Connecting the dots

(This post contains mathematical spoilers. I’ll warn you again just before the reveal.)

Today I want to share a maths puzzle:

Is it possible to arrange an entire set of dominoes in a circle so that touching dominoes have adjacent squares with identical numbers?

Once you’ve experimented with a set of dominoes in which the highest number is six, explore whether it is possible for sets of dominoes where the highest number is different.

This is a favourite puzzle for several reasons: (1) the opportunities to physically explore, conjecture, and justify, (2) the solution is pleasing in its simplicity, (3) it’s an example that being less helpful often leads to a richer task, (4) it taught me to listen to my students because they really are smarter than me.

BEING LESS HELPFUL: At the start

I used to start this puzzle by telling students all about a set of dominoes because usually they have never seen them before. (That’s how I know that I am old.) I told them how many dominoes there are in a set, how the set is constructed, how there are cool ways to count the number of dominoes in any set in which the highest number changes.

No more. Helpful in the short term, perhaps, but definitely not in the long term. Look at all those opportunities that I am robbing my students of to explore for themselves.

Now I ask one question ‘How many dominoes in a set?‘ while brandishing a box of dominoes at them. They can then ask me whatever they like except that same question. Their questions might result in incremental clues, like ‘What does a domino look like?’, ‘What kinds of numbers are on a domino?’, ‘What is the highest number on a domino?’, ‘Are there zeroes on dominoes?’ ‘Is there exactly one of every combination?’ ‘Can a domino have more than one instance of the same number?’. No one has asked me yet for the dimensions of one domino and of the box, but that would be a cool way to approach it.

While they work in groups—discussing, organising their ideas, noticing patterns, justifying their approach, using their own mathematical thinking—I wander around to listen and observe this rich task, brought about by one simple change: ‘Be Less Helpful’.

And, I always wonder why I ever thought it was a good idea to give students the information I thought they needed without having them first decide whether they needed it.

BEING LESS HELPFUL: When solving

If you haven’t tried the puzzle yet, stop and do it now — if only so that you can laugh at with me later on.

Spoilers ahead

The domino problem first entered my repertoire when I started teaching graph/network theory. It is hardly a puzzle if it is presented immediately after a unit on networks with an accompanying expectation that it can be solved that way. Although, it is a nice challenge to see how to model and resolve the problem using networks.

I eventually isolated the domino puzzle from the networks unit and now use it in more general problem-solving sessions, but I didn’t initially isolate my own thinking about the domino puzzle from networks. What I mean is that the only way that I showed students how to resolve the puzzle was to model it as a network: nodes corresponding to numbers, edges corresponding to dominoes, and that finding a circle of dominoes corresponds to finding an Euler cycle through the network.

Urgh. This is one way of thinking about the problem that probably connects with the thinking of zero students in the room — even if they know what an Euler cycle is, how to check for one, and how to pronounce ‘Euler’ without making me scream. When it is the only way being presented as a solution, well, that reinforces to students that mathematics is something that is done to you, not something that you do (Annie Fetter’s Ignite Talk, NCTM 2016).

LISTENING TO STUDENTS

What changed? Now I listen. I really listen. I listen until I feel like my ears are going to fall off. And I hear things like this.

When the highest number is 6, this means that there are eight instances of each number in the set of dominoes. Consider one of the numbers, like 1. For example, there is a 1 on the other end of dominoes with each of the numbers 2-6 and the blank (0), and there are two 1s on the double-one domino. There is exactly one domino with each combination of numbers, so we have accounted for them all to get eight instances of the number 1. Because eight is an even number, it means that we can match each 1 with another 1 and have none left over. This is true for all of the numbers from 0-6, and so it that’s why it is possible to make a circle when the highest number on the dominoes is six.

When the highest number is n, there will be (n+2) squares of each number in the set of dominoes, using the reasoning from before. So, for (n+2) to be even, n must be even. Thus, we will be able to make a circle when the highest number in the domino set is even. We won’t be able to make a circle when the highest number in the domino set is odd.

Simpler. Elegant. It makes sense. And it gives students the chance to connect the dots for themselves.