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Charming little puzzle

I remembered this puzzle from an old Scientific American magazine last night. I think it's interesting because you can solve it fairly straightforwardly by either of the extremes of a purely logical approach, or the brute-force approach of trying all the possible combinations, but it's pretty diabolical if you approach it with trial and error or random playing. In that way it's a good puzzle to use to demonstrate the way a human brain best solves problems compared to how a stored-program computer does, and I kind of like that extra little gift.

The puzzle: In the figure below, place the digits 1 through 8 inside the circles such that consecutive digits are never in circles connected by a line. For example, if "4" is placed in the top circle, then neither "3" nor "5" can be placed in any of the three circles in the second row, because each of them is connected to the top circle. There is only one solution, not counting reflections.

Edit: solutions are being discussed in the comments; if you care to take an unspoiled crack at this yourself, don't look at them.



( 5 comments — Comment )
Apr. 9th, 2007 12:52 am (UTC)
Jon's Thought Process: Well, 1 and 8 are special, because they have the fewest "neighbors" in the sequence. The two circles in the middle are the ones connected to the most other neighbors (6 each), so we get the most advantage from putting 1 and 8 in those two circles. The neighbors to 1 and 8 are 2 and 7, which now have to go in the top and bottom circles, as far away from their neighbors as possible. Then . . . oh, I've pretty much solved it, haven't I? Three and for go to the left and right of 1, 5 and 6 go left and right of 8, and we're done.

Yep, I see what you mean.
Apr. 9th, 2007 04:39 am (UTC)
That's similar to how I did it. My thought process was something like: for the middle two circles, there's only one other circle that they're NOT connected to, so I can't put, say, the 3 there, because then that one unconnected circle would have to have BOTH the 2 AND the 4. So they have to contain the 1 and 8 as the only two with only one consecutive number. From there things proceeded about like they did with you.

I like how you described it more strategically ("because of that, we get the most advantage by doing this") which to me belies your experience with games of strategy, while I went at it more procedurally ("I could do this, but then two things would have to fit in the space for only one, so I can't do that") which maybe belies my own background in CS and T of C.
Apr. 10th, 2007 02:36 pm (UTC)
This was _exactly_ how I approached it. And, szasz, you're right that I was in the exact same mindset as I am in when I'm playing a strategy game while I was solving this.

The 1,8 thing was obvious to me and it took me one round of trial and error before I realized that 7,2 were forced into position by the 1,8 choice and that the rest was then trivial (especially due to the symmetry) if you just pick one of the remaining numbers to place and go from there.

Apr. 9th, 2007 01:14 am (UTC)
My brain immediately goes to the question of how the phrase "no reflections" diminishes the possible solution space. . . but I'm not feeling sufficiently obsessive-compulsive to follow that up.
Apr. 9th, 2007 03:34 am (UTC)
No, "not counting reflections" is standard puzzle phraseology stating that there may be more than one solution, but all solutions are essentially identical because they are all geometric reflections or rotations of the same thing.

You can come up with solutions that look different from what someone else may have done, because they started in a different place, but they will still be the "same."

I'm not phrasing this well; I'm tired.
( 5 comments — Comment )



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