PLEASE STOP READING.

This is not a hints post, and I do not ease you into a solution slowly. This is a post about how I solved the puzzle and some interesting facts about the puzzle I found out afterwards when I began to dive deeper into the solution set. As well as the programming that went with it.

If you would like some proper hand-holding, please see How to solve the 36 Cube puzzle – hints & solution for hints (and samples of the program that got me started down my path).

AGAIN, IF YOU DON’T WANT SPOILERS, PLEASE STOP READING!

Apparently you either don’t care, or have already solved the puzzle…

So when I first got this puzzle, it seemed moderately easy, contrary to the warning on the box. My brother was actually the one to take the first whack at it, and he did pretty well, getting 34 of the 36 pieces on the puzzle, but those last two were giving him issues and he was unable to solve the puzzle before he had to leave. Later that night, I took my first stab at it, and also got 34 of the 36, and after an hour or so of fiddling, and trying to brute force my way into a solution, I looked it up online. Not for a solution, but for more information on the puzzle and what kind of math was involved in solving it, so I could give it a more educated attempt. After reading the wiki page on Graeco-Latin squares, I noticed that Euler had conjectured that there was actually no solution to the order 6 square (of which this puzzle is).

This blew my mind. I could not understand how there could be a puzzle with a solution to which there was (now) mathematical proof that there was no solution. I thought either the proof was wrong (highly unlikely), the inventor had found a solution to the age old problem (also not likely), or the wiki page was wrong (more likely, but still not very).  I then found other wiki pages relating to the puzzle where the inventor said about the order 6 problem, “It struck me as the basis for a potentially great 3-D puzzle, and what eventually became 36 Cube.”  And this is when I became suspicious of the puzzle.

I remembered that when I was initially inspecting the puzzle, I had noted that some of the bases were not the same, even when the base was for the same height piece. I thought this was odd, but thought nothing more of it, chalking it up to manufacturing issues. After become suspicious, I went back to the puzzle and inspected it further. It occurred to me that there might be some chicanery going on, so I placed a few of the smaller, more innocuous-looking pieces on the board (to have a height measuring stick of sorts) and proceeded to try the height 6 pieces on all of the larger piece bases, including the height 5 bases. And lo and behold, when I placed a particular piece of height 6 on a particular base of height 5, it fit. I also tried the height 5 pieces on all of the height 6 bases, and one of them fit as well. This is the trick to the puzzle. Instead of solving the impossible, the inventor modified the puzzle to allow it to “cheat” it’s way to a solution. Once I had the first few trick pieces in place, it was simply a matter of logic to solve the rest, and I had the solution within the hour.

But I couldn’t stop there. I wanted to know how many actual solutions there were to the puzzle, so I sat down with a few example programs and began to pound out my own program to solve the puzzle.

My Program (some functions ported from script written by Daniel Hepper, who wrote the hints post above)
(Full unedited script)

It’s not the most elegant program around, nor does it use the most elegant algorithm, but it works, and it works fairly well. I started out by trying to solve a single color for each row [function solve_color( )], but I soon figured out that this would be difficult to make work for finding all solutions, so I abandoned it. I then wrote a function to iteratively solve each piece size in succession, finding all possible locations for that size, and then iterating further along each solution path until all the pieces were placed, or it failed for whatever reason [function solve_color_by_pieces( )].  If it succeeded in finding a solution, I tacked the solution array on the end, and continued down the next path. If it failed, I deleted the current solution path and continued on with the next one.

When I first ran the function, solving for yellow (which has one of the “trick” pieces), it gave me two solutions. Both of which I had manually proven earlier, which is a good sign for the program. I then added orange into the solution (the other “trick” piece), and it gave me one orange solution for each of the yellow solutions. The odd thing is, both of the yellow-orange solution sets have the same footprint. What this means is, I only have to continue solving for one of the two yellow-orange solutions, and they will both work. I just multiply the total number of solutions I end up with by two, and I have my grand total of solutions.

As I continued, I first tried solving for the whole puzzle, just to see what came out. What came out was an “out of memory” error (I told you it wasn’t the most elegant).  Apparently my program could do with some optimizing. I increased the allowed memory to 2GB(!), and ran it again. I again got the “out of memory” error.

Wow.

Ok, looks like I need to do this with some manual intervention. So I ran it with just the red color. And it quickly spit out 8 solutions. I added those constraints to the initialization one by one, and ran it again for each, this time solving for green (the colors are arbitrary).  It came up with 3 solutions for each of the 8 red solutions for a total of 24 solutions so far. As I was looking at the solutions, I noticed that a lot of them were the same, so I threw them in a lump and sorted them. As I was going through the sorted list, I noticed that there were 3 duplicates of each green solution, which brings my unique green solutions down to 8. I also noticed that there were similarities between the 8 red solutions and the 8 green solutions, so I threw them all in lump again, and sorted that group, and guess what… they were all the same. Each red solution was repeated in the unique green solutions. This struck me as odd when I saw it, but now that I think about it, it makes sense. They are the same solutions, they just move around each other.

This pattern continued with the remaining two colors (blue and purple), blue had 2 solutions for each of the 8 unique red-green solutions, but only 8 of them were unique, and they were the same 8 as the previous red and green. And purple only had one solution for each of the 8 red-green-blue solutions (which makes sense, since it’s the final color and really only has one place to go), and again, they were the same 8 patterns. But when I looked at the final 8 solutions, I noticed that they had similarities within them, so I labeled each of the 8 solutions, and put them together, and it basically ended up being 2 unique solutions, with various permutations and color switching for the others.

So with my final 2 solutions, multiplied by my initial 2 yellow-orange solutions, that’s a grand total of 4 solutions.

I don’t know why, but I assumed that a puzzle of this size would have more than just 4 unique solutions. But I guess that’s part of the game, false assumptions and trickery.

UPDATE: After feeling like my program was incomplete due to it’s inelegance and inability to calculate a complete solution, I went back to it, and tweaked it a bit to get it to properly calculate all the valid solutions.

I finally got it to output all the solutions after initializing it with one of the yellow-orange solutions. It cranked for a couple of minutes, and spit out 48 solutions. I knew that these 48 solutions were just the various color switching permutations of the unique 2 solutions, but I really wanted the program to crank out those unique 4 solutions, and nothing more. So I added the bare minimum of initial parameters (the two “trick” pieces) and set it up so that after the program found each solution, I converted it to an unambiguous state and compared it with any previous solution, and if that solution had already been found, it disposed of that solution and continued on. What I got at the end after the program cranked for a couple of minutes were 4 unique solutions. This not only made me pleased with my program, but pleased with my previous manual calculations.

And just to clean things up, I removed all of the debugging output from the program, and guess what… the program cranked out all 4 solutions in less than a second.

Wow. My program went from taking over 5 minutes to throw an “out of memory” error, to calculating all 4 unique solutions in under a second. I was actually pretty shocked about that, and pleased at the same time.

So I give you the 4 unique solutions:

1. B O G Y P R
   R G Y P O B
   G R P O B Y
   P B O R Y G
   O Y R B G P
   Y P B G R O

2. B Y G O P R
   R G Y P O B
   G R P Y B O
   P B O R Y G
   Y O R B G P
   O P B G R Y

3. B O P Y G R
   P B Y R O G
   G R B O P Y
   R G O P Y B
   O Y G B R P
   Y P R G B O

4. B Y P O G R
   P B Y R O G
   G R B Y P O
   R G O P Y B
   Y O G B R P
   O P R G B Y

Here is my updated and optimized program:
http://36cube.pastebin.com/KiJMjzTa

That is all, I’m happy with this program and the solutions it calculated and (probably) won’t be messing with it anymore.

my parents were in a situation where they were using several user accounts on the computer, but wanted to share the iTunes library across all the accounts. this posed some problems, and I hacked in a workaround, where iTunes stored all the music in the same folder, but it wasn’t elegant, and it didn’t work quite right.

they recently had some computer issues which forced them to reformat their hard drive (ok, fine, forced ME to reformat their hard drive), and they again wanted to share the same iTunes library. so while thinking over how I could more easily get their iTunes library to work across multiple accounts, and be more elegant than the solution I tried last time, I realized that the same solution could possibly work for me.

the solution that I realized for my parents computer and their multiple accounts is as follows:  on a *nix system, there is a thing you can do called a link (more precisely, a symbolic link), where one location actually points to another location. for instance if I had a directory at /bin which has a symlink pointing to it at /home/usr/bin, when I go to /home/usr/bin, I’m actually getting the contents of /bin, but the computer thinks I’m getting /home/usr/bin. a bit confusing i know, but trust me, it’s cool and extremely useful.

if you are using windows, windows doesn’t have symlinks per se, but if you’re using the NTFS file system (and if you’re on anything above win 2000, you should be), there is an almost identical thing you can do called a junction. and it’s made easier by downloading a small utility called NTFS Link.  with this, you can point any empty folder to any other folder on the computer, and when you navigate to that folder, the contents inside are actually in another location on the computer. this gave me an idea…

iTunes has a horrible habit of wanting to store it’s data files in the documents folder of the user. this poses a problem for trying to use the same library for multiple accounts, as each account has it’s own documents folder. what I did, was created a new folder in the root of the C: drive called “iTunes”, and created another folder inside that folder called “__iTunes Data” (the underscores are to keep it near the top of the folder when sorted).  I then opened iTunes with one of the accounts on the computer, which created the folder \My Documents\My Music\iTunes. I then closed iTunes, and moved the contents of that folder into the __iTunes Data folder I just created. after that, I right clicked on the now empty \My Music\iTunes folder and symlinked it to the __iTunes Data folder. now, iTunes goes looking for it’s data in the \My Documents\My Music\iTunes folder, but it’s actually looking in the C:\iTunes\__iTunes Data folder, but it has no idea. I then get into iTunes, and set the advanced preferences and change my iTunes Music Folder location to C:\iTunes.

mission accomplished. iTunes now thinks that it’s accessing it’s data in the documents folder, when really it’s in a different location (__iTunes Data), which can easily be linked to from all the other iTunes “accounts”.  just do the same thing for the other accounts, make a junction from \My Documents\My Music\iTunes to C:\iTunes\__iTunes Data, and set your library location in iTunes to point to C:\iTunes, and everybody is now using the same library. one thing to make sure you set, is while you’re setting the Music Library location, also check “Keep iTunes Music folder organized” and “Copy files to iTunes Music folder when adding to library” this keeps everything nice and tidy, and when one account adds music, it gets copied into the iTunes music folder, and all the other accounts can access it as well.

now on to how this helped me…  I realized that, while this works perfectly for multiple accounts on the same computer, my problem is a bit different. I have only one account, and all of my data is on an external hard drive, including my library data files. so what did I do?  I set up iTunes exactly as I had set up the iTunes for my parents, except that I then emptied out and linked the C:\iTunes folder to the location of my external hard drive library (which is at G:\Music\Benjam’s Music).  so now I have a perfectly functioning iTunes library, with no import required, and to top it off, when I need to wipe my hard drive again, I just set this up as it is now on the “new” computer, and it will behave as if it had never changed. all I might have to change is where C:\iTunes is pointing to.

here are those instructions again, in a nice easy-to-follow list (this only works with NTFS on XP, although if you have another method of creating symlinks in your system (which Vista does, google “mklink”), that would work as well):

1. download and install NTFS Link
2. download and install, or update iTunes
3. create a folder in the root of your C drive called iTunes (or whatever)
4. [Optional step] right click on your new folder and link to the location of your iTunes music
5. create a folder inside C:\iTunes called __iTunes (or whatever)
6. [Skip this step if you have an existing library] open iTunes once (this creates the \My Music\iTunes directory and creates some initial files in there)
7. go to the iTunes preferences, and the Advanced tab, and set the location for the iTunes Music Library to C:\iTunes
8. check both of the boxes below that, save, and close iTunes
9. move all the files in \My Music\iTunes into C:\iTunes\__iTunes
10. alt click on \My Music\iTunes and NTFS link that to C:\iTunes\__iTunes
11. open iTunes, add your music, enjoy

(this also works with pre-existing libraries, just move your music into the C:\iTunes folder, and move your \My Music\iTunes data into the __iTunes folder, and iTunes will see it like nothing changed)

I miss the comradery, the feeling that everyone is there with you, and yet, you are alone on the dance floor with nothing but your feelings and the music that moves your soul. The feeling that nobody cares what you are doing, but everybody cares how you are feeling. I miss the friends, the friends I only saw at raves, but were always there, smiling and feeling the music alongside me.

I miss the lights. The lights that people carried, starting off as innocent glowsticks and then transforming before your eyes into blobs of light that move to the rhythm. Evolving further into the blinky, crazy poi balls and string-worked lightshows that, if done properly, make you wonder just how many arms that photon magician has.

I miss the lasers. The lasers that, at first, were man powered little sticks of “what the… ? do that again…”, and then evolved into the green super-powered music-reactive room that permeated the smoke in such a way that you felt the only people in the world were you and anybody else that was lucky enough to be inside, with that music…

I miss not caring. I’ve grown to be to set in what I like and if it isn’t what I like, it’s not quite good enough. I miss being able to dance all night without needing the music or false energy to push me along, having nothing but the beats in my head long after the party is over. I miss being able to go all night long, and feeling that the night wasn’t even close to over when the DJ hits the power button and the music slows to a crawl and stops. And just then realizing that the sun is rising, and it’s a new day.

I love my life, wouldn’t change it for the world. I love my wife and my daughter, more than either of them will ever know. But I miss raves.

plur

Saturday Cameron took us to downtown Bratislava, in the pedestrian only zone, and then we went to the mall, Aupark, across the Danube. It was a chilly but sunny day, but it was so nice to get out and about finally. I read a few historical things about Slovakia before the move, but it is entirely different to then come here and actually see thousand year old churches and structures;  to be where another countries history took place, such as WWII or the Russian occupation, where it actually happened; to walk down these cobble streets, it makes you wonder who else has walked this path and under what conditions. It is very impressive and it made me feel more a part of our world and long history. I felt unimportant and small, humbled even, but very grateful to experience a tiny piece of another rich history, which really is my history too, since we all reside on one planet.

(more pictures after the break)

the international earth rotation and reference systems service (IERS) has announced that a leap second will be added at the end of 2008, making 2008 one second longer than it should be.

so either start your countdown at 11, or end your countdown at -1, either way…

enjoy the extra long year!  (it has both a leap day, and a leap second)

nope. try again.

I begin to look up information on the file format that the camera uses to record the movies (.MOD), and apparently, nobody supports it. so now, in order to use the file (and subsequently, the camera), I have to either A. use the crappy software that came with the camera (which is slow and difficult), or B. buy some other software (also from the camcorder company, how convenient) that will allow me to convert the movie to some other easier-to-use format.

well, that does it, I’m thoroughly pissed at all software and hardware designers that decide that it’s okay to nickel and dime us by using proprietary file formats that are impossible to use otherwise. even if they don’t want to nickel and dime us, it still sucks when a company tries to “better” the current standards by creating a new format that only their software or hardware uses, and then fail to release the format details so others can use it as well (think MS Word 2007 and the .docx format).  well, at least with that one, OpenOffice.org finally supports it in v3.0, but .odt already existed and is more openly adopted by other word processing software anyway.

anywho…

point is, I doubt seriously that .MOD (which, I found out, is just .MPG with a different extension and some other .MOI files to go with it) was really necessary. why not just use one of the other formats that are already out there and more widely used, why must you create your own?  and if it’s absolutely necessary to create your own, at least release the details so that others may take advantage of your apparent excellence (think .7z files).

and if not, just use a format already available. my current camera records movies in .mov format, and everything I use can read an .mov file, and the pictures are .jpg… easy.

new camcorder… .MOD files… not so much.

NOTE: I finally found a small simple utility to convert my .MOD files to .MPG and set my widescreen marker at the same time: SDCopy

http://news.bbc.co.uk/2/hi/technology/7784908.stm

Done reading?

Now, do yourself a favor and grab one of these:

• Windows
  • Firefox (Highly Recommended)
  • Google Chrome (Also Recommended)
  • Apple Safari
  • Opera
• Mac
  • Firefox (Highly Recommended)
  • Apple Safari
  • Opera
• Linux
  • You’re good, no worries
  • But grab Firefox, just for the heck of it
  • You can also grab Opera, if you want

It talks about linking to a specific time in a YouTube video. I’ll repeat here in case you don’t want to link offsite.

Have you ever recieved links to videos that say something like “It gets really good around 1:47″, or something similar?

Now you can link to a specific moment in that video by putting the time as follows in your link: #t=1m47s

What this does, is when somebody goes to that link, it not only starts at the given time, but it doesn’t buffer all the stuff that comes before it, so you don’t have to wait for the part you’re interested in.

For instance, try the following link:
http://www.youtube.com/watch?v=OP1-5uxZffE#t=5m45s (SFW)

That’s just awesome.

This is the story of those events, and it’s a long one, so I’m breaking it up into several installments.

It begins in the first grade (I know, right? first grade!).

I was in the first grade and I had been doing very well with my studies (studies? like what, drawing horseys?), and was doing so well, that they decided to move me up a grade, right in the middle of the year.  So I guess the story also begins in the second grade. Anywho… on with the story…

My family (consisting of me, my mom, and my little brother), moved from an apartment in Salt Lake City, to a house in Magna when I was in the third grade (but I thought you said this story begins in the first grade?  Well, it does, and we’ll get to that in a moment.)  We had moved into a small neighborhood, with a few kids, but not many.

One day, shortly after moving, I was riding my bike down the street, and there were some kids about my age throwing a tennis ball back and forth across the street.  Naturally, kids being kids, and boys being more so, they threw the ball at me.  I, of course, didn’t really like this, and thought that these kids were being mean to me, and went home.  Later on (possibly that same day, I don’t remember), I met these kids again, and not sure what exactly happened, but contrary to what you might think, we hit it off very well.  One of the kids in particular (Angel), became one of my best friends all throughout the remainder of elementary, jr. high (middle school, for those not familiar), and high school.

Angel, and his brother Jose, were two of the three kids that threw the tennis ball at me, and the three of us were pretty much inseparable from that point on.  Jose was a couple years older than we were (Angel and I were in the same grade, and this is why skipping a grade in early elementary is important to the story, as I don’t think they would have been as important in my life if we hadn’t been in the same grade), but we were always together, all the time.

Fast forward about 6–7 years, the three of us were still inseparable, but we were older now, and all of our fun activities went from playing in the street, or playing Nintendo, to doing things at night.  Because, for some reason, things that you do as a teenager, even if they don’t require darkness, seem so much more fun at night.

We always went to various teenage nighttime events together.  They took my to my first dance club (pretty much snuck me in, because I was only 15 at the time (remember, even though we are in the same grade, I was a year younger than Angel)), which was pretty amazing.  We went on several runs to various places that young boys and girls go to together, like Goblin Valley (not the real Goblin Valley, but the one on 9th south.), and to the top of the world, and ice blocking in Sugarhouse park (and yes, we did have girls with us at the time, it wasn’t just a group of guys).

It was during this period that I got introduced to techno music.  And if you know me, then you know I love techno music.

We (Angel, Jose, and I) were going to pick up some girls, and we were listening to some music in the car.  I was about 14 or 15 at the time, and I’m pretty sure the song that came on was Jump by The Movement.  I loved it.  I asked Jose what type of music that was, and he told me it was techno.

I’m not sure when it was, but it was probably the next day, I ran off to the music store, and grabbed any CD in the store that had the word ‘techno’ on the cover (which amounted to about 3 CDs, because at the time—1992 or so—techno was still very new, and unbeknownst to me, very underground).  I immediately became entranced by the music, and have loved it ever since.  I would listen to it during school… in the car… at home… basically everywhere.

And that is where I’ll stop this installment of the story, and leave you with a blast from the past…

and if you’re wondering… my second love is techno. My tastes have been refined somewhat, for as I go back and listen to some of the stuff I bought back then—the stuff that got me so entranced with the genre—I sometimes wonder what I was thinking. Because even though I loved it at the time, and it therefore still has a place in my heart, it really wasn’t that good. But I still love techno.

On to part 2 →

i downloaded my copy, started it up, and was immediately impressed, and being somewhat of a Firefox fanatic, that’s saying a lot.

it takes no time at all to load, and after reviewing the comic that was released (by accident) a few days ago, it’s easy to see why: every page is it’s own process.  for those of you who don’t know what a ‘process’ is, it’s basically it’s own running copy of the program.  each page is it’s own entity on the computer, so if one page crashes, you don’t have to take down the whole browser to fix it, just close that one tab.  but Google being Google, took it one step further, and made every portion of the webpage rendering it’s own process.  so basically, if you have javascript running on the page you are viewing–it’s in it’s own process; flash running? it’s it’s own process;  quicktime?  own process;  any other plugin? own process.  so if at any time, any of these pieces crashes or slows way down while working on something, it doesn’t freeze the whole system.

this way of doing things also makes it really easy to block malware from infecting your computer.

Google Chrome was basically built for the internet as it stands today, in all it’s multi-media, web 2.0, watch-your-back-for-malware goodness.

the only thing missing from Google Chrome is a Firefox-esque plugin architecture…

because in order for me to switch from Firefox, I’m gonna need to take my Web Developer toolbar and mouse gestures with me.

but if you spend any decent amount of time on the internet, and do most of your browsing in the more modern web 2.0 sites: give Google Chrome a try.

the only thing that sucks about it is that it’s yet another browser that web developers have to code workarounds for.