back
▲ 17 points

A puzzle: dissect a square into congruent pieces, all touching the center

by ColinWright·15y ago·27 comments·view on hn ↗
There's a math problem I've been working on for some time that looks like it's finally been solved. To explain it to people I usually start with a warm-up:

    Cut a square into indentical pieces so
    that they all touch the center point.
As I say, this is a warm up. It's fairly easy to do, and serves to introduce the ideas involved. So we do that, and we move on.

Over the weekend, though, I wondered how many solutions there are to the above problem. I rapidly came up with a small number, and was reasonably content.

Then a friend showed me another. Then another friend found another. Now I have rather more than I thought or expected. Today, a work colleague came up with yet another.

How could I miss so many? What are they all?

I invite the HN community to explore this puzzle/problem. Maybe I still haven't got all the answers.

Who will start me off?

27 comments
I am not very interested in puzzles, math, or much else that this question relates to, but I am incredibly curious-- why is this question being asked?

I don't understand the significance of this problem.

So there are nearly infinitely many ways to accomplish this. What does that mean though?

What are the implications of there being infinitely many ways to do this? Why is it interesting?

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.

>> 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?

By identical, do you allow for rotations - i.e. two pieces are identical if one can be oriented to perfectly overlay the other? If so, there are an infinite number of solutions.
Yes, but simply saying there are infinitely many solutions is not enough. There are infinitely many solutions, but there is more you can say.

It's a shame this will sink without making it to the front page - I'd really like to get the HN input before writing the blog post.

You can make any pinwheel-type shape with four wheels and can vary the line from center to edge in any path. This includes a bunch of spirally-looking things.

(Also you can just cut it in half similarly instead of in four.)

Picture:

http://cl.ly/3K1i2K3O0p2f1U291636/Screen_shot_2011-05-25_at_...

So you have two infinite families - the generality is what I originally missed.

There are more ...

Any straight line through the center point will create 2 identical halves.

Any 2 straight lines through the center point at 90 degrees to each other will create 4 identical pieces.

In both of the above cases each line segment from the center to the edge can be distorted in any way which does not intersect the edge of the square or any of the other lines and the distortion can be rotated 180 degrees in the first case or 90 degrees 3 times in the second to create new identical shapes.

The only special case seems to be using 4 lines to divide the square into 8 pieces which is, I think, the only configuration of 8 pieces.

I think that's every possible solution.

I'm weak at math, but I think that means there are an infinite number of configurations for each of an infinite number of configurations for both of the first 2 solutions, and then there's that 1 extra solution. So does that mean there are uncountably infinite solutions? I'm not sure how you would apply the diagonal method to this.

You've found two infinite families and one sporadic solution.

There is another sporadic solution, and the 8 piece solution you've found is not, in fact a sporadic.

And yes, the infinities are uncountable.

puzzles

Oh, right - you can vary the shape of the diagonal lines, provided each "arm" has rotational symmetry about its mid-point.

OK, so we've got three infinite families and the trivial solution (only one piece - is that your sporadic solution?). I think that might be all: each piece can contain 4, 2, 1 or 1/2 of the original square's corners, since (lacuna) all pieces must contain the same number of corners and further subdividing the corners (into 1/3s, say) would mean some pieces don't touch the centre (another lacuna).

OK, that's now the set of solutions I've got. You've also gone some way to showing them to be complete.

More to do, though.

And now do it for an equilateral triangle.

OK - I've now seen a "solution" with 16 "pieces."

My head hurts.

Ah, yes. You can also rotate the 8-fold solution by 22.5 degrees for another solution.
Right. As I said, I'm weak in math.
You can't be that weak at what I call math, although your experiences with math education might be unhappy ones. You have found more solutions than I first found - that can't be bad.

Half my life is spent helping people discover that they're good at "proper math" even when they think they're bad at "school math."

Hmm. Don't think so.
1) Start from the centre and reach any of the sides following a curve, such that as you traverse the curve your distance from the centre should increase, e.g., a circular arc that is less than a semi-circle. (The construction of this curve gives [at least] an uncountably infinite cubed number of degeneracies.)

2) Now take the reflection of this curve wrt the centre, and attach it to the original curve. This will give two-congruent pieces.

3) You can take the resulting curve and rotate it around the centre by pi/2. The two curves will give 4 congruent pieces.

The restriction on the original curve, ie. its monotonic behaviour, guarantees that the curves in steps 2 and 3 will not intersect.

That gives two infinite families. However, those families contain more examples than your process generates, there is another infinite family, there is a sporadic, and I have just been shown another example for which I am unsure if it is a sporadic, or one member of an infinite family.

Not only do I not know everything, I truly don't know very much at all.

Suppose the square can be dissected into n pieces. Then any symmetry of the square induces a permutation of those n pieces. Hence, the symmetry group of the square (D_8) must be embeddable as a subgroup of S_n. Hence there is no dissection into three pieces, since |D_8| = 8, which does not divide |S_3| = 6.

I suspect this line of argument can be taken further, but wanted to post it before someone else did :-)

Edit: no, that doesn't work, because we already have a dissection into two pieces, and |S_2| = 2. Dammit. Not all of the symmetries of the square must map pieces to pieces.

You can divide it into eight isoceles (pi/4, pi/2, pi/4) triangles, by quartering up/down and then quartering diagonally.
That's one solution. There are more ...
A heart-felt "Thank You" to those of you who have contributed and up-voted. Alas, as I type, this item has now slipped to the third page of HN and is unlikely to be seen by "the masses."

So it is only you, the cognoscenti, who have had the privilege (if it be such) of discussing the dissection.

Again, thank you.

This is #1 on my "Ask" page.
Woah! Cool! Thanks ...

I wonder how many people read the "Ask" page.