The Correntator

I found a rather interesting mechanical calculator in Tynemouth market. It’s called a Correntator, and it adds up old-fashioned money. You add amounts by sticking a stylus in one of the tracks and dragging it down. Little hooks at the ends of the track do the carrying-over for you. It’s rather sweet, in a limitedly-useful way.

I also brought in my 1811 arithmetic textbook to provide a source of problems for people to solve.

We talked for a bit about how much harder things were with old units, meaning people had to spend more time remembering rules for doing simple things and never got on to more advanced and fun topics. Someone said, “Use of the correct concepts helps.” Apart from the switch to metric, someone pointed out that thinking of the trig rules in terms of complex numbers makes a whole load of problems way easier.

Ji pointed out that the Chinese version of the Correntator is the abacus.

An unexisting journal article about a hoax

I mentioned that I had been unable to track down a copy of the paper “An April Fool’s Hoax” by Stan Wagon, purportedly published in Mathematica in Education and Research. It’s widely cited, but the links on the journal’s old website are all broken, and Springer-Verlag denies all knowledge of the journal. I can’t work out whether the article about a hoax is itself a hoax. Anyway, it’s supposed to be about the hoax Martin Gardner famously perpetrated in his Mathematical Games column where he claimed that Ramanujan’s constant is an integer.

Factorising commuters

I mentioned that I had started factorising the number of my Metro carriage on the way into work each morning as a way to pass the time. It took me a long time to factor 4051, because it’s a prime! I’ve written down, “Is there a better method than the sieve of Eratosthenes for primality testing/factoring?” Yes.

More cubes!

Steven brought in yet more weird puzzle cubes…

… and David finally solved the puzzle he started in April.

A random walk on slides

We talked about the rather silly random walk on slides that some Carnegie Mellon students did. We thought it was perfect Big MathsJam fodder, and thought that each local MathsJam could prepare one slide for a presentation to be given at the annual conference in November. Let’s make it happen!

Another way of counting

Cheltenham posted this puzzle on Twitter:

\begin{array}{} 1 & 3 & 5 & 7 & 9 & \cdots \\ 2 & 6 & 10 & 14 & 18 \\ 4 &  12 & 20 & 28 & 36 \\ 8 & 24 & 40 & 56 & 72 \\ \vdots & & & & & \ddots \end{array}

Show that every number appears in this grid exactly once.

After thinking about it for a bit, click here to see the solution.

A rubbish way of calculating $\pi$

John told us about this identity for $\pi$:

\[ \pi = 4 \left( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots \right) \]

A quick googling tells me that it’s called the Gregory series, and it converges really, really slowly. John told us that even if you sum up the first 5 million terms in the series, it’s wrong in the 7th and 21st (and more) digits. He asked if there was an integer sequence for the number of terms you need to calculate to get $n$ digits correct. (Or he asserted that fact but couldn’t give a reference. I can’t remember.)

It’s sequence A126809, and there’s bad news: you don’t get the 7th digit correct until you’ve worked out 18,660,270 terms of the series sum!

David goes showbiz

David arrived quite late. He said that he had passed up dinner with 70s megaband Journey to come to MathsJam. And he wasn’t lying! He left after a while claiming he was bored. David’s too showbiz for us now

Simon!

I forgot to bring Simon the snake! He had been in my back pocket for a month until the morning of MathsJam, but then I took him out and put him on my desk and now I don’t know where he is. Poor Simon.

John’s rather clever games

John brought a couple of games he’d bought from nestorgames. Specifically, he brought Feed the Ducks and Coffee. The game of Feed the Ducks took absolutely ages to play, and eventually I think everyone just gave up.

Abuse of notation

We talked about the bad hexadecimal numbers. I can’t remember how we found them, but the idea is this: instead of using $A,B,C,D,E,F$ as digits in base-16, write $10,11,12,13,14,15$. So $27$ in base 10 is $1 \times 16+11$, which you’d normally write $1B$, but instead we write it as $111$.

Equally appalling is dismal arithmetic. The idea this time is to respond to complaints that students learning to do addition and multiplication on paper can’t do carrying-over by making sure that it never happens. You follow the same algorithms as for normal addition and multiplication, but now whenever you add two digits you just take the largest one, and when you multiply two digits you take the smallest one. It works, as long as you don’t mind about associativity or distributivity or anything like that. I finally satisfied myself that 19 is a prime in dismal arithmetic.

Mad Abel!

We played a few rounds of Mad Abel. Enthusiasm was down on last time we played it. I still think it’s great, anyway.

Wobbly table legs

Edinburgh tweeted about the wobbly garden table problem, which we’ve discussed before. There is a similar problem: can you always find somewhere to land a lunar module with 3 legs so that it’s level? Apparently this is known as Knaster’s problem.

Art!

I talked about my quest to find art to decorate Newcastle University’s maths department. Andrew has written down the name Radmila Sazdanovic.

Surveying

Finally, someone mentioned a device which can calculate the area of a shape after you trundle it around the boundary, using either Green’s theorem or Stokes’ theorem (we weren’t sure which). As far as I can tell, it’s called a planimeter. Pretty clever!

Phew! Done with 30 minutes to spare. Just enough time to saunter down to the Charles Grey pub in the sunshine and get myself a drink.

The first thing I have written down in my notebook for March’s Newcastle MathsJam is a scribbled-upon sketch of what looks like some paths on a torus. I think I might’ve been explaining the definition of the fundamental group of a surface to someone.

Stephen brought in a whole collection of exotic Rubikish shapes. David decided to do art with them.

John brought The Mathematical Coloring Book:

Looking for $a! = b!c!$

Not sure if this was posted on Twitter or brought up by someone at Newcastle. Here’s a solution:

• Let $n := k!$.
• Then $n! = n \cdot (n-1)!$
• i.e. $(k!)! = k! \cdot (k!-1)!$.
• Quod erat to show.

Underneath that, I have couple of interesting identities:

\begin{align} 8!+7!+1! &= 45361 \\ 871 &= 4!+5!+3!+6!+1! \end{align}

That was tweeted by Jim Wilder, and is related to a very short integer sequence.

We must’ve been talking about nice number facts in general at this point, because I’ve written down my favourite number,

\[ 3435 = 3^3 + 4^4 + 3^3 + 5^5 \]

Numbers like this are called Münchausen numbers and belong to an exactly as tiny sequence as the factorions mentioned above. The paper “On a curious property of 3435” by Daan van Berkel will tell you all about them.

Eggs!

Easter happened in March, so Manchester MathsJam’s Katie and I researched ways of drawing mathematically-correct eggs beforehand. It doesn’t look like I kept any of our drawings, so you’ll just have to reconstruct them yourselves. All the literature we found on the subject (eggsegesis, if you will, and I hope you do) came from Germany. Jürgen Köller’s page on egg curves is comprehensive, while Herbert Möller’s paper, “Das 2:3-Ei – ein praktikables Eimodell“, gives a more rigorous treatment.

Inspired by our eggs, John reminded us of this problem that Ji had posed previously:

Given a circle and one point on it, draw an inscribed square with just a compass.

Difficult! I didn’t write down the answer!

I talked a bit about my friend Stacey’s problems with maximal oriented Wicks forms. Stacey copied out part of a proof she was studying on our office blackboard, and it was so attractive I took a picture and sent it in to the What’s on my blackboard? blog.

Next in my notebook is the word “Sarahdohedron”. Sarah made an unusual origami shape and needed a name for it.

We had a go at the jumping frogs puzzle, which the nice people at NRICH have recently made a nice version of. The lengths of solutions to the jumping frogs puzzle form yet another integer sequence, A005563.

We sort of had a conversation with Cardiff MathsJam through our respective Enigma machines. They were using a Pringles paper tube Enigma machine, but we cheated and used the Android app.

Night-night, Simon!

David presented us with a tricky problem of the utmost importance:

Simon the snake is very sleepy. He might fall asleep at any moment, in any position. What’s the smallest blanket can we use to be sure that we’ll entirely cover him, no matter what position he falls asleep in?

There was quite a lot of discussion about this. We started with a semicircle, which David reckoned you could cut a smaller arc out of the top of. I had some crazy cockahoop scheme involving variational methods that I won’t try to remember. Later on, we found out that Simon is the latest in a long line of sleepy slitherers which began with Moser’s worm. The paper “A smaller sleeping bag for a baby snake” by Svante Linusson, and Johan Wästlund has both a charming title and a much better solution than any of us came up with.

Recorded for posterity, here are our efforts:

Simon, pictured here on the edge of Sleepy Towne

Oh no, Simon has fallen asleep! Fetch the blanket!

Simon is moderately, but not optimally, snug underneath this blanket.

Sweet dreams, Simon. Sorry we weren’t clever enough to tuck you in to your exacting requirements.

Simon got a bit frayed towards the end of the night, so John offered to take him home and give him a bit of TLC. He reappeared in April with a neat heat-shrunk head. I assumed custody after that but I’m ashamed to say I have no idea where he is now. Hopefully, he’s having a little kip in the bottom of my backpack. I am a terrible snake-stepfather.

The last note I have is, “Is 10% the same thing as 0.1?”, prompted I think by an argument I was having with someone on Twitter. I’ll say that we agreed it is, because that way I win the argument.

Oh no, it’s the PAPAC-00

I’d hoped the main feature of April’s MathsJam would be the PAPAC-00, a cardboard computer I had assembled based on instructions I found on the internet. It didn’t really work though, so I just had to point to the bits that should’ve been moving when I pulled the control lever. Cereal box is not a precision medium. John said he thought it would work if the bits were made on a laser cutter. I expect to be surprised at May’s meeting.

The Papac-00, in its natural state as a dream too beautiful for this mediocre reality.

The idea is that you set each of the two registers to store either 1 or 0, and then waggle the control lever back and forth. Each waggle of the lever adds the current register to the adder at the bottom. The counter at the top dictates which register is read, and alternates with each waggle. Roughly 13% of those things happened when I tried it.

We talked a bit about colourblindness, because I had just found the most marvellous app for my phone, which both simulates and corrects for colourblind vision through the phone’s camera. Everyone else learned how rubbish my view of the world is, and I learned a lot about what colours things are.

I had brought along the Big, Big, Big Book of Brainteasers, a book which made its first appearance at the third Newcastle MathsJam back in 2011. We tried a few puzzles – I’ve written down numbers 201 and 202 – before deciding they were either too easy, too hard, or too ambiguous to bother with. Eamonn had brought Erwin Brecher’s Puzzles, Brainteasers & Mathematical Diversions, which seemed to be a much better book. I’ve written down number 174, which I think might be a puzzle from Brechner’s book that we enjoyed. Any idea what it was, anyone?

Oh no, more circle construction problems

Here’s one:

A circle is inscribed in a quadrilateral. The ratio of the quadrilateral’s perimeter to the circle’s circumference is 3:2. What’s the ratio of the quadrilateral’s area to the circle’s?

And another:

Given a circle and a point outside it, construct the point’s tangents using only a ruler.

The solution to that one is so hard to find that I commented that many of these puzzles seem to arise when someone chances upon an interesting fact, then phrases it as a question. Asking peopel to do the inverse of that isn’t much fun, in my opinion. (#soreloser, I know)

The run-away train that failed to run away

Well-known interesting fact: $\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots$ doesn’t converge.

Little-known fascinating fact: the same sum but excluding numbers containing a 9 does converge!

In fact, you can pick any string of digits instead of “9″, even a really long one, and the fact is still true. It’s explained in the paper “Summing curious, slowly convergent, harmonic subseries” by Thomas Schmelzer and Robert Baillie.

Finally in my notebook, just this little bit:

$12^2 + 33^2 = 1233$. Others?

To be precise: we want two numbers such that, when you add their squares, you get their concatenation. Turns out Stephen’s done a lot of work on this since it was posed to him as part of an undergrad course (or problem sheet or something). He’s calculated an enormous number of solutions and put them on his website. He explained his method to us – maybe that’ll appear online too one day.

That’s a snapshot of two months’ worth of Newcastle MathsJams. I’ve missed quite a bit because I didn’t take very good notes, as usual. You’ll have noticed that we’re getting a bit hardcore again, without the influence of many non-academic mathmos. Since I get more than my fill of crazy super maths as a result of having to spend every working day with Cushing, and soon also Ji, I’m very keen to do some stress-free fun maths. So if you have any ideas how we can encourage more varied kinds of people to come to MathsJam, please tell me!

The next Newcastle MathsJam will be on Tuesday the 21st of May. If you’re one of the many people from outside Newcastle who read this blog, the MathsJam website lists over 30 cities round the world where MathsJams take place.

January

My art teacher from school appeared in my office at uni one day saying he had a maths problem. He didn’t know I work here, so it was a shock to us both! His problem was:

Can you draw the same asymmetrical figure on all six faces of a cube, so that there’s no symmetry around any edge or vertex?

(He didn’t use exactly those words) Here’s what I mean, in pictures:

So on adjacent faces you can’t have the figure the same way up, or pointing in opposite directions, and you can’t have rotational symmetry around a vertex.

The answer to the question is no, but my proof is just an exhaustion of cases. It’s fun to work out, anyway. Thanks to the MathsJam Rubik’s cube and some arrows drawn on post-it notes, I think everyone eventually agreed it’s impossible, though there were some complaints that I kept “adding rules” whenever someone thought they’d come up with a counterexample.

Then we discussed a problem posted by Tanya Khovanova on her blog as an example of a coffin problem from the Moscow State University entrance exams:

Given four points on a square, reconstruct the square with straightedge and compass.

I can’t remember if we got the solution. It’s possible John showed us the solution he’d been shown, but I’m not sure.

John then showed us a little gizmo he’d made.

It has a number display and a button. The display starts at 1000, and every time you press the button it changes to a random lower number, until it gets to 1. You might want to watch this video I recorded of it in action. John asked us,

What’s the probability that it shows the number 2 before it gets to 1?

We all played with it for a bit (it’s quite compulsive), until after a while I realised I’d answered this problem before. The gizmo is doing the same thing as the seating passengers on a plane puzzle we solved last April. Top memory, CP! I also have written down that we considered the probability that every number is prime (or has some property X), but we must have just thought it up at the end.

Someone told us about this trick a lecturer had performed during a probability class, or something like that:

He asked everyone if they fancy the person sat next to them. It would be cruel to ask people to reveal that information in public, so he got everyone to flip two coins, and tell the truth only if they got two heads. How did he work out how many people fancy the person next to them?

I told everyone about David’s first sequence in the OEIS, sequence A189055: the primes which are the difference of consecutive eleventh powers. David had the idea, but I wrote a Sage notebook to compute the sequence. The notebook is mainly incomprehensible to me; we did some more investigating after finding the sequence but I’ve forgotten what.

I’ve written the question, “N points: how many topologies on them?” but I have absolutely no recollection of what that means.

Finally, I told everyone about my interesting times madness. I’m still suffering from it and it’s proven contagious, so I’ll spare you.

February

We started with a tricky puzzle I’d seen tweeted earlier in the day:

Yikes! @mathsjam “@tobiasteo: Here’s to screw your brain for the day: twitpic.com/c4ubbp“

— Christian Perfect (@christianp) February 19, 2013

Then I mentioned, for who knows what reason, Tarski’s high school algebra problem: there exist identities using only addition, multiplication and exponentiation on the positive integers that can’t be proved using the algebraic rules a typical high-school student is taught.

Eamonn showed us his odd Rubik’s cube. It’s one of these:

Stephen scrambled it and set to solving it, assuming Eamonn could put it right if he got stuck. Eamonn couldn’t, but fortunately it turned out that Stephen could.

John posed us a problem:

I have an infinite line of coins, all facing heads-up. I give you a template consisting of a finite number of arrows, which you can move along the line any number of times, flipping every coin pointed to by an arrow. Is it possible, no matter what the template is, to end up with exactly two coins facing tails-up?

Here’s an example of a template:

The proof involves the bane of my existence, generating functions. Yuck.

We discussed the coffin problems yet again. Possibly someone posed one of them not knowing where it came from. Then, there was some discussion of infinities. I wish I’d written more down!

I can very clearly remember doing some origami, but I have no written or photographic record of it. Instead, here’s one I made earlier:

It’s an icosahedron inside its dual, a dodecahedron.

For our last puzzle, I asked this unexpectedly easy question:

How many anagrams of the number $123456789$ are prime?

Finally, we talked about written versus spoken maths, after I’d discussed it on the All Squared podcast with Katie Steckles and Edmund Harriss.

Phew! I’m sure I’ve missed loads, but I’m glad I’ve finally written something down for the last two MathsJams.

Twitter just switched the switch to let me download my tweets archive, so I immediately did that and grepped it for my many many new business names. Here they are:

Buying jangly things for my new sensory stimulation and meditation chamber, “We have nothing to hear but here itself”

Locating premises for my new Newcastle-based cutlery steam-cleaning service, “Fog on the Tine”

Buying tiger balm for my new physiotherapy practice in Gosforth, “Knee Bother”

Buying cages for my new exotic birds shop, “Super Mario Plovers”

Buying a PRS licence for my new musical instruments shop, “If you’ve got it: flautist.”

Getting security clearance for my new spa strictly for defence officials, Intercontinental Holistic Missile.

Acquiring a PRS licence for my new cellar bar under the Newton Institute, “Fermat’s Last Beer-Room”

“The customer is always right.” That’s something we strongly believe in at my new pest extermination company, Sick Of Ants

Hiring bar staff for my new underground opera hall, “Grotto Voce”

Scouting for locations for my new concept café with a chess board at every table, “En Croissant”.

Buying stock for my new kids’ clothing shop, Newcastle upon Tiny.

Soupremum. Simpleggs. Identitea. All on the menu at my new mathematical bistrho.

Trying to get Fearne Cotton to endorse my new back scratcher design. I’ll call it the Fearne Itcher.

In Chinatown looking for premises for my new laser hair removal business, Singe-A-Pore

looking for angel investors for my new estate agency specialising in flipping run-down urban homes: Quain’t.

seeking funders for my new-Romantic fireworks show, “Sturm und Bang”.

now taking orders for my new range of Provençal crockery, S’il te Plate.

please buy my new range of plates featuring images of the abyss, a navel, the circle of life and the night sky: “ContemPlates”

looking for investors in my new chain of kosher Argentinian steakhouses, “Gaucho Marx”

Lost a favourite mallard recently? Then come down to my new duck cloning service, Quacksimile!

Meeting the pope later to discuss my idea for a convent based on an airship, called “Nun of the Above”

Drawing up a health and safety plan for my new plush toy cleaning service, Teddy Gross Felt

Well, I like them.

I’ve written a few posts here in the past about the Grzegorczyk hierarchy, computable groups, and so on, but I think this might be the first time I’ve presented my work to real people (apart from an impromptu hour-long braindump when one of the real seminar speakers cancelled and I decided to test my memory).

As usual, I wrote my slides using my deck.js template, which I’ve finally got round to putting on GitHub. I’ve uploaded the slides – you can see them here – but I thought I’d have a go at using my laptop to record the audio of the talk as well. Amazingly, the little built-in microphone picked up my voice quite clearly, and you can even just about hear the questions people asked.

We had a few discursions about tangential topics; maybe it’s good that the recording only picked up my attempts to draw everyone back to the maths.

At some point I’ll gather up the energy to have another go at finishing off the paper. For now, it’s in my phd-notes repository on github. Here’s a direct link to the PDF of the paper in progress.

Update 13/02/2013: The seminar organisers have set up a page with the schedule of talks.

I was thinking about that this morning, so I decided to go through and play a few of my old games. I feel just as much a need to show them off now as I did when I made them, so I’ve recorded a few screencasts. Camtasia broke for Unknown Reasons and it turns out Jing is sort of terrible at recording games, so they’re really jerky and the volume’s quite low, but I’m not going to not share them now I’ve made them.

Here they are:

• Bubble worlds
• Poption
• Trenchant
• Others

I’m quite disappointed that a few of my favourites didn’t run. I’m sure if I recompiled them they’d work. All of my code is in a repository on github, and compiled versions of everything are on my original homepage, somethingorotherwhatever.com. Here’s a list of links to the ones that work at least a little bit:

Trenchant, Poption, Fencing, Bubble worlds, Trader, Schroedinger’s Ghost, Beards, Rowing, Surroundemup, Clockwork Shooter, Schizophrenic Pacman, Museum, Horsey (browser version), Kleinsteroids

There were eight of us in November, including two very welcome new Durham undergrads.

John began with a printout of Brent Yorgey’s factorisation diagrams. Several people battled against the poor pub lighting to try to discern the scheme behind the colouring.

Ji briefly mentioned a puzzle from his coding theory course – how to weigh twelve balls on a set of scales in the fewest weighings? I don’t have the answer written down, and I won’t work it out! The puzzle might actually be to separate one light ball from the uniformly-heavy others. I can’t remember.

Katie, organiser of Manchester MathsJam, had asked the organisers of all the MathsJams around the world to fill in a survey so we can get an idea of what’s going on everywhere else. There were a couple of Fermi-problem type questions at the end, asking for estimates of how many ping pong balls would fit in our pub, and how many dominoes were in a bag in a photograph. Our answers were, respectively, “probably quite a few”, and “exactly 253″, both calculated by taking arithmetic means of everyone’s guesses. I’m definitely going to fill the rest of the survey in after I finish this post. Definitely.

Someone at another MathsJam tweeted this question:

Two robots land at random locations on a sphere. What identical path should they both follow so they collide?

We couldn’t think of an answer that made sense, apart from “inwards”.

Eamonn had bought a compendium of Puzzlebombs at the big MathsJam conference. It includes a sneaky preview copy of December’s sheet, which includes a puzzle by me, CP! After lots of reassurances that it wasn’t as hard as it could’ve been, a few people cracked it.

Picture-hanging

Following my recreational maths seminar on the subject, I thought I’d pose the picture-hanging puzzle in real life:

How do you hang a picture from $n$ nails so that removing any one nail causes the picture to fall?

I demonstrated the solution on two nails a few times, using empty glasses as stand-ins for nails. A couple of people got it fairly quickly. Amazingly, Andrew said he already knew the solution because he’d solved basically the same problem in a serious knot theory paper.

Here are Andrew and Ji making their attempts:

I’ve had quite a few goes at the 5-nail puzzle and I haven’t got it to work yet. 2,3 and 4 were all straightforward, so I must be doing something wrong. Maybe I’ll work it out.

I showed everyone the Wow-Card that Mark Carney gave me at the big MathsJam. Everybody was suitably impressed. Marjane claimed to have worked out how it was made, but I’m not sure.

Eamonn had brought a pack of Set cards, so we played a few games. Andrew was way, way better than the rest of us. I pled colour-blindness.

Car-sharing puzzle

I’ve had this puzzle saved in my “things for MathsJam” file for ages and ages. I finally used it:

My brother Morty and I took a drive up to the mountains this weekend to see the Golden-breasted Flooglebird. Morty drove the first 40 miles and I drove the rest of the way. We saw the Flooglebird right away (or something closely resembling it) and immediately returned home on the exact same route.

On the return trip Morty again drove the first part of the trip, and I drove the final 50 miles this time.
In total who did the most driving, me or Morty, and how many more miles than the other brother did that driver do? (yes, there’s enough information to reach an answer!)

Ji’s cellular-ish automaton

Ji asked us a question about a rule on patterns on a grid. The setup goes like this:

Take a demi-semi-infinite grid (or the top-right quarter of an infinite grid, if you like) and fill in the $2 \times 2$ square in the bottom-left. The Rule is that, whenever you have an (oriented) L-shape with the corner filled in and the ends empty, you can swap it for an L-shape with the ends filled in and the corner empty.

Ji’s question was: what patterns can you make in the bottom-left $2 \times 2$ square? I quickly found a couple of patterns that Ji reckoned couldn’t be done, but there are 16 possibilities. Andrew came up with a rather marvellous argument that some patterns are impossible.

Write numbers in each of the squares, starting with $1$ in the bottom-left corner, $\frac{1}{2}$ in the squares next to it, and further powers of $\frac{1}{2}$ in the rest of the diagonals. So the bottom row of the grid reads $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$. The sum of the whole grid is exactly $4$. I worked this out by summing the rows individually, but I think Andrew got there a different way, or it just looked like it should be 4.

The sum of the filled-in squares is an invariant across all patterns. That’s because the squares above and to the right of a square labelled $x$ are both labelled $\frac{x}{2}$, so when you apply the Rule to the square $x$, you take away $x$ and add $\frac{x}{2}+\frac{x}{2} = x$. So no matter how many times you apply the Rule in whatever order, the filled-in squares will always add up to the same amount.

At the very start, the sum of the filled-in squares is $1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{4} = 2\frac{1}{4}$. That’s more than half of the available score so no matter what you do, you can’t end up with the bottom-left square completely empty. QED!

Eamonn presented us with the Movie Math Quiz from Spiked Math. I don’t like it.

Finally, I have in my notes that we talked about space-filling curves. I can’t remember why!

I enjoyed myself, as always. We decided December will be a non-MathsJam month, because we have better, more turkey-oriented, things to be doing. The next Newcastle MathsJam will be on the 22nd of January, TWENTY THIRTEEN!

David asked what the smallest and biggest “A” series sizes of paper could be. Fudging the aspect ratio a bit, he decided that the smallest bit of paper would be the width of an atom (roughly $10^{-10}$m), and the largest would be as wide as the observable universe (about $10^{27}$m). That would mean there are just over 120 sizes of “A” paper in-between.

David then posed a prisoners-in-hats problem. He’s written about it on his new blog.

Stepping round a circle

The prisoners-in-hats problem got us talking about sizes of infinities, and David asked this question as a motivating example:

If you walk around a circle, with each step taking you $x^{\circ}$ round at a time, will you eventually have stepped on every point on the circle?

Clearly, if you step $90^{\circ}$ at a time, you’ll only hit four different points. Similarly, any integer and any rational value of $x$ will only give you a finite number of points. But even if $x$ is irrational, the answer is still no: the points on the circle are equivalent to the interval $[0,1)$, which has the same cardinality as the set of real numbers $\mathbb{R}$. But the set of steps taken is only as big as the set of natural numbers, $\mathbb{N}$, because the steps can be enumerated. The set of real numbers is strictly bigger than the set of whole numbers, so there will always be points you haven’t stepped on, no matter how long you keep going.□

I have written in my little notebook that David asked “What’s the point of proof/maths?”. We must’ve discussed it; I assume we came up with some compelling arguments.

In celebration of Martin Gardner’s 98th birthday, I made a hexahexaflexagon from first-principles, using a compass and ruler.

I read out the problem from Tanya Khovanova’s paper, Conway’s Wizards. Sadly, nobody took it on. It’s a very very tricky puzzle.

Sarah and I had a go at making some of the knot patterns from a book I bought at Barter Books, String Figures of the Tuamotus. It took a while, but Sarah was more successful than I was. First she made a female turtle…

… which then became a male turtle

And then Sarah graduated to string figure master with this very complicated pattern:

A couple of coin puzzles

John asked this probability question:

I have $N$ coins. I place each of them in turn in piles. It’s equally likely that I’ll place a coin on any of the existing piles, or start a new pile. How many piles should I expect there to be when I’m done?

We didn’t come up with an answer. David, Stephen and Ji set to work computing numbers for small values of $N$. David demanded that I take a photo of his “maths explosion”.

Stephen and Ji were concentrating too hard to notice me take this picture.

David asked a question about coins. It goes like this:

I have a biased coin, but I don’t know what the bias is. How can I use it to make a fair decision between two choices?

If you want to think about this before reading the solution, here’s your warning that the next sentence will spoil it for you. The solution is to flip the coin twice until you get two different results, and take the result of the first flip as your answer. It works because, if the probability of getting heads is $p$, the probability of getting heads-then-tails and of getting tails-then-heads are the same: $p \cdot (1-p)$.□

Hanging pictures just about safely enough

When David got hold of the wool we’d been using to make the string figures of the Tuamotus, he asked us this:

How do you hang a picture from $N$ nails with a single piece of string, so that removing any one of the nails makes the picture fall?

David showed us the solutions for two and three nails (or fingers of a willing participant in our case), which involve wrapping the string in a particular pattern around the nails. I first saw this trick at a Maths Busking event, but I can never remember the solution. On googling, it turns out the Demaine brothers and others have written a big paper on the subject.

Somehow, my pair of scissors ended up with some string tied very tightly to one of the handles. Since we had no other cutting implements, we agreed that David would have to run relativistically quickly near a black hole with the scissors to cut the string off.□

In preparation for my PGF talk the following week, I went through the zero-knowledge sudoku protocol with John.

Finally, we learned that googlewhacks don’t exist any more, because somebody has created two pages each containing every pair of words. I haven’t checked that fact; I hope it isn’t true.

The next Newcastle MathsJam is on the 20th of November, two days after the big MathsJam conference at Wychwood Park. I’m looking forward to both!

Click here to see the slides.

I used deck.js, along with the Computer Modern web fonts and MathJax, to make the slides. I think it looks pretty nice! I’ve also uploaded my template deck, in case you want to build on it for your own presentations. It’s a bit big because it contains all the files needed to display the Computer Modern fonts on any browser.

September’s MathsJam was very enjoyable. We had some magic tricks, debated the application of game theory to the penal system, almost played a game of rhythmomachy, and of course we solved a few puzzles.

David’s card tricks

David knows a lot of card tricks. He showed us a few, but I think the Magic Circle would do bad things to me if I explained them here. I did record David doing a different, very mathematical, card trick a while ago, so here’s that:

Never start a sentence with a proposition

David asked a provocative question: what if, when a prisoner killed another prisoner, they got the murdered prisoner’s sentence taken off their own? What would happen? Would anyone live long enough to serve out their sentence?

A bit of mathematical modelling followed. We assumed that prisoners’ ability to kill each other is directly proportional to the length of their original sentence, and for the sake of simplicity assumed one fight to the death per day, between a randomly chosen prisoner and his target. Because David had a deck of cards out, we used it as a source of random numbers, to give everyone an initial sentence, to pick prisoners and to decide who won fights.

In our simulated prison, Vicky, Ji and Stephen all died; I was released ten days early for killing Vicky; David was released having fought nobody, and John was released after only two days with a bone-chilling seven-day imprisonment credit. We can only hope he doesn’t decide to use it.

David had actually asked me his question earlier in the day, when we were supposed to be doing proper work, so I wrote a Python script to simulate the game. I’ve assumed that prisoners will choose to fight the opponent with the maximum expected sentence reduction. When sentences range between 1 and 20 days, just over half of all prisoners are released, on average, which I think counts as evidence that our policy is humane. As the range of sentences gets bigger, however, the proportion starts to decrease, as you can see in this chart I made of the results.

The obvious next question to look at is whether length of sentence is correlated with length of stay in prison, either positively or negatively. I’m really not sure what the answer would be.

John’s puzzles

John (or Twitter, I’m not sure) showed us this fun fact:

If you start with a square grid of 25 dots and pick 5 of them, there will always be at least one pair of picked points such that the centre of the line between them lies on a dot.

John also told us the rules of a rather fun guessing game. Player 1 picks a number, and keeps it secret. He then announces a “guess”, which could be any number at all. Player 2 announces a guess, and Player 1 says if it’s closer or further from the secret number than his own. Player 3 then makes a guess, and Player 1 says if it’s better or worse than Player 2′s. I think the aim is for the last player to guess as close as possible to the secret number.

Ji’s puzzles

Ji posed us a couple of nice puzzles. The first began with him emptying his coin purse onto the table.

He then explained: he and anyone who thought they could outsmart him would take turns picking piles to remove any number of coins from. The last person to pick up a coin would be the winner. I think most of us recognised it as Nim straight away, but David played along anyway.

Another game is to take turns picking numbers in the range 1 to 9, with the first person to make 15 the winner. This game is equivalent to noughts and crosses because of the following picture:

Ji ended with a much harder challenge. He claimed that most of the time, when you pick 4 cards from a deck, you can make 24 by combining all of them through addition, subtraction, multiplication or division. We thought about one instance of this a few months ago – can 8,8,3,3 be combined to make 24?

We had a few goes, and found a couple we could make 24 with and a couple we couldn’t. So the jury’s still out. We couldn’t do this set; can you?

When I got home, I wrote a Python script to find solutions, if they exist. It finds the solution for 8,8,3,3 but not for the 4,9,12,13 pictured above, so I reckon it works. It’s a pretty stupid exhaustive search though, so I’m not going to set it off deciding how often sets of cards have solutions.

Back at MathsJam, a few of us started trying to enumerate all the possible expressions on four numbers. We started drawing binary trees, until Ji pulled a pad of paper out of his bag and put an end to the conversation.

More games

I got my homemade Rhythmomachy set out again, but nobody wanted to play after seeing how it sent Eamonn and me mad when we played it for the first time in July.

While I was trying to goad people into playing Rhythmomachy, David started trying to get people interested in the definition of amenability for Banach spaces. This was not a highlight of the evening.

There was a quick game of Tic-Tac-Othello, which I didn’t get a picture of. I made an online version of the game a while ago. The rules are pretty much what you’d expect them to be.

Put a Wieringa roof over your head

I followed Adam Goucher’s instructions for constructing a Wieringa roof from golden rhombi. The Wieringa roof is an aperiodic tiling of the plane with just one kind of tile, but it gets round the fact that nobody knows if one of those exists by allowing the tiles to tilt up and down, producing a corrugated surface whose projection onto the plane looks like the Penrose rhombus tiling.

It turns out I got mine wrong, so it looked a bit odd, but it worked as a hat to protect me from David’s Banach space nonsense. Vicky captured the scene in a photo that definitely didn’t take 90 seconds to pose properly.

I claimed that the secret spy listening posts that are dotted around our countryside use Penrose tilings in their construction to minimise funny diffraction effects as radio waves pass through their shells, but I’ve had a look and I can’t find any evidence to back me up. I’m sure I read it somewhere.

I’ve since had another go at constructing a Wieringa roof, which is going much more successfully so far:

Jokes

We ended the night with a mix of mathematical yo mamma jokes, Gauss facts, and discussion of ways to outwit bookies and lotteries, none of which I can remember.

I had a lot of fun! So much that it prompted me to get back in the habit of writing these recaps. I hope October’s MathsJam is just as good.