### The Double Back puzzle is a nice mix of Towers of Hanoi and Solitaire

This post is both a video and text. The content is largely the same in both versions, so you can pick one to look at.

I’d like to show you a puzzle, or game for one person, that Ed Kirkby came up with. Ed showed this to me at the Big MathsJam gathering last year.

I was walking around MathsJam and people were doing all sorts of stuff and Ed was quietly sat at a table with some cards in front of them, and a couple of people were gathered around with thoughtful faces, so I thought “oh that looks interesting” and I walked over and Ed explained the rules of this game to me.

And like most things at MathsJam, I didn’t immediately know the solution but it really got to me, so I went away and tried to find a solution myself. It was really fun and it was one of those pure experiences of mathematical discovery that you really remember.

I kept grabbing other people and saying, “have you seen Ed’s puzzle? Come over here. Ed, explain this puzzle to this person!” I did that for the duration of the weekend, and for a little while after I got back home.

I’ll show you what the puzzle is.

Take two sets of cards numbered 1 to 4, and lay them out in two rows. The game is to get them the other way round, so each row goes 4, 3, 2, 1.

The things you are allowed to do are to swap two cards that are adjacent numbers and in adjacent columns.

So you can swap the four and three here:

You can also swap to the other row.

You can’t swap the four and the two here because they’re not adjacent numbers.

It’s a permutations puzzle. And with a bit of thinking, a lot of just making random moves, eventually I worked out how to do it for four cards (or two lots of four cards, to be precise).

It took me a few more goes to be sure that I’d found the shortest solution — the fewest moves — but then five really stumped me.

When I tried six cards, I wasn’t sure it was possible at all: there was a point where I kept getting stuck.

So I was really unsure, and Ed was unsure as well – they didn’t know if there was a solution for every number of cards. They’d written some code that very slowly churned out solutions for small $n$, but didn’t have a general solution.

So I did my usual problem solving technique which was, if I couldn’t do it with four cards, do it with three cards, and very briefly with two cards. Then after a while I sort of started seeing a pattern.

I came away from Big MathsJam not knowing how to solve this puzzle.

I kept playing with it when I should have been working, and eventually I think I’ve sort of got the vibe of how it works and I’m pretty sure now I can solve it for any number of cards. I haven’t written down a really rigorous algorithm for doing it but I’ve got a general idea.

### Interactive version

Something that often happened when I was playing with cards was that I’d forget which way I was going, because when you don’t have a strategy you’re sort of making lots of moves to see where they’ll go, and then maybe you want to backtrack a bit, and you go like wait wait — which end did the aces start at? You’re continually picking up cards and moving them, and if you’re as clumsy as I am they scatter all over the place and you end up unsure if you’re in a valid state.

So it won’t surprise you to learn that I made an online version of this.

Big MathsJam happens in November, and by the time I’d got this online version working it was getting close to Christmas and I thought, “we need something that’s not playing cards to be the pieces – why shouldn’t it be baubles? They’re circles, they’re easy to draw.”

So I spent some time drawing a bauble image and making it look Christmassy. I didn’t quite get it ready for last Christmas, so I spent some more time improving the interface and I made it fully keyboard accessible.

And then I made a note to write a post about it in December 2023. And here we are!

I think this is a nice Christmas puzzle: I think the set-up and the rules are easy to remember, and it only needs a pack of cards, which you’ll probably have lying around.

So over Christmas, if there’s a quiet moment, if you get some playing cards in your Christmas cracker or just some pieces that you can put in an easy to identify order, try showing this to someone.

So there you go, that’s the Double Back puzzle, invented by Ed Kirkby, online version made by me.

### Hexaflex yourself!

Hexaflexagons are great.

If you haven’t seen one before, you’re about to have a lovely time.

If you have seen them before, the reason I’m writing about them is that I’ve made a webpage that helps you create a template for a hexaflexagon with your choice of picture on each of the three faces.

### We all need some space!

A couple of days ago, a question occurred to me:

What’s the furthest I’ve ever been from anyone else?

### What Can Mathematicians Do? Recordings of ten talks by disabled mathematicians

In December I organised a series of online public maths talks called What Can Mathematicians Do?

The recordings of the talks are now online, free for anyone to watch. You could go to the official page I put up on Newcastle University’s website, or you could just watch them here!

### How to fold and cut a Christmas star

This week and last I hosted a series of public maths talks featuring disabled presenters. I’ll post about how that went later, but for now I just want to share this clip of me filling time spreading Christmas joy.

This is a party trick that Katie Steckles showed me: you can fold a piece of paper and then make a single cut to produce a five-pointed star. I showed how to do it by following the instructions I’d been told, and then recreated the steps just starting from the insight that when you make the cut, all the edges of the shape need to be on top of each other.

Maybe you’ll show someone else how to do it during the Christmas holiday?

This doesn’t only work for stars: there’s a theorem that you can make any polygon by folding and a single cut. Erik Demaine has made a really good page about the theorem, with some examples to print out and links to research papers. Katie can cut out any letter of the alphabet on demand, which is impressive to witness!

### Integer sequence review: A101544

It’s nine years since the first integer sequence review, and six years since the last one. We’ve grown as people, and in CLP’s case, grown people. The world has changed, but our love for the Online Encyclopedia of Integer Sequences hasn’t.

#### A101544

Smallest permutation of the natural numbers with $a(3k-2) + a(3k-1) = a(3k)$, $k > 0$.

1, 2, 3, 4, 5, 9, 6, 7, 13, 8, 10, 18, 11, 12, 23, 14, 15, 29, 16, 17, 33, 19, 20, 39, 21, 22, 43, 24, 25, 49, 26, 27, 53, 28, 30, 58, 31, 32, 63, 34, 35, 69, 36, 37, 73, 38, 40, 78, 41, 42, 83, 44, 45, 89, 46, 47, 93, 48, 50, 98, 51, 52, 103, 54, 55, 109, 56, 57, 113, 59, 60

### Presenters wanted: a series of public maths talks by disabled mathematicians

I’m organising a series of online public maths talks through my work, the School of Maths, Stats and Physics at Newcastle University.

The point is that talks will be delivered by disabled presenters. This came about because I and some other disabled people who do maths talks got tired of missing out on opportunities to do outreach because it involves travelling. Not every disability makes travelling harder, but we felt that there were enough people excluded by in-person events that it would be nice to put on a more accessible event.

My aim is for this to take place in December 2022, near to the International Day of People with Disabilities. Talks can be any mathy topic, or about your experience as a disabled mathematician.

I need speakers!

To give some idea of what I’m looking for, I’ll use myself as an example. I count myself disabled at least four ways: I’m colourblind, autistic, dyspraxic, and have Ehlers-Danlos syndrome.

• Use of colour in mathematical communication, for example how red chalk makes chalk-and-talks inaccessible.
• How ambiguously-worded maths problems have stymied me in the past, and how to write them more clearly.
• Integer sequences, which is just something I’m interested in.

The sessions will be delivered over Zoom, with live captions written by humans and a BSL interpreter. (If you can recommend a BSL interpreter with experience of interpreting maths talks, please get in touch!)

We’ll be advertising the talks to the general public, both grown-ups and schools, so I’m not looking for talks about high-level maths or education research.

This is open to anyone around the world, but if you’re a long way from the UK bear in mind that we’ll schedule the sessions at a convenient time for a British audience.

If you’re interested in taking part, please email christian.perfect@ncl.ac.uk.

I’d like to have a list of presenters by the end of September, to leave plenty of time to arrange whatever needs to be arranged and to advertise the talks.