The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>Well, his name was James Stewart and he died last December, so now Integral House is up for sale, for $23m.

Yes, textbooks are *that* ridiculously expensive in North America.

The Daily Beast has written an article about the sale, and there’s a good thread on MetaFilter with a mix of discussion about the house itself and lots of griping about the American textbook racket.

]]>Good news, logic fans! The *Open Logic Project* is a project to write an open-source textbook on logic. And if you read it, you’ll find tautologies like the last sentence completely thrilling.

The book is aimed at a non-mathematical audience, mainly computer science and philosophy students, so it assumes very little knowledge of the basics. The project was instigated by Richard Zach, who’s Professor of Philosophy at the University of Calgary. The rest of the project team consists of Aldo Antonelli, Andy Arana, Jeremy Avigad, Gillian Russell, Nicole Wyatt, Audrey Yap, and Richard Zach. They’re aiming to cover first-order logic, sequent calculus, soundness and completeness theorems, computability theory, and incompleteness. If things go well, they want to add material about model theory, computability and Turing machines (that’s already in progress), and some stuff on philosophy of language and mathematics.

A high-quality textbook for free would be pretty good on its own, but what’s really nifty is that the source code has been set up so the book is configurable to your tastes: you can say what kind of notation you’d like, and even adapt theorems and lemmas to use different proof systems.

The Open Logic Project official website

Get the source code and contribute on GitHub

@OpenLogicProj on Twitter

]]>In the latest Taking Maths Further podcast (Episode 19: Computer games and mechanics), we had a puzzle that we say could be answered roughly, but the precise answer 23.53 (2 d.p.) required a little calculus. On Twitter, @NickJTaylor said

Not sure the @furthermaths podcast Ep 19 solution "requires calculus" to arrive at 23.5cm Just use v² = u² + 2as and solve for s @stecks

— Nick Taylor (@NickJTaylor) May 11, 2015

The question was: “Susan the Hedgehog runs at 20cm/s across the screen while the run button is held down. Once the run button is released, she slows down with constant deceleration of 8.5cm/s^{2}. Will she stop within 32cm more of screen?”

Taking the position to be $x$, we have constant acceleration $x^{\prime\prime}=-8.5$ and initial speed $x'(0)=20$. Therefore we get, w.r.t. time $t$,

\[ x’ = \int x^{\prime\prime} \mathrm{d}t = -8.5 t + 20\text{.} \]

Setting $x’=0$ gives $t=\frac{20}{8.5}=\frac{40}{17}$ when Susan has stopped.

Now we can integrate again to get position and, since we can decide $x(0)=0$, we can omit the constant:

\[ x = \int x’ \mathrm{d}t = -4.25 t^2 + 20 t\text{.} \]

Putting in $t=\frac{40}{17}$ gives

\[ x = -4.25 \left(\frac{40}{17}\right)^2 + 20 \left(\frac{40}{17}\right) = \frac{400}{17} \approx 23.53\text{.} \]

@NickJTaylor is suggesting that we use the fact that “$v^2 = u^2 + 2as$” or, using the notation above, $(x’)^2 = u^2 + 2ax$, where $x'(0)=u$ and $x^{\prime\prime}=a$ is a constant. This is okay, and it works, but to me it still uses calculus.

To get to this, we start with $x^{\prime\prime}=a$, $x'(0)=u$ and $x(0)=0$, and obtain

\[ \begin{align}

x’ &= \int x^{\prime\prime} \mathrm{d}t = at + u\text{;}\tag{1}\label{1}\\

x &= \int x’ \mathrm{d}t = \frac{1}{2}at^2 + ut\text{.}\tag{2}\label{2}

\end{align} \]

From (1), we rearrange for $t$ to give, for non-zero acceleration,

\[ t = \frac{x’-u}{a}\text{.} \]

Substituting this into (2), we get

\[ \begin{align}

x &= \frac{1}{2}a\left(\frac{x’-u}{a}\right)^2 + u \left(\frac{x’-u}{a}\right)\\

&= \frac{1}{2a} (x’-u)^2 + \frac{1}{a}u(x’-u)\\

&= \frac{1}{2a} ((x’)^2-2x’u+u^2) + \frac{1}{a}(x’u-u^2)\\

&= \frac{1}{2a} ((x’)^2 – u^2)\text{.}

\end{align} \]

So

\[ (x’)^2 = u^2+2ax\text{.}\]

Setting $a=-8.5$, $u=20$ and $x’=0$ gives

\[ 0 = 400-17x\text{,}\]

so we see $x=\frac{400}{17} \approx 23.53$.

If you are happy to accept $v^2 = u^2 + 2as$ as a given, or to work out the area under a graph of the velocity to get displacement, then you could say there’s no calculus needed. I’d say that deriving the formula, or knowing that the area gives the displacement, uses calculus. And if you’re doing a calculus question on my exam, you should expect to have to show me the calculus.

]]>Puzzlebomb is a monthly puzzle compendium. Issue 41 of Puzzlebomb, for May 2015, can be found here:

Puzzlebomb – Issue 41 – May 2015

The solutions to Issue 41 will be posted at the same time as Issue 42.

Previous issues of Puzzlebomb, and their solutions, can be found here.

]]>Following on from the resignation of the editorial board, CUP has announced that it’s not publishing the Journal of K-Theory any more. The new journal started by the former editors, *Annals of K-Theory*, aims to start publishing papers online this year.

If you can’t handle K-Theory withdrawal for that long, here’s “Time Heals Nothing” by popular music artist K Theory.

*via David Roberts on Google+*

Reader Danial Clelland wrote in to tell us about his new calculator app for iPhone, CALX.

None of us owns an iPhone, but I borrowed someone else’s for a while and had a brief look at the app.

Obviously I do all my sums in my head, but when I do use a calculator, I like to do it in RPN – reverse Polish notation. In RPN, you *push* numbers onto a stack, and each operation *pops* one or more numbers off the stack, does something with them, then *pushes* the result back onto the stack. Among many other benefits, this means that you don’t need to worry about brackets or operator precedence. You do have to think backwards compared to normal, hence the ‘R’ in ‘RPN’.

On my Android phone, I can do that with the formidably adequate RealCalc. RealCalc has everything you could possibly want in a calculator, since it tries to emulate the workhorse HP calculators of old, even down to a faux-LCD display.

CALX is another RPN calculator, but it takes advantage of the iPhone’s big touch screen to make using it lots easier than an old-fashioned physical calculator.

You use the keypad to input numbers, and the up arrow button pushes onto the stack. +, -, × and ÷ are next to the digits, and you’ll find the rest of the functions you’d expect in a scientific calculator when you swipe left. If you swipe right, there are buttons to calculate the sum, mean and standard deviation of the whole stack – so much more natural here than on a standard Casio calculator!

The big advantage of CALX over RealCalc is that you can manipulate the stack with swipes and taps – tapping an item duplicates it on the top of the stack, and swiping right removes it from the stack. You can scroll up and down with your finger to see the whole stack – the biggest problem I have with RealCalc is that you can only see the top two or three items, and it’s easy to get lost in the middle of a long calculation.

There are a few customisation options – you can pick a different font, or change the colour scheme. Apart from that, it’s a very simple app. If I had a phone that could run it, I think CALX would be my calculator of choice.

A warning: if you’re not used to RPN already, it might not be worth the effort of retraining your brain to use it effectively, but on the other hand it’s worth learning for the geek cred alone.

At the moment, it’s $1.99/£1.49 on the App Store

]]>The Aperiodical turned three on Saturday. I was away attending my brother’s wedding, but I couldn’t let the birthday pass without mention.

In three years we’ve published 1,462 posts (make that 1,463 including this one) by 32 authors, read by 713,000 visitors.

Thanks for reading!

]]>US organisations the Mathematical Sciences Research Institute (MSRI) and the Children’s Book Council (CBC) have founded a youth book prize, called *Mathical: Books for Kids from Tots to Teens**.* The prizes, awarded for the first time this year, recognise the most inspiring maths-related fiction and nonfiction books aimed at young people. This year, they’ve awarded a set of prizes for books released in 2014, as well as honouring books published been 2009 and 2014, plus two ‘hall of fame’ winners from the further past.

The selection committee was chaired by maths author Jordan Ellenberg, mathematical sciences professor Rebecca Goldin, and young people’s literature ambassador Jon Scieszka. The panel comprised maths specialists, teachers, computer science professors, and chairs of organisations related to maths and young people.

The awards were classified into pre-K (under 4), grades K-2 (roughly UK KS1), grades 3-5 (KS2), grades 6-8 (KS3) and grades 9-12 (KS4/5). The list of winning books, and more information about the award, are available on the Mathical Books website.

Mathical Books: Award Winners Announced

]]>**C:** $K_A m; \\ K_B d.$

**A:** $\neg K_A d; \\ m \vDash \neg K_B m.$

**B:** $d \not\vDash K_B m; \\ (K_A(\neg K_B m)) \vDash K_B (m,d).$

**A:** $m \wedge K_B(m,d) \vDash K_A (m,d).$

Albert, Bernard and Cheryl have had a busy week. They’re the stars of #thatlogicproblem, a question from a Singapore maths test that was posted to Facebook by a TV presenter and quickly sent the internet deduction-crazy.

First of all: no, it’s not meant to be answered by an average Singaporean student. It’s a hard question from a schools Olympiad test.

You’ve doubtless seen multifarious debates about the correct answer on your Facebook/Twitter/Friendface feeds. The various newspapers of the world saw an opportunity for some easy page views and each wrote basically the same story: *When is Cheryl’s birthday? Only maths weirdos can possibly work it out.*

By far the best treatment, in my opinion, is by top pop maths chap Alex Bellos in the Guardian. He’s written a few pieces: first asking “can you solve” (with an encouraging number of fallacious arguments for each answer in the comments) and then explaining “how to solve” with his model solution. Later on, Alex was invited onto BBC TV to present his solution.

Over on Radio 4, the *Today* programme got carried away with logic puzzles and presenter Sarah Montague managed to read out the wrong answer to a hats puzzle.

**STOP PRESS: **while this post was working its way through our editing process, Numberphile posted its own explanation of the problem.

Following a discussion on Facebook where Phil Walker of Leeds University gave a firmer epistemic reason for the mainly-incorrect answer of August 17, Alex invited James Grime to write about this alternate line of reasoning. I’m still not having it though. And because everything written on the internet has to subsequently become a video, James has posted his explanation of the different interpretations to his YouTube channel.

If you’re still unsure, or still being bombarded by relatives who want an explanation, have a look at this fantastic interactive explanation by Mark Josef. When you click on the day you think is Cheryl’s birthday, it walks through the conversation (as well as Albert and Bernard’s internal monologues) and highlights any statements that would be inconsistent.

By the way, my wife^{1} asked why the characters in the original puzzle were called Albert, Bernard and Cheryl. In case you’re also wondering that, it’s a convention in this kind of thing to give the characters names whose first letters work through the alphabet, so in your working-out you can abbreviate them to A,B,C,… Famously, in cryptography problems Alice and Bob have a whale of a time sending messages to each other.

Now, with the original question definitively dealt with and the populace at large going mad for logic problems, the world’s mathmos are scrabbling to show off their favourite deduction problems.

Kit Yates is jumping on the bandwagon with this version, which he says was used as an interview question for several years:

@alexbellos @jamesgrime While people are in mood #thatlogicproblem Lets try to send another one viral #neighboursprob pic.twitter.com/OKtOK1lXBn

— Kit Yates (@Kit_Yates_Maths) April 16, 2015

Tanya Khovanova is the master of this sort of thing. Conway’s wizards on a bus puzzle is the apotheosis of the genre and Tanya has written a great paper about its solution, along with a generalised version. Probably inspired by the current hullabaloo (or maybe not – she is usually thinking about these things), Tanya posted an extremely concise puzzle to her blog yesterday.

Deduction is normally done by prisoners either in hats or with boxes. If you really want to blow your brain sockets, try the blue-eyed islanders puzzle. Terry Tao posted it to his blog once and a gargantuan comments thread full of people not getting it ensued.

Meanwhile, Tim Gowers is all “I hear you like deduction, so I put some deduction in your deduction so you can deduce while you deduce” with this transfinite version. Although Joel David Hamkins turned up in the comments and shared his version, which looks much nicer to me. Later on, he added yet another transfinite epistemic logic puzzle.

With all this talk of deduction, you might remember the QI-worthy observation that Sherlock Holmes doesn’t do deduction – he does abduction (hey, whaddya know: I did learn that on QI!). *Deductive* reasoning is the process of working from a set of premises to a certain conclusion by fixed rules of inference. *Inductive* reasoning involves using premises which provide evidence for a conclusion which is *probably* true. Finally, *abductive* reasoning is when you try to find the simplest explanation for an observation.

“CP”, you say, “that’s all well and good, but can you put all this in joke form?”

Yes.

Since this is a maths site, I should think about what all these puzzles have in common and how to formalise them. Clearly they’re logic puzzles, but the deductions based on announcements can’t be written down in classical logic. It turns out there’s a thing called public announcement logic which provides a whole load of new modal operators such as “I know $\phi$”, and “$\phi$ is not refutable”.

My go-to resource for explaining public announcement logic is *The Muddy Children:*

*A logic for public announcement*, a set of slides by Jesse Hughes.

There’s a whole load of notation around public announcement logic, involving things like Kripke frames. The whole thing got going with Jan Plaza’s paper “logics of public communications”. For a paper introducing a new field of logic, it’s surprisingly readable!

The notation at the top of this post is a mangled mash of various things that sort of represent the conversation in #thatlogicproblem but no self-respecting logician would recognise it.

- I got married on Sunday! Hooray!