There are two ways that spells are rolled for in D&amp;D 5e, depending on the spell: attack rolls rolled by the players and saving throws rolled by the DM.  But as far as I can see, saving throws could be made as attack rolls instead. I know D&amp;D 4e did it this way, with players making attacks against AC/Fortitude/Reflex/Will Defenses instead of the DM making Fort/Ref/Will saving throws and players making attack rolls against AC.  I don't really see why the designers of D&amp;D 5e would choose to have two different ways to roll something that, as far as I can tell, could be simplified into one mechanic (and ), so I wondered if I'm missing something important that distinguishes the two methods.  I guess it boils down to: Is there any difference in probabilities or practical concerns between one or the other method, or are they interchangeable?  This is related to  and  This question is focused on if there is any practical difference, regardless of why they actually made that design decision.
In D&amp;D 4E as with many RPGs, characters have a fixed armour value or AC. When attacking, the attacker must roll a die to overcome the target's AC. In 4E offensive magic works the same way: the target has a fixed Fortitude/Reflex/Will defence determined by their stats, and in order to hit, the attacker rolls to overcome whichever defence is relevant.   However it was not always like this. Some spells in 3E and now again in 5E work the opposite way. In those editions the target is the one who rolls against a fixed value determined by the caster's stats, in order to avoid spell-damage.  Both methods compare the attacker and target's stats, and have an element of randomness. I personally like the 4E system as it simplifies combat without leaving anything out. Why was the decision made, when designing the early D&amp;D editions (and still when iterating upwards into (A)D&amp;D 1e&ndash;3e), to use saving throws to control spells?  Answers should consist of designer quotes and not personal opinions.
Find all functions $f$ such that $f:\mathbb{N}\rightarrow\  \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$   Let us plug in $n=1$  $f(f(1))+f(2)=3$  Since the function is from $\mathbb{N}$ to $\mathbb{N}$, $f(2)$ can only take the values $1,2$. Now we divide the problem into cases.  Case-1: $f(f(1))=2,f(2)=1$  We can assume that $f(1)=c$ for the time being. Then plugging in $n=3$ and using $f(2)=1$ gives $$f(3)=4-c$$ and again since the range of the function is positive integers,then $4-c$ has to be positive and hence $c$ belongs to  {$1,2,3$}. Now, $$f(1)=c$$$$\implies f(f(1))=f(c)$$$$\implies 2=f(c)$$ by the assumption that $f(f(1))=2$ . Now,since $c$ can only take the values $1,2,3$,we start treating cases. If $c=1$,$$f(c)=2$$$$\implies f(1)=2$$ but we know from the deinition of $c$ that $f(1)=c=1$,a contradiction.If $c=2$,then $2=f(c)=f(2)$ but $f(2)=1$ by assumption. Finally,if $c=3$ $$2=f(c)=f(3)$$ but $$f(3)=4-c=4-3=1$$ which is once again a contradiction. Therefore there are no such functions in this case.  Case-2: $f(f(1))=1,f(2)=2$  Again assuming $f(1)=c$ and using $f(n)\le n$ along with plugging $n=c-1$ will give us that $f(1)=1$ and then it is easy to prove that such a function exists by recursion. I can only give a "sort of recursive" way to define the function. Here it goes $$f:\mathbb{N}\rightarrow \mathbb{N}$$$$f(1)=1$$$$f(n)=n+1-f(f(n-1))$$  But this case is harder to deal with.Some help will be appreciated. Tax considerations for selling a property below appraised value to family? I own a property that recently appraised for $350,000 in the state of Maryland that I purchased for $300,000 7 years ago. I wish to sell this property for a value of $240,000. Here are my questions:   Is this legal? The state of Maryland has a transfer/recordation tax of 1.5% for each, the buyer and seller. Would this be computed on the appraised or sale value? Will this pose any problem if the buyer needs financing? Would I be able to deduct a capital loss on my tax return?     From comments: I should have mentioned - the buyer is a sibling. Preventing console window from closing on Visual Studio C/C++ Console application File being used by another process after using File.Create() I want to merge two shape files in which one is of point geometry type and other is polygon type in order to get a single layer containing both points and polygon in a single map. Browsing the internet from the command line How can I browse the Internet from the command line in Ubuntu? Can 'AMD-only' RAM really be restricted to AMD chipsets, and if so, why? I was browsing for a RAM upgrade earlier for my system, and came across 'AMD-only' RAM (aka, RAM that only works on AMD chipsets).  This RAM can , and seems to be slightly cheaper than 'normal' RAM.  The RAM prompts a question: is this RAM really restricted to AMD chipsets?  When considering RAM for a system, there are certain considerations to make such as:   Speed (1066MHz, 1600MHz, etc); CAS (latency, lower is better); Generation (DDR2, DDR3, DDR4, etc)   However there is usually no such consideration as chipset compatibility. Sure, it's possible to guarantee that a given stick of memory will work if it's within the bounds of the chipsets listed in the information given by the seller/manufacturer.  For instance if a series of chipsets supported RAM between 1600MHz and 2400MHz, with 9-12 CAS, and on DDR3, it would be possible to list a product as being fully compatible with that series of chipsets if the said RAM was 2000MHz, 10 CAS, and DDR3.  I suspect that's partially the story here. However, is it possible at all for a given stick of memory to be 'AMD-only' or 'Intel-only', and if so, how? 80's Animated movie: An old man joins mission to save a boy stranded on a hostile world Given $(a_n)$ with $a_n \geq a_{n+1} \geq 0$ show that if $\sum \sqrt{a_n a_{n+1}} < \infty \implies \sum a_n < \infty$ Is there a W3C valid way to disable autocomplete in a HTML form? When using the xhtml1-transitional.dtd doctype, collecting a credit card number with the following HTML  &lt;input type="text" id="cardNumber" name="cardNumber" autocomplete='off'/&gt;   will flag a warning on the W3C validator:     there is no attribute "autocomplete".   Is there a W3C / standards way to disable browser auto-complete on sensitive fields in a form? Can it be proven in $\sf ZF^{\neg\infty}$ that the sets $x_n=\{x_{n+1}\}$ do not exist? Report sites that use Stack Exchange content without following attribution rules here NB The following is a community-generated list of websites that republish Stack Exchange content without attributing it properly. It is no longer being maintained, because the procedure for reporting such sites has changed; see the duplicate for more information. There are a number of license-violating clones of Stack Exchange sites popping up that use Stack Exchange's  data without following our Creative Commons attribution terms. Those terms are , and are also included as a .txt file in every data dump we produce.  Google now has  in search results. Also, you can :  The option to block a site appears when you click a search result and then navigate back to the search results page. Click the &quot;Block&quot; link next to that result to block all pages within the site's entire domain.   This post was created as an attempt to organize information originally posted . You may also be looking for . Someone's car hit our metal bar gate, so one of the rods is bent as shown in the photo. What kind of a tool can I use to bend this back, so it's straight? Thank you! Calculate difference between two dates (number of days)? Dublin transit visa requirements for Indian citizens Draw somewhat random function I am trying to reconstruct this image here    I am not very happy with the code, as it uses hacking and adding white boxes and alike. My problem is now how do I draw the somewhat arbitary function in the figure? I tried to do a polynomial fit, but that did not work. I got dimensions to large, would a spline solution work better?    \documentclass{standalone}     \usepackage{tikz}     \usetikzlibrary{patterns}     \usepackage{pgfplots}     \pgfplotsset{compat=1.6}     \begin{document}     \begin{tikzpicture}      \begin{axis}[         xmin=-0.1,xmax=5,         xlabel={z},         ymin=0,ymax=4.5,     xtick={2.5,4},     xticklabels={$v_1$,$v_2$},     ytick={2.5,3.5},     yticklabels={$u_1$,$u_2$},     xlabel={$u$},       ylabel={$v$},     axis lines=middle]  \draw [fill=gray!40!white,thick,dashed] (axis cs:0,0) rectangle (axis cs:4,3.5); \draw [color=black,fill=white,thick,dashed] (axis cs:0,0) rectangle (axis cs:2.5,2); \draw [fill=black] (axis cs:0,0) rectangle (axis cs:0.01,4.45); \addplot+[black,thick,domain=0:5,no marks] {0}; % \node at (axis cs:2.5,0)    [anchor=south west] {$A$}; \node at (axis cs:4,0)      [anchor=south west] {$A'$}; \node at (axis cs:2.8,2.2)  [anchor=north]      {$B$}; \node at (axis cs:4.2,4)   [anchor=north]       {$B'$}; \node at (axis cs:0.25,2.5) [anchor=north]      {$C$}; \node at (axis cs:0.25,4)   [anchor=north]      {$C'$}; \node at (axis cs:0,0)     [anchor=south west]  {$O$}; % \end{axis} \end{tikzpicture} \end{document} Regex for numbers only I haven't used regular expressions at all, so I'm having difficulty troubleshooting. I want the regex to match only when the contained string is all numbers; but with the two examples below it is matching a string that contains all numbers plus an equals sign like "1234=4321". I'm sure there's a way to change this behavior, but as I said, I've never really done much with regular expressions.  string compare = "1234=4321"; Regex regex = new Regex(@"[\d]");  if (regex.IsMatch(compare)) {      //true }  regex = new Regex("[0-9]");  if (regex.IsMatch(compare)) {      //true }   In case it matters, I'm using C# and .NET2.0. Can I convert a C# string value to an escaped string literal? In C#, can I convert a string value to a string literal, the way I would see it in code? I would like to replace tabs, newlines, etc. with their escape sequences. If this code: Console.WriteLine(someString);  produces: Hello World!  I want this code: Console.WriteLine(ToLiteral(someString));  to produce: \tHello\r\n\tWorld!\r\n Does the output bypass capacitor of an LM7805 double as a decoupling capacitor? I'm using a 5v linear regulator (specifically an LM7805) that outputs directly to an ATMEGA328P. According to the  (page 23) input and output bypass capacitors should be used, as seen below, to tame peaks and ensure stability.    It is good practice to also include a decoupling capacitor in front of an IC, in this case an ATMEGA328. Does the 0.1μF capacitor on the output side of the LM7805 act as a decoupling capacitor if the regulator feeds into the IC directly after the output bypass capacitor? Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation