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I'm playing this addictive little compass and straight edge game: http://www.sciencevsmagic.net/geo/

I've been able to beat most of the challenges, but I can't construct a square in 8 moves.

To clarify a move is:

  • Drawing a line
  • Drawing a circle

Extending a line is not a move.

Lines can only be drawn to and from the root points and any intersections. Circles can be drawn with their center at any existing node and their edge must touch another.

My method for constructing the square in 9 moves is as follows:

  1. Draw circles from each point to the other, and the connecting line
  2. Use the intersections of the circles' edges to drawn a line perpendicular to the first
  3. Draw a smaller circle with its center at the intersection of the two lines and its edge touching the two root nodes.
  4. Connect the four points on the small circle's edge to form a square.

The end result looks like this:

enter image description here

I can't find a way to refine this method to get to eight moves, and I've failed at finding another method. Is it even possible?

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    $\begingroup$ Thanks to you, I do not expect to get any more work done today! $\endgroup$
    – BlackAdder
    Jul 8, 2013 at 14:36
  • $\begingroup$ @yanbo My day has already been a complete write-off... $\endgroup$
    – fredley
    Jul 8, 2013 at 14:37
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    $\begingroup$ Hum, +1 for the game. How do you know that it' possible to do with 8 moves? Another question: the four edges will be counted as four moves? If so, then we'll need to determine the four vertices with 4 moves? $\endgroup$
    – Sigur
    Jul 8, 2013 at 14:40
  • $\begingroup$ @fredley extending a line is indeed a move. You should also note that you don't need the horizontal line at any point in this construction. $\endgroup$
    – Robert Mastragostino
    Jul 8, 2013 at 14:41
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    $\begingroup$ It appears that extending a line does not count against yuor score!? $\endgroup$
    – Hagen von Eitzen
    Jul 8, 2013 at 14:42

1 Answer 1

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Your solution fails because you give up a move creating the center point of your bounding circle. To complete the challenge in eight moves, you need to use one of the preexisting intersections as your center point, and then rely on the fact that extending a line is not considered a move to create the fourth vertex of the square.

The following diagram illustrates these steps, where step 4b is the non-move extension of the line:

square in bounded circle in eight moves

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