Upon Amie and Cathy‘s request, I am writing a blog post about a problem we worked on at One Hundred Factorial recently. In fact, in order to do so I am creating a whole new category for the blog called One Hundred Factorial, so I can talk about the things that happen there. (Just so you know, One Hundred Factorial is the name of our regular puzzle-solving gathering of staff and students here at the Uni of Adelaide. I tweet about what we do under the hashtag #100factorial.)

Anyway, a few weeks ago we became interested in problems involving removing spots from dice. A good place to start is the following puzzle, which we created during our investigations:

spotless-die-1

In order to solve this, it might be useful to know the ways that we envisaged the dots on a die are arranged, which are shown below. We don’t really care about the orientation of each face, only the relative arrangement of the dots. Also, it might be very useful to know that on a standard die the faces opposite each other add to 7. (Go look at a real one and you’ll see it’s true.)

spotless-die-faces

If you have a go at the puzzle, you’ll probably find yourself doing a bit of quite sophisticated reasoning using various if-then statements. I can imagine quite an interesting exercise of writing down all these statements and arranging them into a proper proof of what the original die must have looked like. If you’ve got a group of people, see if you can come up with multiple different proofs that follow different lines of reasoning.

But that’s just the starter! The real fun for us happened when I asked how I might change the question. In the problem above, we removed four of the spots and were still able to tell which face was which. Just how many of the spots can we actually remove?

spotless-die-2

This problem is very interesting indeed, especially because there are multiple ways to interpret the question. If you own the die and are familiar with how the faces were arranged relative to each other before you removed the spots, you might get a different answer than if you were a new user of the die who only knew it was a die with some spots removed. As you work through them, you can create some puzzles of your own like the intro puzzle (that’s how the intro puzzle was created in the first place, but I thought it’s a good way to get started with the concept).

There are several other questions you can ask that we didn’t even begin to delve into yet, but I would love to think about one day:

  • How many spots can you remove and still be able to tell what number was on each face? (Did this one already)
  • How many spots do you have to remove to make it so you can’t tell what some faces are?
  • How many spots can you add to the die and still be able to tell what number was on each face?
  • How many spots do you need to add to make it so you you can’t tell what some faces are?
  • How does all of this change if you don’t require/know that opposite faces add to 7? (thanks for this Fred Harwood.)
  • How would all of this change if the faces of the die were some other numbers from 0-9?

PS: I should say that the inspiration for this puzzle was from seeing Eric Harshbarger’s wonderful dice collection, though the versions of the puzzle here are all my own (with the help of some of the other puzzlers from One Hundred Factorial).

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If you google “fundamental trig identity” you will get many many images and handouts which all list the fundamental trig identity as:

sin2 t + cos2 t = 1

This is, of course in the wrong orderĀ and it should really have cos firstĀ then sin, like this:

(cos t)2 + (sin t)2 = 1

“But David,” you say, “it’s addition, so it doesn’t really matter what order it’s in does it?” Of course it does! Mathematically it’s the same, but psychologically it’s different. If it really wasn’t different then you would sometimes write cos first and sometimes write sin first, but I can bet you always write it in a particular order. And if you write it with sin first, then you’re making it harder for yourself.

Let me explain.

The reason we have the fundamental trig identity is because the angle t there is a piece of the circumference of a unit circle, and cos t and sin t are the coordinates of the points on that unit circle. If I asked you to write down an x-y equation for the unit circle, you would naturally write x2 + y2 = 1 with the x first. But the x-coordinate of a point on the unit circle is cos t, and the y-coordinate is sin t, so of course that means it’s (cos t)2 + (sin t)2 = 1. Writing your trig identity with the cos first makes it easier to make the connection with the equation of the unit circle. If you write it with sin first you’ll have to continually switch it round!

Also, the order does matter if you’re using hyperbolic trigonometry. Then the formula is (cosh t)2 – (sinh t)2 = 1 and having sin first would be definitely mathematically wrong. For years, I had great trouble remembering which way around this was supposed to go until I realised that the cos and sin were in alphabetical order. From that point forward I always wrote my ordinary trig identity in the same order as the hyperbolic trig identity (in alphabetical order) so that through force of habit I would never get the hyperbolic one wrong.

So, I recommend you start writing your fundamental trig identity in the right order. It might help you remember and make connections to other things!

PS: You may be wondering about the way I put brackets in my trig identities. Don’t even get me started.

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In the Drop-In Centre, the majority of students visit to ask for help learning in a very small number of courses, mostly the first-year ones with “mathematics” in the title. Of course, any student from anywhere in the uni can visit to ask about maths relating to any course, and we do see them from everywhere, but the courses called Maths 1X have between them a couple of thousand students per semester and that’s a lot of people who might need help to learn how to learn.

Anyway, the upshot of this is that I help people with the same topics semester after semester, year after year. Sometimes people ask me “Do you get tired of the same topics?”

Short answer:

No.

Long answer:

I actually really love the topics in first year maths. Row operations and the fact that they help to solve equations and decide independence and find inverses are fascinating. Nutting out how to do an integral is a fun game. Eigenvalues are the Best Thing Ever. And don’t get me started on conics and quadrics. To me, seeing them every semester is like watching the Muppet Christmas Carol every December. I get to be reminded of a story I love, and notice something little I had never noticed before every time.

Also, it’s not just the topics I get to see each semester, it’s the students learning the topics. So many of them have a perfectly appropriate and successful way of understanding it that never occurred to me and these make the topics fresh again. Who ever thought of checking vectors are parallel by making sure that cos of the angle between them is 1 or -1? Not me until yesterday when a student did it.

And then finally, I get to be there at the moment everything clunks into place and see the light in their eyes as they feel the buzz of understanding it for the first time. And that never gets old!

Short answer:

Hell no.

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After writing the previous blog post (Finding errors by asking how your answer is wrong) and rereading one I wrote three years ago (Who tells you if you’re correct?), I got to thinking about how students are supposed to learn how to check if they are right.
It occurred to me that, at least at university, […]

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One of the most common situations we face in the MLC is when a student says, “I’m wrong, but I don’t know why”. They’ve done a fairly long calculation and put their answer into MapleTA, only to get the dreaded red cross, and they have no idea why it’s wrong and how to fix it. […]

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Yesterday the Maths 1M students handed in an assignment question that asked them to prove a property of triangles using a vector-based argument. It’s not my job to help students do their assignment questions per se, but it is my job to help them learn skills to solve any future problem. This kind of problem, […]

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Did you know that cats have scent glands just inside their bottoms that are constantly being filled with liquid and are squeezed as their poos come out, and if their poos are too skinny the glands are not squeezed enough and get over-full making them very painful and inflamed? Neither did I, until my cat […]

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One of my friends and a past MLC staffer graduated from her PhD yesterday (congratulations Jo!). One of my strongest memories of Jo is when she told me something about my teaching that I never knew I was doing, but that she saw as an essential part of what I was trying to achieve at […]

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There are two terminologies in probability which many students are confused about: “independent” and “disjoint”. The other day I was working with a student listening to their thinking on this and I suddenly realised why.
In your standard introduction to probability notation, various notations and terminologies are introduced, usually with reference to the meaning of the […]

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At the end of last year, the MTBoS (Math(s) Twitter Blog-o-Sphere) introduced me to this very interesting task: you have a cross made of four equal squares, and you are supposed to colour in exactly 1/4 of the cross and justify why you know it’s a quarter. I call it “Quarter the Cross”.

(Apparently, this problem […]

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