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## Hello, I'm Martin And I'm An Algebraic :-)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 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
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 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" |