## Pages

### The Third Reich Is Listening

As Bletchley Park's "Enigma Man*" I was often asked the question "... but what were the other side doing?" The museum doesn't cover this side of the story other than very fleetingly and in passing so it was always tricky to answer, but it is nevertheless a very pertinent - and thoroughly interesting - question.

 This is my copy of The Third Reich is Listening and you can't have it (but you're welcome to buy your own, or borrow it from a library, of course)
This post is, essentially, a review of Christian Jennings' The Third Reich is Listening.

Its subtitle "Inside German Codebreaking 1939 - 1945" tells you the main focus of the book, but the story does start earlier than that, summarising work done by key characters in World War I and the inter-war years.

BP's story is utterly fascinating whether you have the briefest of overviews or a more in-depth understanding. As you wade deeper you find that every part of it is much more complicated and involved, from a historical perspective, than you had ever thought possible. Every aspect of allied operations in World War II has the site's work somewhere in its background and the story has an almost fractal nature to it: pick any part you like, zoom in, and you'll see it explode into so much detail you'll wonder why nobody has started a museum dedicated to just that bit.

You'll also notice, as you learn more and more, that almost every aspect of the story is steeped in myth and misconception. So many things that are common knowledge, routinely-told anecdotes when scratching the surface turn out to be simplifications, half-truths, or downright incorrect as you pick apart the evidence.

One such myth is that the codebreakers were working to reverse-engineer an almost completely unbreakable cipher. This, when dipping your toe into the story, is not incorrect: Enigma** is a tough nut to crack. But as you wade into deeper waters you find there's an unspoken caveat: if it's used properly. Bletchley's codebreakers had a helping hand, and not just from the Polish mathematicians who figured so much of it out before the war even started: Germany itself proves very helpful in the fight against Enigma. At the lowest level, mistakes were made by Enigma operators that were of great aid to their counterparts in the brick blocks and wooden huts of Bletchley Park; as were many features of the procedures put in place at much higher levels.

One key theme in favour of allied codebreaking was Germany's disbelief that its ciphers had been (or could be) broken. As Jennings puts it, "the Third Reich might have been listening but increasingly it had its fingers in its ears."

All this is just a very brief, very anglocentric view of what was going on in Germany.

Whilst there are still many and varied parts of allied codebreaking in World War II still to be uncovered, clarified or corrected in the public mind, the mirror within which we may see Germany's point of view is cloudy and smudged, and hasn't been dusted in a long time. Jennings' book takes a damp rag to it and does a surprisingly good job of clearing away some of the grime.

And what we can see is just as fascinating as the story we already know. There are parallel figures in that world, characters even more forgotten than the codebreakers of Bletchley Park, such as Wilhelm Tranow, and internal fights against an oppressive and militaristic regime to rival those depicted in The Imitation Game. There are revelations of mistakes made by British encryption procedures and communications staff that equal those made by Germany, and which similarly aided German cryptanalytical victories. There are disastrous decisions that reduced its effectiveness and eroded the viability of what could have been a genuinely effective rival to Bletchley Park.

If you're interested in secret messaging during World War II and, so far, you have only read about Bletchley Park you have restricted yourself to less than half of the story. The Third Reich is Listening is a surprisingly comprehensive introduction to the parallel world that is German codebreaking in World War II. It takes us out of the narrative that they're just the Bad Guys in the Bletchley Park story and shows us the human ingenuity, struggles, victories and defeats of a whole new, largely unexplored dimension of the history of cryptography and cryptanalysis.

* I used to visit schools (and other places) with an Enigma machine on behalf of the Bletchley Park Trust. I would often introduce myself at reception, with the response "oh, you're the Enigma man!"

** Of course, another myth is that Enigma was the only, or toughest, cipher being broken at BP.

### The Most Important Thing in Your Pencil Case, or Calculators Ain't Bad

I have a friend who's starting secondary school in September. Her mum sent me this:

I need your opinion on something. B is laughing at me because I said that choosing a new pencil case is THE MOST IMPORTANT thing about a new school year. I messaged my English dept, but she said they’re bound to agree and I needed to message a Maths teacher. Thoughts?

I responded with:

I nearly agree... THE MOST IMPORTANT thing is assembling your pencil case kit, which includes choosing a new pencil case, but you have to make sure the right kit is in there - non-exploding pens, for example. The stuff that goes inside is important because that will help to inform which pencil case is the right one. It might feel right for a moment, but if it doesn't fit what you need, or allow you to arrange it how you want to then it's not the right one.

 Two calculators that have stood the test of time from my own school days. I've just noticed that the batteries haven't, though.

I then listed my essential pencil case kit. That's not really what this post is about, but of course I have included it at the bottom for all and sundry to disagree with.

The final thing on the list, THE MOST IMPORTANT thing to go in your pencil case, is a calculator.

And that's something I really do believe.

## Calculators are not the enemy

### Teaching to the Void

Following a conversation sparked by one of their training events GEM, the professional network for people interested in learning through museums and heritage, asked me to write a guest post loosely based upon (and an expansion of) the ideas in this post. It's here:

## Teaching to the Void at GEM.org.uk

### Is Zero Odd or Even?

It's been a while since I posted something mathsy, but I've seen a few people being wrong about this one lately, so here goes...

Let's start with the popular-but-wrong response; the one that would get the QI alarm awooga-ing: "it's neither!"

Wrong!

Now I've lit the blue touchpaper for a few ranty comments or tweets, let's look at the facts. A good place to start is to take a step back and ask...

## What exactly is an even number?

The first port of call for almost all questions must surely be the great* Wikipedia, which says that "an even number is an integer that can be divided by two and remain an integer or has no remainder."

Wolfram|Alpha, possibly a more reliable source of mathematical information in many eyes, says "an even number is a number in the form $n = 2k$ where $k$ is an integer."

### Does zero fit here?

Wikipedia first:
• An integer is a number with no fractional part (i.e. no decimals; a whole number): ✓
• If we divide zero by 2 we get zero (with no remainder): ✓
And Wolfram|Alpha:
• If we can find an integer $k$ such that $0=2k$ that means $0$ is even:
$k = \frac{0}{2}$, so $k = 0$. ✓
Both Wikipedia and Wolfram|Alpha agree:
$0$ is even.

## Need some more?

• Back at primary school you may have learnt a quick way of identifying whether a number of any size is even by glancing at its units digit: any whole number that ends in $0, 2, 4, 6$ or $8$ is even, $0$ ends in... $0$. So it's even.
• Additionally, you might also recall the rule that if you add two even numbers together the result is always even, for example: $2+2=4$ $120 + 346 = 466$
Well, $-2 + 2 = 0$, so $0$ is even.
• It's also true that two odd numbers added together ($3 + 7$, for example) give an even number ($10$).
What's $-3 + 3$? Oh, it's $0$, so that's even.
• Multiplicatively, any even number multipled by any integer results in another even number, e.g.:
$2 \times 3 = 6$ $84 \times -37 = -3,108$
$28 \times 0 = 0$, so $0$ is even.
• We're all familiar with the number line, stretching from $-\infty$ to $+ \infty$ . Let's look at a section of that line with $0$ in it:
... , -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, ...
And let's colour in the even ones:
... , -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, ...
Without $0$ included there's a big hole and the pattern is broken. Why would we do that to a perfectly decent pattern?

## Still not sure?

Many a maths fan's bible, the On-line Encyclopedia of Integer Sequences (OEIS) naturally includes an entry on the nonnegative even numbers and this begins with $0$. Are you going to argue with the OEIS?

* And often, I admit, fallible

### Stony Stratford: Things to See and Do

 If you're looking for things to do in Milton Keynes I've recently discovered* Sophie etc, a blog covering just that topic. It focuses on food (she's pointed me towards some GREAT takeaways during lockdown) but also posts now and then about other adventures that are up for grabs in the area.There's a post (this one, in fact) about all of the fantastic things you can do in Stony Stratford, a gorgeous former market town up in the furthest reaches of north-west Milton Keynes.Actually, she doesn't quite cover all of the things that are worth seeing, though the one I'm going to tell you about is a bit niche.

## A little background

EDIT (06/07/2020): Now with added PowerPoint!

I've taken part in a good number of online discussions, seminars, conferences and the like since our Lockdown period started and on the whole it's all been very good, with incredible use of broadly (and often freely) available use of resources to turn an experience that could be flat and impersonal into a very positive one. There are a few things that niggle me, however, and I've been looking into overcoming them, especially as I've been moving into this online teaching lark as well. I thought I'd post about some of the things I've found out and include a few "how to..." walkthroughs. This is largely so that I can find these things again but I realise there may be others out there who might benefit.

 Thoughtful transitions can be used to make slides clearer and less cluttered. Losing them can make a big difference.

If that's you, fill your boots! Also ask questions if I'm at all unclear, miss out an important step, or say something you don't understand. If I get something wrong correct me; if there's an easier way let me know!

The other day I posted...
... and a couple of people asked me how to do it.

To some of you this may be the most obvious thing on the planet, but I only discovered it earlier this year and - my oh my - has it saved me some time. My day* job sees me visiting various places around the country and I keep track of it all in my calendar. As any self-respecting nerd is driven by unknown forces to do, I often put a bit of effort into finding out how I can get the tech around me to do some of the more boring jobs so that I have more time available for doing the interesting things. I'd list some potential use-cases for this spreadsheet-to-calendar trick but I figure that if you've read this far you already know what you want it for.

Before we get started, a disclaimer: Yes, there are plenty of other articles detailing how to do this and I visited many of them. I haven't referenced any of them because it was about six months ago so I've forgotten which ones I used, none of them gave me every piece of information I needed, and there were a few important stumbling blocks that weren't mentioned at all.

### Another Puzzle for the Today Programme!

I set another puzzle for Radio 4's Today programme and it was featured this morning (Monday, 6th April 2020).

It goes like this...

"Doc note, I dissent. A fast never prevents a fatness. I diet on cod."

Bletchley Park Codebreaker Peter Hilton constructed this impressive 51-character palindromic sentence, allegedly in response to a challenge set by a fellow Codebreaker. Palindromic numbers work in the same way: the same digits in the same order whether you read them backwards or forwards.

What is the 51st Palindromic number?

You can find the puzzle (with a link to a solution) on the Today Programme's website, here.

As with many puzzles the solution is debatable depending on some starting assumptions. Feel free to discuss your assumptions (and how these might change the answer) in the comments. Teachers, I'd love to hear how you might use any of the ideas or topics featured (even tangentially) in your own teaching!

### Research Poster: Digital Mathematics Resources and Museums

Back in September I asked for some maths teachers to take part in the trial of a mathematics resource that I had created and was trialling as part of the work towards my PGCert in Digital Leadership.

With massive thanks to the teachers and students who took part, I completed the project which formed the second assignment towards the first of two modules in the course. Our brief was to present our report in the form of a research poster (with a creatively loose definition of what that means).

Here's my final submitted piece (click to view it embiggened).

 Page 1: Academic Poster. Click to enlarge.
 Page 2: Appendices

### M2, M4, M6, M8, Motorway!

I've always* wondered how far it was possible to go staying only on consecutively numbered motorways in the UK. I've also been looking for a reason to fiddle around with Graphviz since the eternally helpful Colin Wright pointed me towards it a while ago.

So I found a database of the UK's motorways and created a graph:

 Graph generated using dreampuf.github.io/GraphvizOnline

Labelled nodes represent the motorways themselves and edges represent a shared junction between two motorways. It strikes me that this is almost exactly opposite to the way that motorways actually work.

From this diagram, it appears that the answer to my original question, "how far is it possible to go staying only on consecutively numbered motorways in the UK?", with the further refinement that "far" means the number of motorways experienced rather than an actual distance, is rather more boring than I was hoping for.

Unless I've missed something (please correct me if so) it appears to be...

... wait for it...

 Photo by Jack Hunter on Unsplash

That is:
,  , .

Sorry, folks.

Another interesting** thing that pops out of the graph above is that there are no disconnected subgraphs: it's possible to visit all of the UK's named motorways without spending time on a road that isn't a motorway.

What else have I missed? Can anything interesting be gleaned from the graph above? Does anyone have an ideas about how it could be used educationally? Are there any questions (deep, shallow, important or who-cares) that might be clarified by such a graph?

* Well, not always, but, y'know.
** To me, that is. If you find it interesting too, I'm chuffed to bits. If you don't, I don't need to hear about it thanks. What are you even still doing here?

## My Blog List

• - As part of Cambridge Mathematics' *Interviews and Intersections* series, Energy & Environment Project Coordinator Zoe is interviewed about her work and l...
• - *Question posed by Daniel, who is six and absolutely bonkers about physics. This might be a little ahead of where he's got to so far, but it'll just give h...