I've been invited to a maths themed Xmas after party. I need to prepare a selection of interesting, and relatively simple fallacious proofs which other guests will try and find the flaw in. I'm trying to avoid very well known ones, of course.  Each proof needs to fit comfortably on a large whiteboard.  Assume undergraduate level of maths knowledge, though the true cause of flaw does not have to be!   A place where i've previously found suitable material is 'Mathematical Fallacies and Flimflam' by Edward J Barbeau. In there is a good example of a simple fallacious proof of 1=2 using telescoping series, which can also be found here (p.2):   Feel free to share personal favourites, and make it as annoying to spot the flaw as possible!
If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:    Let $a = b$.  Then $a^2 = ab$  $a^2 - b^2 = ab - b^2$  Factor to $(a-b)(a+b) = b(a-b)$  Then divide out $(a-b)$ to get $a+b = b$  Since $a = b$, then $b+b = b$  Therefore $2b = b$  Reduce to $2 = 1$   As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.
How does the Mounted Combatant feat interact with Infernal War Machines? R version 3.1.1 (2014-07-10) -- "Sock it to Me" &gt; bl &lt;- c(140, 138, 150, 148, 135) &gt; fu &lt;- c(138, 136, 148, 146, 133) &gt; t.test(fu, bl, alternative = "two.sided", paired = TRUE) Error in t.test.default(fu, bl, alternative = "two.sided", paired = TRUE) :  data are essentially constant   Then I change just a single character in my fu dataset:  &gt; fu &lt;- c(138, 136, 148, 146, 132)   and it runs...  &gt; t.test(fu, bl, alternative = "two.sided", paired = TRUE)      Paired t-test   What am I missing here? How to use unicode characters in Windows command line? Installing MinionPro fonts on Ubuntu 10.04 running TexLive 2012 I followed the instructions to install MinionPro fonts from this website.   However, when I tried to run the tex file at the bottom, I have this error message:  ! LaTeX Error: File `MinionPro.sty' not found.   Running the command kpsewhich MinionPro.sty gives the following output:  /usr/local/share/texmf/tex/latex/MinionPro/MinionPro.sty   So the system knows where MinionPro.sty is, so why can't the file compile? I am familiar with the set-theoretical definition of functions.   A function $f : A \rightarrow B$ from a set $A$ to a set $B$ is a subset $f \subseteq A \times B$ of the Cartesian product $A \times B$ such that the following conditions hold:   $\forall a \in A : \exists b \in B : (a,b) \in f$ $\forall a \in A : \forall b, b' \in B : (a,b), (a,b') \in f \rightarrow b=b'$   Is there a purely set-theoretical definition of the function evaluation $f(a)$ for every $a \in A$? I get as far as defining  $f(a) = \{ b \in B : (a,b) \in f \}$ but that defines rather the set of values of a multi-valued function. Contemporary written usage of "whom" in objective case Possible Duplicate:          I was writing a LinkedIn recommendation one day, and ended up pondering for a while which of these forms to use:     … is a great developer whom I always   found easy to work with.      … is a great developer who I always   found easy to work with.   Both are basically correct in contemporary English, right? But is one or the other preferable, and if so, why?   (In this case I went with the latter, as it seemed more common (Google)  and I wanted to avoid sounding unnecessarily “archaic”, although I’m not sure whether that would have been the case. Also note that I try to write in a “friendly professional” style instead of overly formal one. :-) Debt condition abuse? The rules states that you cannot have two of the same conditions.  You can take a debt condition to gain two additional successes on your influence test to gain assets from the reserve. Thus if you already have a debt, does that mean that you automatically gain two additional successes for each gain asset actions? Is a continuous function on a bounded set bounded itself? If $f\colon A\to\mathbb R$ is continuous and $A$ is a bounded set, does it necessarily follow that $f$ is bounded? What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? If $x\sqrt{1+y}+y\sqrt{1+x}=0$ find $y'$ Find $\frac{dy}{dx}$ if $x\sqrt{1+y}+y\sqrt{1+x}=0$ for $-1\leq x\leq 1$   My Attempt $$ x\sqrt{1+y}=-y\sqrt{1+x}\implies x^2(1+y)=y^2(1+x)\implies x^2+x^2y=y^2+xy^2\\ 2x+2xy+x^2\frac{dy}{dx}=2y\frac{dy}{dx}+y^2+2xy\frac{dy}{dx}\\ \frac{dy}{dx}\Big[ x^2-2y-2xy \Big]=y^2-2x-2xy\\ \frac{dy}{dx}=\frac{y^2-2x-2xy}{x^2-2y-2xy} $$ How do I proceed further and find the derivative ? Why do the $n \times n$ non-singular matrices form an "open" set? Which colors make you hungry I am working on a new brand, which is all about food.   What colors should we use for the logo and website?   I keep reading that red makes you hungry, but i would like to stay away from that color. All of our competitors must have read the same articles. So I'm making this summary of a course where I've only been getting information from one book. Sometimes I want to show what page from the book a statement is taken from but I can't figure out a clean way of doing it.  Basically I would like to have something (in function) similar to:  "Statement[Book p123]"  Do you have any suggestions? 'Good' review audits can be misleading Which is correct: "__ is different from __" or "__ is different than __"? As someone who learned English later on in life, I was taught that different from is the correct grammar to use: this is different from that. However, it seems these days everyone uses different than instead. I know it's incorrect usage, but does the language evolve if the majority wins?  Edit: Some commented different than is American English and different from is British English, both are OK. IMO, this isn't like color versus colour, though.   Than is used after a comparative adjective, e.g. taller than, whiter than. The word different isn't a comparative adjective, unless used in more different than. Logically, it makes no sense to say different than. How to get video duration, dimension and size in PHP? One-sided Fisher's exact test and its complement I had applied for a PLAB (Doctors) Visa for the UK to appear in PLAB test. I got refused on the basis of two recent large deposits in the bank statement I provided (my mothers). In reality, these are my own savings which I had deposited in her account to make it more strong. The only objection I got was that I must provide evidence for the source of these deposits. Other than this, I have previously travelled to the UK and USA as well as other parts of the world and my travel history is fine. Can someone please advise on how to proceed and re-apply? How do I use an NBT editor to modify item properties? Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group. Let $G$ be a group and $c$ be a fixed element of $G$. Define a new operation $*$ on G by:   $$x*y=xdy$$   for all $x$ and $y$ in $G$, where $d$ is the inverse of $c$. Prove that $G$ is a group under the operation $*$.    If you want to check, this problem is in the book “A Course In Group Theory”, by John F. Humphreys (problem $3$, page $28$)   I know what are the axioms that $(G,*)$ must obey in order to be a group and I did show that all the axioms where obeyed except the inverse axiom.  My problem is that, whenever I try to verify what would the inverse be for some element $x$, I get an equation where the $x$ is cancelled in both sides.  Can please someone guide me?  Thank you!  (P.S.: This is NOT a homework assignment, I’m studying Group Theory by this book and got problems solving this question)