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Some puzzles don't have "significance", they are simply puzzles. Some people find some (but not all) puzzles intriguing. Why do people do Sudoku? Or crosswords?

This particular problem is also an excellent example of learning to think "outside the box," insofar as almost everyone comes up with the same solutions, and almost no one comes up with all the solutions.

Including me.

Finally, this is just a warm up to an unanswered question in math/puzzles. Solving that may turn out to be mundane, but it may lead to the development of new techniques and insights. You never know. It's like the Collatz Conjecture. Simple to state, no one knows if it's true, no one know how to prove (or disprove) it, and who knows what techniques may be developed to answer it.

You say there are "nearly infinitely many ways to accomplish this." Can I ask, what do you mean by "nearly infinite"? Anything that's not infinite is infinitely far away from being infinite, so I'm a little confused by your statement.

And learning to think about these things is an excellent exercise in logic and reasoning. Explaining solutions is superb training in communication.

And for some, it's fun.

1 comments

>> not infinite is infinitely far away from infinite

Nit picky in my opinion, but the reason I didn't assign it a number and rather opted for the more generic "nearly infinite" is because there are a ton of possibilities.

Here is my idea for you and will eventually come up with most solutions (might miss some outliers, this is my <5 minutes of thought on the topic):

Write a program that shows a square and it's center point.

A single line increment out from the center point going at a random starting angle between 181 and 360 (in other words, they generally tend toward going left). The line moves the smallest amount possible each iteration (1 pixel for example).

After the line has moved one pixel, a new angle is picked for the line move, from 0-360, and it increments 1 more pixel. If the randomly generated line crosses the center line of the square (if you folded the square in half from left to right) force the line to randomly pick a new angle before incrementing.

Do not allow the line to cross over itself.

Now you have your first line. Take this line, replicate it, rotating it 90 degrees, then 180, then 270.

This is your first square. Set the program to keep generating new squares, add rules as you see fit.

Now you have more solutions to this puzzle than you can count.

Edit: This is pretty badly explained, but it is based on the idea of a single line leaving the midpoint, moving randomly until it reaches an edge. You then replicate that single line 3 more times, each rotating by 90 additional degrees.

In essence, take any line from the center to the edge, and take three copies (giving four in total) rotated by 90 degrees. Provided they don't cross, that's a solution.

And that's one infinite family of solutions, one that I didn't (initially) find.

There are more.

Initially I had 5 actual solutions, and I thought I had them all. Then someone produced this infinite family, and I suddenly had my mind expanded. I've since found what I hope - but have not yet proved - is all solutions.

Can you find any more? You might not care, but that's the challenge. I find it akin to the best sort of programming, except it doesn't, in the end, actually do anything.

I won't find any more solutions. I will let the program find them, which, given enough time should find every single possible solution.

Any line drawing variant is just the concept I've already outlined expanded.

So I guess my question is: Is the challenge in this finding other ways to generate solutions, even though a method that will generate all possible solutions has already been found?

I must have mis-understood you. The method you have outlined - as I understand it - definitely will not find all solutions, and I don't see how you can think it would.

Perhaps you should explain it again. It seems like you find a squiggly line, rotate it 4 times by 90 degrees, say that's a solution, do it in all possible ways, and claim that's everything.

Have I misunderstood?