This post is about a puzzle I’ve been tweeting about for the last couple of days. I got it originally from a book I was given back in the 1980’s called “Ivan Moscovich’s Super Games”. In the book, Ivan calls this puzzle “Bits”, but I don’t think that’s nearly descriptive or cute enough, so I asked my daughter what it should be called and we have come up with the much better name of PANDA SQUARES.

The puzzle starts out like this.

You have 16 square tiles, each made up of four small squares. These smaller squares are coloured either black or white. In fact, the 16 squares cover all possible colourings considering different orientations as different. (Figuring out all the possibilities and how many there are is a puzzle in itself, really, and I imagine would make a lovely starter for this puzzle if you were to do it in a classroom.)

You can download a Word document or PDF document with each square on a different page. Printing it 16-to-a-page works well to have individual sets, or you can do what I did and print them big then laminate them!

Anyway, the puzzle is this: arrange the 16 tiles into a 4 by 4 square so that the colours match along the edges of tiles next to each other. There are lots of possible solutions, so I don’t think there’s any danger in showing you one here:

And that’s it. One thing that’s nice about it is that after getting a solution you can scramble the pieces and very easily get another one that isn’t the same, or someone else can have a go after you. At the Celebration of Mind event yesterday we had the attendees at the barbecue try it out, and today at our regular One Hundred Factorial session we all had several turns making our own. (The one above is my favourite from today because I think it looks vaguely rabitty.)

Only that’s not it! If you look at several different solutions, you may find that you notice a few things, and that several questions occur to you, just like happened to us! In order to help us in our investigations, we had photos of all our solutions on one of our phones so we could flick through them. We also had 8 by 8 grids on paper so we could copy down our solutions and draw all over them. Plus, as always, we had a big whiteboard to draw our thoughts all over too.

Here are some of our noticings and wonderings and where we’re up to with them. If you don’t want spoilers I suggest you don’t look too closely!

  • We wondered: How many possible solutions are there?
    A student, James, asked this question yesterday and came back today having written an algorithm to find them all. He’d found 10904, not counting rotations as different. He said he hadn’t proved his algorithm actually did cover all the solutions exactly once, but I wasn’t going to let him discount the awesome work he had done!
  • We noticed: The four little squares surrounding the intersection of four tiles are the same colour.
    And we did manage to prove that yes this must always be the case.
  • We wondered: Is it possible to have a solution where all the identical tiles are in a different orientation?
    It is, because a student, Lewis, found a solution today with this property!
  • We noticed: There are two black and two white squares in the four corners of the big square.
    Then we found some other solutions where this wasn’t the case. But Finn did manage to prove that the only other possibilities are all black or all white big-square corners.
  • We wondered: Is it possible to have only one connected black region? (Connecting only by a corner doesn’t count)
    We’re pretty sure it’s not possible, but we haven’t got a proof we all find convincing yet.
  • We noticed: The number of black regions and the number of white regions is either the same or one different.
    Again, we’re pretty sure this is actually the case, but again we didn’t get any argument that was convincing beyond a doubt. Lewis and Daniel had this idea about the edges of these regions having to cross at the diagonal-two-black tiles, but it wasn’t getting us anywhere yet.
  • We wondered: Is it possible to draw a cross on a solution so that a quarter of the cross is black (or white)?
    Well, I wondered that anyway, but you know my obsession with Quarter the Cross, don’t you? It turns out the answer is yes you can, but not all the time – I was quite proud of my inspiration to trace the cross on a plastic sleeve and move it around on a drawn version of the solution to see if I could make a quarter. Everyone worked together to come up with the surprisingly severe restrictions on how you can do it if the squares in the cross are whole tiles.
  • We noticed: At least one of the all-black and the all-white tile were always on the edge.
    And yes this does in fact have to be the case. I think it was Michelle who had the inspiration for the proof of this one. It was quite clever and involved the fact that each of these tiles forces a surrounding border of the same colour squares.
  • We wondered: Is it possible for the all-black and the all-white tiles to be in opposite corners?
    We’re still not sure on this one. We did try a few times but couldn’t make it work, but then again we don’t have a proof that it’s not possible.
  • We noticed: None of the solutions are symmetrical.
    And there were several different proofs of why none of the solutions could possibly be symmetrical with different types of symmetry.

I’m sure there were several other things we were working on that I wasn’t there for since there were about ten of us going simultaneously! (Also one of the reasons I may have gotten the attributions for ideas a little mixed up — sorry!) And we haven’t even started on other arrangements of the tiles or other sized tiles!

I wasn’t quite prepared for just how fabulously rich this puzzle was, with all sorts of great noticings and wonderings. On top of that, it’s just so aesthetically pleasing and it’s still a sheer joy to see all the beautiful solutions that come up.

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It seems like ages ago — but it was only yesterday — that I wrote about differentiating functions with the variable in both the base and the power. Back there, I had learned that the derivative of a function like f(x)g(x) is the sum of the derivative when you pretend f(x) is constant and the derivative when you pretend g(x) is constant.

Since then I have realised that this idea actually dictates ALL of the differentiation rules where two functions are combined through an arithmetic operation! It’s everywhere! Look:


Oh my goodness – it’s the product rule!


Oh my GOODNESS – it’s the quotient rule!


OH MY GOODNESS – it’s the addition rule!

So it’s everywhere! This is totally awesome!!

In the comments in the last post, it was mentioned that the reason all this happens is because of the multivariable chain rule, which is all about holding all but one variable constant and adding together the different results. When I realised this I thought to myself “duh!” – I was using the same words to tell myself what to do in both situations.

The issue is that when students learn these rules, they don’t know multivariable calculus so this can’t be used as a way to explain what’s happening. But I have a good feeling that it might be able to help make sense of the multivariable chain rule when they do learn it because it will connect perfectly so something they already know. I’ll let you know if I ever try.

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Here in Australia, we are at the tail end of a reality cooking competition called “Zumbo’s Just Desserts“. In the show, a group of hopefuls compete in challenges where they produce desserts, hosted by patissier Adriano Zumbo. There are two types of challenges. In the “Sweet Sensations” challenge, they have to create a dessert from scratch that matches a criterion such as “gravity-defying”, “showcasing one colour” or “based on an Arnott’s biscuit”. The two lowest-scoring desserts from the Sweet Sensations challenge have to complete the second challenge, called the “Zumbo Test”. In this test, Zumbo reveals a dessert he has designed and the two contestants try to recreate it. Whoever does the worst job is eliminated.

I find it very interesting that the Zumbo test is the harder of the two tests. In the Sweet Sensations challenge, the contestants can choose to use whatever skills they are already good at, and design their dessert in a way that they can personally achieve. In the Zumbo Test, the contestants have no control over the techniques that are required, and must try to do things they are not familiar with in ways they may not have seen before.

And why am I talking about this? Because my medical students find themselves in similar situations. Our medical students have two projects to do as part of their research curriculum during their third year. One project is a research proposal: they work in a group with a supervisor to plan a hypothetical research project, including ethics, literature review and (this is where I come in) statistics. The other project is a critical appraisal: they work in pairs to analyse a published article, including where it fits in the research, the writing, the importance and (again where I come in) whether the statistics is appropriate.

I have noticed over the years that in terms of statistics, the critical appraisal is harder than the research proposal. A meeting with students about the critical appraisal usually takes twice as long as one for the research proposal, and twice as much preparation for me. Many more students come to me to talk about the critical appraisal, and the ones who do come are more worried about the statistics they find in the critical appraisal than the statistics they need in the research proposal. Why is this?

When watching Zumbo’s Just Desserts, it occurred to me that the reason why is the same as the reason the Zumbo Test is harder than the Sweet Sensations challenge.

When doing your own research you can choose to only investigate questions in such a way to use the statistical methods that you understand. Even if you need a new statistical method, you just need to learn that one. Either way, you have complete control over your own decisions and know the things you are measuring and what they mean. It’s just like in the Sweet Sensations challenge the contestants get to make all the choices and use methods they are familiar with.

On the other hand, when reading someone else’s research, you have no control over the wacky statistical methods they choose to use. Even if they are the appropriate ones (they often are in medicine, actually), the paper almost never describes how the researchers decided to use them — it just says what they used. And they often measure new things in new ways that you don’t deeply understand. It’s just like in the Zumbo Test the contestants have to do things that are new to them in ways that are new to them.

It’s much much harder to understand the statistics in someone else’s research than it is to make your own.

Let’s just hope we don’t eliminate all the students by asking them to do it with less support.


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In Maths 1A here at the University of Adelaide, they learn the following theorem (this is taken from the lecture notes written by the School of Maths here):

It says that, given a function of x defined as the integral of an original function from a constant to x, when you differentiate it you get the […]

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I was talking to a student about his calculus last week. He was trying to differentiate xx. (Actually he was trying to differentiate x ln(x) and had decided the best place to start was to raise e to the power of it, thus producing xx.) At first he tried this:

I asked him what he thought […]

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It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the […]

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Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff like this:

There were several small issues a lot of them had, which combined to stall their calculations. One of the more important problems was about how the calculator interprets what they type […]

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I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I’m not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind […]

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Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, […]

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In maths, or at least university maths, there are a lot of statements that go like this: “If …., then …” or “Every …, has ….” or “Every …, is …”. For example, “Every rectangle has opposite sides parallel”, “If two numbers are even, then their sum is even”, “Every subspace contains the zero vector”, […]

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