## 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" |

1Willie commented PermalinkNice post; enjoyed reading it I am now even more impressed by you than I was before. You see, I am one of "those" who can't add much less do algebra. In fact, I can still hear my high school math teacher telling by parents they should send me to trade school to learn a skill like welding. Fortunately, they did not listen.. . and I went on to fail his course miserably. <div> </div> I do enjoy Sudoku though and work out their puzzles quite frequently on take off and landing.. if I stay away that long. You see, I am that "unfriendly" person on the plane that talks to no one and wear headphones for as much of the flight as they will allow. <div> </div> I am now also a but jealous of you. I wish I could work out math problems.. <div> </div> BTW, were do you find your equations to solve? It seems once you have worked out the solution, working it out a second time may not be as much fun... (smile)... <div> </div> Thanks for sharing <div> </div> Willie

2MartinPacker commented PermalinkFirst, you're very kind, Willie. Second, you ask a good question about where to find good algebra problems: I happen to have kept many of my High School level maths books (and beyond). And, third, you're right I probably did do this one the first time round the "wrong" way. :-) That's why I'm susceptible to the idea the method I presented could be improved on.

3bobleah commented PermalinkI'm pointing my friend Peter to this blog post... he loves this stuff!

4PeterYim commented PermalinkAgree. Very elegant math. This is fun. Please post some more of these problems. Thank you.

5MartinPacker commented PermalinkThanks, Peter (and Bob)! Will try to oblige.

6CJS commented PermalinkGood Stuff.....I like to apply the formula (a - b)^2 = a^2 -2ab + b^2 and (a + b)^2 = a^2 + 2ab + b^2....nice observation though.....all in all....your post brings back those memories we had solving algebraic equations in math class.

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