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[SPOILER] The Pipe Cleaner Puzzle

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This is the spoiler thread for the pipe cleaner puzzle.

Spoiler warning:

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All of the parts to the puzzle are difficult, but each is difficult in a different
way.  If you are stuck on any of the parts, you should try the other parts.  You may
even inadvertently stumble on the solution to the part you were stuck on!

My lowest solution had more than 5 triangles.  My middle solution had more
than 10 triangles.  My highest solution had more than 15 triangles.  If your
numbers are lower than these , you may want to reconsider your solutions.
You're definitely missing something.

Also, make sure that you're not accidentally making unintended triangles for
the third part.  If you do, then you may just invalidate your pipe cleaner
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My solutions:

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Solution #0:  84 triangles
This is making as many triangles as possible with 9 lines, dealing with any triangle
as opposed to just equilateral triangles. The solution is any arrangement in which
every line crosses every other line. [mathematically: (9*8*7)/(3*2*1) = 84 unique
sets of three lines out of 9; since all lines cross, any three make a triangle]

Solution #1:  27 triangles
This is making as many triangles as possible with 9 lines, dealing only with equilateral
triangles. Since we only want equilaterals, only three different orientations of lines are
used ( angled such as "-", " / ", and " \ "  ). To maximize number of triangles I split them
evenly using 3 lines of each of the 3 orientations. The solution is any arrangement in
which each line crosses every line of a different orientation.[mathematically:
3A*3B*3C= 27 ABC where A,B,and C are lines of the three different orientations]

Solution #2: 12 triangles
This is making as many same-size equilateral triangles as possible with 9 lines. This is
the same as solution #1, only with the lines spaced evenly so as to maximize same-size
triangle creation. [unfortunately, I couldn't come up with a good simple mathematical
model for this due to the need to not count overlapping and such.  However, my solution
fits nicely on a triangular grid, and each line is part of four different same-size triangles.
So it's likely as efficient as it can be.]

Solution #3: 7 triangles
This is making as many same-size equilateral triangles as possible with 9 lines while
avoiding making other sized triangles.  The first two lines do not create any triangles,
but each line after can make exactly one more. It isn't possible to use them more
efficiently because to do so would create larger triangles.
[mathematically: 9 lines - 2 starting lines = 7 triangles]

Note: edited to correct solution #3.
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You are *very* close...

The responses:

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Solution 0:  This is correct.  The sample solution I have is the 9 pointed star.

Exactly one of your other solutions is different than what I got in both number and
arrangement.  The other two both have correct numbers and matching
arrangements to my solutions.

As stated in the previous spoiler, each of these sub-puzzles is difficult in it's own
way, and because you are attacking the problems from a certain perspective
(namely, trying to find a theoretical maximum, then seeing if a solution of said
theoretical maximum exists), one of the parts is going to be *much* harder, if
not impossible, for you if you continue to use this "magic formula".  You should
consider trying these problems from a different logical paradigm.

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Ah...  I found the problem.  And by the way, that was one that previously did not match a theoretical maximum. Now it does assuming the limitation I listed is correct (which as far as I can tell it is for the reason I listed).  It still bugs me that I don't have a mathematical answer for the one solution though. hmm....  have to work on more maybe.

You're still off.  There's an even more efficient solution to part 3.


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