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        riddles >> hard >> Arrange the 13 pieces
        
(Message started by: SWF on Dec 19^th, 2002, 7:25pm)

Title: Arrange the 13 pieces
Post by SWF on Dec 19^th, 2002, 7:25pm

The last time somebody put a dissection problem in the Hard section, it was criticized for being too easy and being an old problem. This one will prove more challenging, and it should be unfamiliar, since I just made it up.

The problem is to find different arrangements of the 13 pieces to form two different figures: a (non-square) rectangle and a square. Each shape must use all 13 pieces as part of the shape. A larger figure than shown below which is more suitable for printing and cutting out pieces can be found at this link. (http://www.ocf.berkeley.edu/~wwu/images/riddles/13pieces.gif)

http://www.ocf.berkeley.edu/~wwu/images/riddles/13pieces_small.gif

Title: Re: Arrange the 13 pieces
Post by Rezyk on Oct 24^th, 2003, 2:35am

This seems like a solution...I guess this would be the non-square rectangle, though it's pretty close to square (there's a gray square in the background).

Title: Re: Arrange the 13 pieces
Post by SWF on Oct 26^th, 2003, 2:57pm

Excellent, Rezyk! I was beginning to think that this riddle would set a record for the longest time with zero replies. You are correct in that this is the (non-square) rectangle. Making the other figure (a square) is more tricky.

Title: Re: Arrange the 13 pieces
Post by Lightboxes on Oct 28^th, 2003, 9:22pm

I tried it as well for 1-2 hours. With an attempt to logically put them together. However, with that many pieces I gave up eventually.

I posted this reply because I wanted to let you know, SWF, that people did try it, but did not succeed, even in coming up with a strategy to post (like me). And also to say that it is a very nice puzzle that you made that I liked but that I'm unfortunetly not too good at.

Title: Re: Arrange the 13 pieces
Post by Rezyk on Oct 29^th, 2003, 12:00am

Personally, I was way too lazy to actually print and cut out anything. I imported the gif into Adobe Photoshop and fiddled with the pieces in there.

In case anyone wants to be even lazier and not bother with the conversion process from the gif, here's the file (psd format) with the pieces ready to drag around. I'm not very proficient in photoshop, so use at your own risk.

The gray square in the background should be the right size for the square solution -- I got it by having photoshop sum up the 13 areas pixel-wise.

Oh yeah, I left this in the state of the solution for the nonsquare rectangle. I can jumble it up if anyone would like.

Title: Re: Arrange the 13 pieces
Post by Rezyk on Oct 29^th, 2003, 1:02am

Here's some preliminary thoughts for the search for a square solution. Includes spoiler for the nonsquare rectangle.

All the angles involved can be reduced to variations of 2 angles x and y. In the attached picture, I've labeled all the non-right angles with their difference from a multiple of 90. Note that around any vertex, the differences sum to 0.

What bothers me is that if that condition holds true for the square solution, it's hard to see how the square could qualify as "more tricky". The 6 pieces that have angles involving y would have to combine with each other into one or more figures without any y-type angles at all. When taking into account some of their side lengths (for example, lengths BD and CF seem to be unique), this allows for trivially few possibilities for handling the y-cuts -- they'd pretty much have to be in the same configuration as the nonsquare solution (I think).

I think it would be impressively clever if the solution involved violating the differences-sum-to-0 condition. For example, if x was actually equal to 360/7, then you could fit 7 +x corners together nicely.

If that is the trick, there must be some discrete nontrivial combination of x and/or y that sums to 360. I checked if x or y were themselves equal to 360 divided by an integer (they aren't). Then I found something quite interesting: it appears that 8x+4y=360, or 2x+y=90. So my guess is that the trickiness lies in one or more vertices that use multiples of (2x+y) to maintain ortogonality.

That's about as far as I've gotten at the moment. I think it could also be useful to start measuring and categorizing sidelengths to determine what can and can't match with each other. AH appears to match the side of the solution square, for example.

Title: Re: Arrange the 13 pieces
Post by Rezyk on Oct 29^th, 2003, 6:00pm

More thoughts... examination indicates that JK=KL=2ML, which means that x is the smallest angle of a 1-2-[sqrt]5 right triangle. That would make a lot of sense, especially since then 90-y=2x would be the middle angle of a 3-4-5 right triangle.

This makes it easy to determine many of the sidelengths (relative to each other), shown here. There are quite a few that don't seem immediately determinable analytically though -- perhaps these represent a degree of freedom in maintaining squarity during puzzle construction, and so should mostly be oriented parallel to each other in the square solution?

Title: Re: Arrange the 13 pieces
Post by SWF on Oct 29^th, 2003, 8:11pm

Quote:

I tried it as well for 1-2 hours.

That is less time than I spent designing this, but I appreciate anyone who gives it a try. It is demoralizing to spend time making an original problem only to have it ignored for 10 months. The majority of the problems on this site are not original, and a fresh problem offers the rare opportunity to the be first person to ever solve it.

On the other hand, maybe this problem is not deserving. I am concerned that the tricky solution might be considered unfair by some, but also don't want to ruin it by giving too many clues.

Rezyk is making some progress in finding the dimensions of the shapes. Some dimensions are correct but I think others are not quite right. Are those being measured from Photoshop, because they are numerically close to what I have for dimensions? Rezyk, the lines in your figures are a little bit too thin to show up well for me. I am curious if you had much trouble solving the rectangle. I had a few friends try it, and nobody came close to a solution. But they give up easily.

Title: Re: Arrange the 13 pieces
Post by Lightboxes on Oct 30^th, 2003, 5:54pm

After the square solution is found...
I'd like to know how you made this yourself! Did you mean to have the puzzle make a rectangle and square on purpose...or kinda stumbled on it, SWF?

Quote:

but that I'm unfortunetly not too good at.

This is one thing I'm better at...clear and concise competition.
The race is on Rezyk!

ADDED:MAN!...I assumed we couldn't flip any over. Sheesh!

Title: Re: Arrange the 13 pieces
Post by Rezyk on Oct 30^th, 2003, 10:35pm

on 10/29/03 at 20:11:48, SWF wrote:

I would ascribe the 10 month dry period to a few other reasons -- mostly that this type of problem isn't very open to progressive attacks. Until you reach the solution, you generally won't know if you're at all close to it, making it difficult to comment any earlier (in addition to having to compose an image for it).

As far as being "not deserving", I don't find that true at all. I found it particularly impressive that this puzzle could have such a resilient solution, much less two of them. I'm also having more fun attacking the square solution with a more analytical approach (I haven't spent more than a few minutes brute-forcing it) even though noone is joining in yet...

Quote:

Some dimensions are correct but I think others are not quite right. Are those being measured from Photoshop, because they are numerically close to what I have for dimensions?

They were all either matched by eye (within Photoshop) or analytically determined. I double-checked them and found that the sides (of the small rectangle) that I had matched to an analytic [sqrt]5 was actually not an exact match.

The remaining eye-matched sides are those of length 1 and 2...so if any of those are off, I think it's fair to let me know. :)

Quote:

I am curious if you had much trouble solving the rectangle. I had a few friends try it, and nobody came close to a solution. But they give up easily.

It wasn't as much trouble as I initially expected, but in hindsight I was extremely lucky. The very first thing I did (after figuring out basic Photoshop) was to guess the 2 big pieces into the correct positions. Some fiddling with the second-biggest pieces filled out the left side soon after, and the middle formed up once I figured to match the y-angle pieces together. The end got tricky mainly because, like Lightboxes, I wasn't sure if it was legal to flip pieces over.

So basically I just made a series of guesses, 2-4 pieces at a time, that all turned out to be correct.

on 10/30/03 at 17:54:43, Lightboxes wrote:

The race is on Rezyk!

??? Aren't we all in this together? :)
I'll tell you what...I'll consider this a race as long as you join me in giving updates in the progression of your strategy. ;) This solo verbosity has been feeling awkward...

Title: Re: Arrange the 13 pieces
Post by Lightboxes on Oct 31^st, 2003, 6:44pm

:)
You know...I think I have a great and very easy strategy that anyone can use. I had trouble following your analysis Rezk because of its complexity to me.

Anyway, your solution to the rectangle gave me an area of 4.5" X 4.8" = 21.6 in^2. So the square should be 4.648" on each side. So all you have to do is (spend a lot of time) measure all the sides of all the pieces. Put them in categories that you think would be easiest to determine the combination of sides, of the pieces, to make up a side of the solution.

I really doubt SWF made the square solution with two pieces going into one corner at the same time. (i.e. two triangles that have a 45 degree angle making up a 90 corner) The reason I say that is because SWF said it took less than 1-2 hours to make the puzzle. Doing two pieces in one corner can be very complex to allow a rectangle and a square (I think).

P.S. As a woodworker, I have the ability and perfection to see 1/100 of a inch on any ruler. It's even easier when the ruler has very small fractions. Anyway, my point is, I'm not sure where you stand when measuring the pieces relative to eachother.
****Rounding off the numbers?
****If so, by how much?
I actually did round my numbers because my pieces were printed out and cut out. Even though I'm a master at cutting pieces to perfection :) I believe there is a problem to where to cut. On the outside of the ink line, in the middle, or in the inside. I should analyze it some more, but I believe rounding off from .01 - .03 of an inch should be okay.

And one more thing: I'll only be tuning in every once in a while because I don't have the time to do such a puzzle like this (I'm taking 18 credits)

Title: Re: Arrange the 13 pieces
Post by Lightboxes on Oct 31^st, 2003, 7:11pm

K,
Having measured piece #3, after cutting in the middle of the ink line, I got:
1.342" and .675"
(referring to SWF's orginal picture: top left side, and top left side respectively, LOL)

Title: Re: Arrange the 13 pieces
Post by Lightboxes on Nov 1^st, 2003, 11:03am

I don't think you've posted this yet Rezyk:

Referring to the trickiness:
I kept thinking of position of pieces when it's really the lack of pieces in the center of the square solution.
Use the rectangle solution and just rearrange the large set of section that fit nicely? Maybe its a square solution with a rectangle hole in the middle?

Title: Re: Arrange the 13 pieces
Post by Rezyk on Nov 1^st, 2003, 1:49pm

on 10/31/03 at 18:44:30, Lightboxes wrote:

Anyway, your solution to the rectangle gave me an area of 4.5" X 4.8" = 21.6 in^2. So the square should be 4.648" on each side.

Nice, those rectangle sides are in perfect agreement with what I have (within a normalization factor).

Quote:

So all you have to do is (spend a lot of time) measure all the sides of all the pieces. Put them in categories that you think would be easiest to determine the combination of sides, of the pieces, to make up a side of the solution.

Hmm, applying this to what I have so far, this would mean that every side of the square has to have an edge from pieces {4, 6, 7, 10, 11, 12}, because no combination of the other pieces by themselves will give me the correct square side length.

Quote:

The reason I say that is because SWF said it took less than 1-2 hours to make the puzzle.

Actually, [e]she[/e] said that 1-2 hours was less than how long it took. ;)

on 11/01/03 at 11:03:24, Lightboxes wrote:

Referring to the trickiness:
I kept thinking of position of pieces when it's really the lack of pieces in the center of the square solution.
Use the rectangle solution and just rearrange the large set of section that fit nicely? Maybe its a square solution with a rectangle hole in the middle?

Yeah, I've been worrying about what SWF said about the solution possibly being considered unfair. So far I only have 2 cases for the source of unfairness that might not be ludicrous:

A square with a orthogonal square hole in the middle. (The argument would be that "it's a thick outline of a square")

An imperfect square -- negligible cracks or overlap or slightly-off side lengths.

================================ My update:

I've renormalized the side lengths by a factor of [sqrt]5 for nicer numbers.

Noticing that the rectangle bottom's 2 edges matched each other allowed me to determine many of the major edges, but in the process I also noticed a discrepancy. This led me to reexamine the angles, and I found that my previous result of 2x+y=90 was slightly off. So all of my y-angles are back to being "unknown", leaving me with exactly 3 degrees of freedom left to determine.

Now that I have the dimensions of the rectangle (15x16), I know the square must have sides of length 4[sqrt]15 which, assuming approximations aren't legal, seems almost impossibly hard to achieve (which might be good or bad). Half the pieces definitely lack any factor of [sqrt]15 and the remaining half is mostly "stuck together" to eliminate y-angles (unless there is some relationship between y and x that I've missed). This makes it very difficult to distribute all the "possibly [sqrt]15-related" distances along the entire height and width of the square.

It seems like this will either let me zero in on the solution or is indicative that I've made an invalid assumption earlier. Hopefully I can find some [sqrt]15 term in the remaining undetermined sides.

Title: Re: Arrange the 13 pieces
Post by Lightboxes on Nov 1^st, 2003, 9:51pm

I always assume SWF was a she.
Because of the kitten...
Or is it a manly kitten?

Title: Re: Arrange the 13 pieces
Post by rmsgrey on Nov 2^nd, 2003, 3:56am

on 11/01/03 at 21:51:52, Lightboxes wrote:

I always assume SWF was a she.
Because of the kitten...
Or is it a manly kitten?

I always assume SWF is a she because of the singles columns: SWF==Single White Female

Title: Re: Arrange the 13 pieces
Post by SWF on Nov 3^rd, 2003, 10:16pm

on 10/30/03 at 17:54:43, Lightboxes wrote:

Did you mean to have the puzzle make a rectangle and square on purpose...or kinda stumbled on it, SWF?

The pieces for this were not something I just stumbled upon. My effort was concentrated on exactly this problem, but also tried to make it difficult to assemble. Also, it took more than 2 hours, not less. I thought flipping over pieces is typically allowed in problems like this unless othewise specified, and I apologize if this was unclear.

Rezyk, it looks like your latest values given for the side lengths are correct so far. When I have tried to remember the solution without looking at my notes, I always recall how those two large pieces fit together for the rectangle, but even knowing that I have always given up. I wasn't sure if it was difficult to do or knowing the solution was written down right next to me caused a loss of motivation.

Quote:

So far I only have 2 cases for the source of unfairness that might not be ludicrous:

I think the tricky solution is even less ludicrous than those examples, but thinking along those lines should be helpful. Where the question says "Each shape must use all 13 pieces as part of the shape.", I was trying to disqualify forming the outline with the shape being a hole in the middle.

Title: Re: Arrange the 13 pieces
Post by Rezyk on Nov 5^th, 2003, 7:46pm

Alright, I've come up with a solution for the square. My answer image has been zipped+attached to avoid spoiling. I also included a complete proof that it is a perfect square -- no approximations.

Title: Re: Arrange the 13 pieces
Post by SWF on Nov 6^th, 2003, 6:23pm

Very good, Rezyk. That does resemble a square figure, but it is out of proportion and does not have the traditional shape. There is still room for improvement.

Title: Re: Arrange the 13 pieces
Post by Icarus on Nov 6^th, 2003, 7:24pm

on 11/05/03 at 19:46:36, Rezyk wrote:

Alright, I've come up with a solution for the square. My answer image has been zipped+attached to avoid spoiling. I also included a complete proof that it is a perfect square -- no approximations.

http://www.ocf.berkeley.edu/~wwu/YaBBImages/laugh.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/laugh.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/laugh.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/laugh.gif Well done!

Title: Re: Arrange the 13 pieces
Post by Grimbal on Jan 8^th, 2005, 4:58pm

I found this in the "unsolved problems" thread.

I'd like to propose the following improvement on Rezyk's solution.

Title: Re: Arrange the 13 pieces
Post by Icarus on Jan 8^th, 2005, 5:25pm

Is this an indication that I ought to get off my butt and update it again? I thought I would let another year or two go by first! ;)

Title: Re: Arrange the 13 pieces
Post by SWF on Jan 11^th, 2005, 5:27pm

Bravo, Grimbal! That is exactly what I had in mind. I have been waiting more than 2 years for somebody solve this.