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I crossed the "dork line" last night.
I grabbed a book of brain teasers as I crawled into bed last night and glanced at the first couple of puzzles in the book. The first two were pretty easy word problems in the "find a word that contains the letters wkw in that order" vein. The third problem in the book, however, involved creating a magic square.
It may astound some of you know learn that I have never actually tried this before last night.
If you're not familiar with a magic square, it's a 3x3 grid into which you insert the numbers 1 through 9 in such a way that the sum of the digits is 15 in all directions. The book said, "You could do this by trial and error, but it would be more fun if you could come up with some rules in building the magic square."
I decided that an even better challenge would be to develop the rules and construct the magic square in my head. It turns out that it wasn't nearly as hard as I'd thought it would be.
Some of you probably have better, more elegant solutions than mine, but this is what I came up with.
I quickly decided that the lines intersecting the centre were a given. 5 obviously had to go in the middle, with the pairs of 1:9, 2:8, 3:7 and 4:6 placed around it in the four directions that intersected the middle. The trick was in figuring out how to arrange those.
The next thing I figured out was that 9 couldn't go in a corner, since there are only 2 valid combinations that combine with 9 to make 15. (2:4 and 1:5). Thus I quickly constructed the base square in my head.
? ? ?
9 5 ?
? ? ?
As I lay there contemplating on this layout, it suddenly occurred to me that 9, 4, and 2 were the only numbers whose placement I could vary, and at that there were only 8 possible magic square configurations. (I am probably wrong in this assumption, but I was tired at the time and that's the way it appeared to me).
The placement of the 1 is self-obvious:
? ? ?
9 5 1
? ? ?
I placed the 2 and 4 arbitrarily:
4 ? ?
9 5 1
2 ? ?
From there I realized that I didn't even have to think for the rest of the numbers. The top right number had to be an 8 and the bottom right one had to be a 6.
4 ? 8
9 5 1
2 ? 6
Lastly I only had to figure out where to put the 3 and 7. That took a nanosecond of my time.
4 3 8
9 5 1
2 7 6
For the record, here are the 8 possible configurations that I came up with:
4 3 8 | 2 7 6 | 4 9 2 | 2 9 4 | 8 3 4 | 6 7 2 | 8 1 6 | 6 1 8
9 5 1 | 9 5 1 | 3 5 7 | 7 5 3 | 1 5 9 | 1 5 9 | 3 5 7 | 7 5 3
2 7 6 | 4 3 8 | 8 1 6 | 6 1 8 | 6 7 2 | 8 3 4 | 4 9 2 | 2 9 4
Am I overlooking something fairly basic and fundamental? Are there other configurations?
It may astound some of you know learn that I have never actually tried this before last night.
If you're not familiar with a magic square, it's a 3x3 grid into which you insert the numbers 1 through 9 in such a way that the sum of the digits is 15 in all directions. The book said, "You could do this by trial and error, but it would be more fun if you could come up with some rules in building the magic square."
I decided that an even better challenge would be to develop the rules and construct the magic square in my head. It turns out that it wasn't nearly as hard as I'd thought it would be.
Some of you probably have better, more elegant solutions than mine, but this is what I came up with.
I quickly decided that the lines intersecting the centre were a given. 5 obviously had to go in the middle, with the pairs of 1:9, 2:8, 3:7 and 4:6 placed around it in the four directions that intersected the middle. The trick was in figuring out how to arrange those.
The next thing I figured out was that 9 couldn't go in a corner, since there are only 2 valid combinations that combine with 9 to make 15. (2:4 and 1:5). Thus I quickly constructed the base square in my head.
? ? ?
9 5 ?
? ? ?
As I lay there contemplating on this layout, it suddenly occurred to me that 9, 4, and 2 were the only numbers whose placement I could vary, and at that there were only 8 possible magic square configurations. (I am probably wrong in this assumption, but I was tired at the time and that's the way it appeared to me).
The placement of the 1 is self-obvious:
? ? ?
9 5 1
? ? ?
I placed the 2 and 4 arbitrarily:
4 ? ?
9 5 1
2 ? ?
From there I realized that I didn't even have to think for the rest of the numbers. The top right number had to be an 8 and the bottom right one had to be a 6.
4 ? 8
9 5 1
2 ? 6
Lastly I only had to figure out where to put the 3 and 7. That took a nanosecond of my time.
4 3 8
9 5 1
2 7 6
For the record, here are the 8 possible configurations that I came up with:
4 3 8 | 2 7 6 | 4 9 2 | 2 9 4 | 8 3 4 | 6 7 2 | 8 1 6 | 6 1 8
9 5 1 | 9 5 1 | 3 5 7 | 7 5 3 | 1 5 9 | 1 5 9 | 3 5 7 | 7 5 3
2 7 6 | 4 3 8 | 8 1 6 | 6 1 8 | 6 7 2 | 8 3 4 | 4 9 2 | 2 9 4
Am I overlooking something fairly basic and fundamental? Are there other configurations?
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Actually, I've seen this puzzle before, and my train of thought matched yours exactly, even down to figuring out all 8 "possibilities". Wow, I'm a geek too. Yay!