Squares of Differences II: Changing Direction

[Note: this is a continuation to the first Squares of Differences post. Read that before continuing.]

A week after introducing her class to squares of differences (see  for the first post on this lesson), one of Katherine’s students walked in with a list of squares that they claimed would continue for 20 steps before reaching all zeroes. Here’s an example from that list.

Try it out. Does it work?

We had two major questions from before:

  • Will this game always end with a square of zeroes in the center?
  • What’s the most number of steps you can go before arriving at a square of zeroes?

If you’ve messed around with the squares of differences, you’ll probably have noticed that most of them end with a square of zeroes after six or seven steps. To construct a square of twenty steps is quite a claim. Was there a subtraction mistake? Or is there some method to build squares of any depth?

One thing mathematicians like to do is work backward. That might be a helpful strategy here. For example, if you ended up at all zeroes, where might you have come from? What about the turn before that?

One of my students had a thought that maybe it would be possible to create a kind of map of possibilities, working your way out from zero. We found, though, that you can’t always work your way outward. For example, if you start here,

you can find a square that came one level up from this one. In fact, there’s quite a few options of how to make one. Here’s one possibility (I’ve rotated the original square here):

Here’s another possibility:

That first square very nicely went outward (or upward). But it was kind of special; it just so happened that there entries (1, 4, 7, 10) could be differences of other squares. Are either of the squares above as nice? Check out (1, 0, 4, 11). Can those be the differences of another square? Try it out! What about (100, 99, 103, 110)?

Now here’s a little conundrum. Neither of those squares, it turns out, allow another level above them. But they were just two choices we could have made. What about (2, 1, 5, 12)? Or (1.5, 0.5, 4.5, 11.5)?

A little algebra holds the key to solving this puzzle. If we want to be general, the level above (1, 4, 7, 10) must look like this:

Convince yourself that n could be any positive number, and this would still work.

Now here’s the interesting question: can you find a value for n that allows you to go up another level?

And if so, what does this suggest about how far up (or down) these squares of differences can go?

A pedagogical aside: another reason I like this question so much is that it motivates algebra in a very real way. Master the algebra, and you can figure out what the next level up should be to allow another level after that. However, it is possible to figure this out without algebra, so don’t let it stop you if you or your students are unfamiliar with it. But using algebra in this kind of context is pretty empowering, in my opinion. Doing it for no reason always seems like drudgery, but setting up and solving an algebraic equation here allows you to pluck the one correct answer from an infinite collection of erroneous ones; it’s like you called the pearl up from the recesses of the ocean floor. No wonder algebra has been called the greatest labor-saving machine every created by the human mind.

Questions? Comments? Observations? Put ’em in the comments. I’ll be back with more possible directions for this lesson later. Until then, let me know how it goes with your students, and what you’ve figured out, and ideas for improvements.

For Part III, click here.

Comments 5

    1. Post

      I’ll look forward to the updates, Josh! I’ve been playing with this problem a lot since I saw it in Houston (though I remember seeing it earlier. Do you know the origin of the problem?). I think I’ve got a lot of different angles on it at this point, but I don’t think I understand every detail of where it goes and what it connects to. Do you know of anywhere else that it’s analyzed?

  1. Dan

    for extensions, how about n-gons instead of squares? or other operations instead of subtraction?

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