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Thread: An exercise in reason, logic, and intuition

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    An exercise in reason, logic, and intuition

    A sublime problem for both the intuitive and logical.

    1. Draw 3 squares in a line on paper with spacing.

    2. Under each square draw a circle.

    3. From the left circle draw 3 lines one to each square from the circle without physically crossing any lines .

    4. Repeat for the other two circles without physically crossing any lines.

    How many tries did it take you to find a solution?

    Please no comments from the peanut gallery.

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    Veteran Member Wiploc's Avatar
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    One try.

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    Wow, I first encountered this problem about fifty years ago, as it was popular in recreational maths books and elementary topology articles. It was generally cast in the form of three houses being connected to utilities (electricity, gas and water).

    It is unsolvable using a flat plane and can be only solved via a third dimension - usually a torus (donut).
    So zero tries from me.


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    The peanut gallery spoiled the problem. I was hoping to engage our resident philosophers and logicians.

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    Contributor Speakpigeon's Avatar
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    Quote Originally Posted by Spacetime Inhabitant View Post
    Wow, I first encountered this problem about fifty years ago, as it was popular in recreational maths books and elementary topology articles. It was generally cast in the form of three houses being connected to utilities (electricity, gas and water).

    It is unsolvable using a flat plane and can be only solved via a third dimension - usually a torus (donut).
    So zero tries from me.
    Figured it out in broadly 30 seconds

    Still, there's a solution in two dimensions only using curvy lines... Do you agree?
    EB

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    One try and found a solution, and then a second try and found a fundamentally different solution.

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    Quote Originally Posted by steve_bank View Post
    The peanut gallery spoiled the problem. I was hoping to engage our resident philosophers and logicians.
    Why do you feel the peanut gallery spoiled the problem? You say it's for the intuitive and logical and philosophers -- but the only spoiled problem is the problem for graph theorists. Sure, your exercise looks like a graph theory problem; and sure, if you read it as a graph theory problem it's one of the two basic impossible-to-draw graphs that all non-planar graphs must contain; but that just makes the puzzle into a challenge to read the problem statement more carefully, and find a tricky way to solve it with intuition and logic and philosophy instead of with boring old graph theory.

    Here's one of my solutions:


    Draw the lines with a pencil. After step 3, use the eraser to make a gap in one of the lines you drew. Then one of the lines you draw in step 4 can go through the gap. It doesn't physically cross any line, and the problem as stated never said you can't erase lines. I'm not a lawyer, but I play one on the web.


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    the baby-eater
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    Had three tries and couldn't solve it.

    I've now read the other solutions above and realised my mistake.

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    Contributor Speakpigeon's Avatar
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    Quote Originally Posted by Bomb#20 View Post
    Quote Originally Posted by steve_bank View Post
    The peanut gallery spoiled the problem. I was hoping to engage our resident philosophers and logicians.
    Why do you feel the peanut gallery spoiled the problem? You say it's for the intuitive and logical and philosophers -- but the only spoiled problem is the problem for graph theorists. Sure, your exercise looks like a graph theory problem; and sure, if you read it as a graph theory problem it's one of the two basic impossible-to-draw graphs that all non-planar graphs must contain; but that just makes the puzzle into a challenge to read the problem statement more carefully, and find a tricky way to solve it with intuition and logic and philosophy instead of with boring old graph theory.

    Here's one of my solutions:


    Draw the lines with a pencil. After step 3, use the eraser to make a gap in one of the lines you drew. Then one of the lines you draw in step 4 can go through the gap. It doesn't physically cross any line, and the problem as stated never said you can't erase lines. I'm not a lawyer, but I play one on the web.
    Yeah, I thought about something like that: use dashed or dotted lines.

    But I KNEW it would be acceptable in this venue.
    EB

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    Contributor Speakpigeon's Avatar
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    Quote Originally Posted by bigfield View Post
    Had three tries and couldn't solve it.

    I've now read the other solutions above and realised my mistake.
    Well, that probably isn't going to count because, no offense intended, I'm sure this thread somehow doesn't count you as one of "our resident philosophers and logicians".
    EB

    - - - Updated - - -

    Quote Originally Posted by Wiploc View Post
    One try.
    Perfidious logician.
    EB

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