(Originally posted 2011-07-09.)
If you’re sat next to me on a plane you’ll probably notice at take off and landing I do algebra puzzles. You may not have heard of the term "algebra puzzles" before and perhaps think the juxtaposition of the two words to be odd, but I think it apt…
(You may also think this whole post to be showing off, but that’s a risk I take in sharing a passion I have.) 🙂
A classic problem with take offs and landings is what to do given you’re not allowed to use electronic equipment. I’ll readily agree that staring out the window is a good one – which is why I prefer a window seat. I love staring out the window. I love maps – and to me looking out of an airplane window brings maps to life. And figuring out what I’m seeing is another great puzzle. But sometimes there’s nothing to see. So what do you do?
I started by taking puzzle books with me. I’ve done Sudoku (but not recently), Kakuro, Futoshiki, Hashi, Kenken and any number of others. I enjoy them but each one lacks variety. (And I’m disappointed that by far and away the most common puzzle books are Sudoku.)
But I find the best puzzles of all are algebra problems. I still have a copy of my "high school" Further Mathematics textbook. I don’t know why, I just do. 🙂
I actually think it’s the elegance of expression and the neatness of the right shortcut that appeal to me. As I’ve said many times I’m a sucker for ingenuity. Below is an example of a neat shortcut that I’d like to share with you. I hope you’ll see what I mean.
One of the nice things about mathematics in general is that you’re perpetually "standing on the shoulders of giants". Some of them well known (Newton, Leibnitz, Euclid, Gauss, etc) but many are anonymous. In the example below I’ve no idea who thought of the shortcut first. (I’m just pleased I understand it and can see its applicability.)
A Simple Example Of Elegance
Problem: Solve (x – 3)² – (x + 2)² = 0
It looks like a difficult puzzle to solve. Of course if it were I wouldn’t be offering it here. 🙂 You could multiply everything out and gather terms but that’s horrid. Thankfully, there is a more elegant way:
Observe x² – y² = (x – y) (x + y) . (Check it if you don’t believe me!)
If you substitute a for (x – 3) and b for (x + 2) you get:
a² – b² which, of course, can be rewritten as (a – b) (a + b) .
I think you’ll agree working out what (a – b) and (a + b) are is easy:
a – b = (x – 3) – (x + 2) or -5
a + b = (x – 3) + (x + 2) or 2x – 1
Multiply them together and you get:
-5 × (2x – 1) = 5 – 10x
So (x – 3)² – (x + 2)² = 5 – 10x which = 0, as the original problem stated.
If 5 – 10x = 0 then 10x = 5 and so x = ½.
See, that wasn’t so hard, was it? I think people think mathematics is hard. I don’t think algebra is hard. I do thing topology is hard – because of the abstractness of the concepts. I do think proving things is hard – because of the need to not miss any loose ends and to know whether you’ve actually proved anything. But algebra is, to me, pure puzzle solving. And elegance is important: In the above example I could quite easily have made a mistake if I’d not known the trick. With the trick I’m much less likely to.
Now someone will probably come along and point out a few things about the example, including a further trick. If they do I’ll be delighted. This "old dog" loves learning new tricks. 🙂 And if I am sat next to you on the plane at least I won’t be muttering to myself as I manipulate those symbols. 🙂