The Pipe Cleaner Puzzle

We have a trifecta this month! There is still a storyline, but the information about the puzzle itself is relatively consolidated (last month's was just a hair long...). Nonetheless, the relevant information is color coded as before. If you don't want the storyline, just read the light green parts as well as the part after the second line of dashes, and you will have all the puzzle relevant information. As always, enjoy!

I apologize beforehand if I made a mistake in constructing this puzzle in terms of it's solution. I have been refining it over the past month or so, and am now about 99% sure (through semi-formal proof and sound logic) that my solutions for the first 2 parts are right and 85% sure (through only semi-justified explanation) that the answer for my third part is right.

Also, I realize that most of you don't have pipe cleaners at your disposal for the purposes of a visual aid. Straightened paper clips can act as a excellent surrogate, and pencils/pens are an adequate and more immediate (though sub-par) alternative. Just remember the caveat of pipe cleaners outlined at the end if you choose to use pencils/pens. In short, pencil != pipe cleaner.

And finally, please do *not* post ideas/solutions/etc. here for the courtesy of other puzzle solvers. Instead, feel free to post in the corresponding spoiler thread.

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John and Jane were running their customized pet tag store again when a guy walks in the door on an otherwise slow day. The man introduced himself to Jane as Bob, and asked to discuss with her details about a potential business order. Jane promptly escorted Bob into her office, and the two of them began their discussion.

"I just want to say that I really appreciate the timeliness that you exhibited when you made the pet tags for me and my friends."

"Don't mention it. Are you here looking for more pet tags?"

"Actually, no. Billy and I wanted to have some employee name plates for our co-workers' offices. Can you equipment make such plates?"

"Of course! All I need is the details for each plate, and I'll be able to give you an estimate."

"Well, Billy was *supposed* to be here with the co-workers' choices for font and color, but it seems that he isn't here right now, and his phone appears to be turned off..."

"We can give him a few minutes, if you'd like. Today's a slow day, and I have nothing better to do."

"Neither do I."

There was a short pause. "So, why is it that you specifically requested for me when..."

"Billy didn't want John in charge of this order."

"Ah." Jane realized that Billy was trying to avoid last time's fiasco.

The pair of them waited about five more minutes for Billy to show up. Out of boredom, Bob gives an offer to Jane. "Since we have nothing better to do, why don't I pose a puzzle for the purpose of entertainment?"

"Yours or mine?"

There was a pause. "Well... a little of both." Bob then reached into his briefcase, grabbed nine straight pipe cleaners, and put them on the table in the form of three triangles.

"I don't believe I've told you this, but I've always had a fascination with triangles.

"Triangles? Why... triangles, of all things? And why on earth do you have pipe cleaners in your briefcase?"

Bob took a moment to look at the gargantuan assortment of objects in his briefcase that couldn't be justified as a part of an office environment. "Well... what can I say? I guess I'm just a hard-core packrat."

"That's beside the point. Regardless, you mentioned something about a puzzle. What is it, anyway?"

"As I was saying, I have a fascination with triangles, particularly equilateral triangles..."

"Isn't that the kind where the three sides are the same length?"

"Indeed, it is. The first part of your puzzle is this: Rearrange these nine pipe cleaners without bending them so that you make as many equilateral triangles as possible."

After a little bit of thought, Jane took the nine pipe cleaners and rearranged them in a pattern on the desk. "Is this about right?"

"That... would most certainly be the correct arrangement. I'm surprised you found it out so quickly."

"I assume that there's still another part to this puzzle?"

"There is. This time, try to get as many equilateral triangles with one inch (3 cm) sides as you can. Again, you must not bend the pipe cleaners."

"That shouldn't be too hard." Sure enough, Jane had come up with another fancy arrangement in a matter of minutes. "Is this it?"

"Well... yes, that is it. How are you getting this so quickly?"

"I'm just good at puzzles. Is that the end?"

"No, there's one more part. Again, arrange the pipe cleaners without bending them to maximize the number of equilateral triangles, but this time, make every equilateral triangle that is formed the same size as every other equilateral triangle."

"Doesn't this setup meet those criterion?"

"No. The small triangles may happen to be the same size, but when you put small triangles together..."

"...they make bigger triangles, and different sizes of triangles is bad. Got it." With a little bit of ingenuity, Jane was able to get the pipe cleaners into the appropriate configuration.

"Impressive. You got all three right."

"It wasn't that hard."

At that moment, Billy walked into the office. He spied the pipe cleaners. "Are you trying to fry her brain, Bob?"

"I was, but she got the puzzle in the..." Bob took a glance at the clock. "...forty minutes that it took you to get here."

"I'm sorry! I forgot to set my alarm clock back from the time zone change... Wait... You said she got the puzzle in forty minutes?"

"I did."

"...That's better than my time!"

"I know."

Billy turned to Jane. "Let me just say, your coolness factor just went up a hundred fold. Congrats!"

"Thanks. Shall we get started on your order?"

"Let's."

With that, the three of them went through the details of the employee name plates, and the three of them then returned to their pleasant but otherwise uneventful days.

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Your puzzle is this: What were the solutions to Bob's puzzles? More specifically, how many triangles were formed in each solution, and what was each configuration?

For the purposes of this puzzle, it is assumed that pipe cleaners have a finite, uniform length of approximately 12 inches (30 cm) and *negligible* thickness. Therefore, if a number of pipe cleaners all overlap, crossing at a certain point, the pipe cleaners are considered to coexist at that point, possibly resulting in them being in the same plane (2D area), which means that the "top" pipe cleaner(s) can still form triangles with the "bottom" pipe cleaner(s).

*BONUS* If you find the above to be too easy, you can try to figure out the maximum number of triangles of *any* kind (even those that are not equilateral) that can be made with 9 pipe cleaners without bending them. The solution to this variant is a bit trickier...