf(e.range.columnStart) != -1) {    e.range.offset(0, 7).setValue(new Date()).setNumberFormat("dd.MM.yyyy at HH:mm");    }   }     I have a second sheet "Sheet 2" which I need to run the same script on but against column 5.  function onEdit(e) {    if ([5].indexOf(e.range.columnStart) != -1) {    e.range.offset(0, 7).setValue(new Date()).setNumberFormat("dd.MM.yyyy at HH:mm");    }   }     I've tried .getSheetByName("Sheet1") but just cant seem to get it working correctly. Which are the spec bits in this cache? I have a direct mapped cache of size S with the line size L. The cache is physically indexed and tagged. The physical address is 50 bits, numbered from 0 to 49 (with 0 being the least significant bit). The machine has a word size of 4 bits and the memory is byte-addressable. Which bits of the 50 bits are compared with the address tag? How do I deal with a compromised server?