The Nineteenth Byte

5:06 AM

Hmm. There are 9! = 362880 ways to fill a 3x3 square with 0..8. I can divide this by 16 because of rotations, flips, and inversion (subtract from 3^2-1), which is 22680, a little more than 60 times 376.

You know what would be really helpful? A group action and a way to know if it preserves sturdiness.

Alex A.

@nhahtdh Oh hey, you're n̴̖̋h̷͉̃a̷̭̿h̸̡̅ẗ̵̨́d̷̰̀ĥ̷̳

El'endia Starman

Wait, no...I need to know what the group of 3x3 squares filled with 0..8 is, to start with.

The answer is, of course, that there are multiple such groups. So now...how do I pick a useful one...

feersum

I know what are groups but what is a group "action"?

El'endia Starman

It's been about half a year since I learned about them, so gimme a couple moments.

I may have meant simply operation.

Okay, I think this is right: a group action is where each element of a group is an operation on some set. So composing two such operations gives another operation in the same group, that sort of thing.

I'm not totally sure this is the right idea though.

I'm trying to figure out how to use Burnside's Lemma.

Okay, yes, "group action" is right.

feersum

What does this theorem tell us that's interesting if we could use it?

El'endia Starman

5:18 AM

It lets you not count duplicates by counting fixed positions.

So like, let's say you have a cube where you can color each side with one of three colors.

How many distinct colorings are there?

up to rotation?

yes

You can look at every possible rotation and count how many colorings are fixed, that is, don't change under that rotation.

You add up these, and then divide by the number of possible rotations.

And ta-da! You have the number of distinct colorings!

feersum

Oh, well I knew that.

That doesn't stop us from having to examine all the rotational duplicates, does it?

El'endia Starman

5:22 AM

Mmm? Not sure what you mean?

Alex A.

Mmm!

feersum

Using this lemma, you still had to look at all 6-tuples of colors.

El'endia Starman

Yes and no.

For instance, consider the rotation where you spin the cube around an axis that goes through two corners on opposite ends.

Not on the same square.

All of the faces adjacent to one corner must be the same color, and likewise for those adjacent to the other corner. So you only have 3*3=9 possibilities.

Sp3000

Man this \n\n rule :/

El'endia Starman

I really don't like it.

Sp3000

5:27 AM

Doesn't yours do \n\n as well?

El'endia Starman

.....yes.

It's hard to actually do without also keeping track of how many grids have been computed so far.

feersum

What's nice for writing faster programs is easily-computed rules that allow me to identify a grid as the sole representative of its equivalence class under the group operations (Peter called it the 'canonical' grid).

El'endia Starman

feersum and don bright's answers also have \n\n.

feersum

Oh, is that really a rule?

El'endia Starman

Comments on the question.

I knew of it and decided not to bother with it.

feersum

5:34 AM

lol

nhahtdh

@AlexA. Yup.

El'endia Starman

It would add something like 12 bytes or so.

Sp3000

+17 for me :/

nhahtdh

(As for the avatar, you might want to visit Maid Cafe chat room)

Sp3000

Yuru Yuri?

nhahtdh

5:38 AM

Yep, Toshinou Kyouko from Yuru Yuri, but the thing is, our room is having an avatar change event, where people participate by changing their avatar to that of Toshinou Kyouko

At one point, all our mods change their avatar to this exact same picture, and also their handle to Toshinou Kyouko

It's hilarious looking at the list of moderators

Sp3000

Ahaha nice :P Haven't popped by the cafe for a while (only really lurked tbh)

El'endia Starman

@Sp3000 Looks like you beat me anyway, so props on that.

Sp3000

You should from itertools import* btw

El'endia Starman

...that saves me 1 byte.

lol

Sp3000

:P (import ... as ... is basically useless for single imports, fyi)

El'endia Starman

5:48 AM

Also, isn't Python syntax highlighting ?

Sp3000

lang-py is what I usually use

El'endia Starman

Ah, the problem was that I didn't put -all.

The last three squares all have four numbers different from the first one.

Peter Taylor

@El'endiaStarman Yes. An m x n square has (m-1) x (n-1) blocks, which between them cover most of the cells 4 times, the edge ones 2 times, and the corners ones 1 time. You can get a lower bound for (m-1)(n-1)S by assuming that the corners are the four largest numbers, and the edges are the next largest; and an upper bound by assuming that the corners are 0 to 3, the edges are 4 to ...

That's what the case-split calls to SetBounds in the gist I posted do.

El'endia Starman

ahh, gotcha

Peter Taylor

@NathanMerrill I removed that comment because I replaced it with a much more general one.

El'endia Starman

6:03 AM

Oh. Looks like I was wrong before.

8 1 6
0 7 2
5 4 3

8 4 6
0 1 2
5 7 3

> Things that won't work: swap two numbers, rotate a sub-square, ~~shift one row, shift one column~~.

feersum

3x3 is a special case because the center square doesn't matter.

El'endia Starman

That'd apply to odd squares.

feersum

What I mean is that the center square is in all of the 2x2 boxes, so it can be changed to an arbitrary value without breaking equality.

El'endia Starman

riiight

The top and bottom middle values just increased by three.

Oh, yeah, that would only work for 3x3.

You wouldn't be able to increase the sum of all blocks simultaneously by adding a constant to two values.

Okay, in this pastebin are all of the squares where there is another ("greater") square that shares six of the same values. "Ref:" refers to the referent square; the following one or two squares are those that are three value-changes away.

Most squares do not have such a "neighbor".

Specifically, 216 squares do not have such a neighbor, 128 squares have just one neighbor, and 32 squares have two neighbors.

.... very "nice" numbers...

6^3 (or 2^3 * 3^3), 2^7, and 2^5, respectively.

feersum

3x3 seems a bit too small...could be misleading to study it

El'endia Starman

6:19 AM

mmm, perhaps

Unfortunately, though, I can't generate all 20,352 5x5 squares at the drop of a hat. :P

feersum

that's the number of 4x4

El'endia Starman

whoops

All 80,000+.

feersum

Do you want to use the C code?

There are 51840 of 5x5.

El'endia Starman

You'd probably work with the C code a lot better than I.

I'm just investigating connections.

Like this: {'[2, 1, 1]', '[2, 2, 1]', '[2, 1, 2]', '[1, 2]', '[1, 1]', '[0]'}

So, I counted how many neighbors each square has. Then for each of these neighbors, I counted their neighbors. This set above is all of the possibilities that show up.

feersum

test

El'endia Starman

6:30 AM

So, if a square has two neighbors, those squares cannot both have two neighbors each. If a square has one neighbor, that neighbor might have one or two neighbors.

@feersum: Do you understand what I'm doing here?

feersum

Not really.

El'endia Starman

hmm

feersum

Have you verified that neighbors even exist for 4x4?

El'endia Starman

Haven't tried yet.

Hey, is it possible to add a constant to all elements and then take a modulus? Or, equivalently, are squares ever cyclic permutations of each other?

Some more stats:

376
(1, 2) 32
(0,) 216
(2, 2, 1) 16
(2, 1, 1) 8
(2, 1, 2) 8
(1, 1) 96

Numbers on the right are how many of each there are.

flawr

7:02 AM

@AlexA. I like displaying my old versions, I think it is interesting to see how the length developed and what tricks were used!

El'endia Starman

@feersum The answer is YES! Where neighbors are those that share 12 values (so 4 are different).

20352
(4, 8, 4, 4, 4) 256
(4, 4, 4, 4, 8) 1024
(4, 4, 8, 8, 4) 256
(4, 4, 8, 8, 8) 3072
(4, 4, 4, 4, 4) 3072
(4, 8, 8, 4, 4) 256
(4, 4, 8, 4, 4) 1024
(4, 8, 8, 4, 8) 3072
(8, 4, 4, 4, 8, 4, 4, 4, 4) 2624
(8, 4, 4, 4, 4, 4, 4, 4, 8) 96
(4, 4, 4, 8, 8) 256
(4, 4, 4, 8, 4) 1024
(4, 8, 4, 4, 8) 1024
(8, 4, 4, 4, 4, 4, 8, 4, 4) 896
(4, 4, 8, 4, 8) 1024
(8, 4, 4, 4, 4, 4, 4, 4, 4) 96
(0,) 384
(8, 4, 8, 4, 4, 4, 4, 4, 4) 896

.... fascinating!

New Main Posts

7:20 AM

0

Q: Convert a string to alternating upper/lower case letters of the (english) alphabet

PHP Goal Write a PHP program that takes a parameter as input and outputs a filtered version with alternating upper/lower case letters of the (english) alphabet. Non-letter characters have to be printed but otherwise ignored. Example Input: Hello World! Hallo Welt! Output: HeLlO wOrLd! ...

El'endia Starman

7:37 AM

Gonna leave my computer working on 5x5 overnight.

feersum

8:30 AM

@El'endiaStarman I don't get what the numbers mean.

There are a lot of symmetry operations for 4x4 though.

Standard square symmetries (x8), column rotation (x4), row rotation(x4), subtraction from 15(x2).

If none of these are redundant then that's 256 elements in a symmetry group.

Would explain why there are so many powers of 2.

Anonymous

Wow my internet is laggy as all hell tonight

Martin Büttner

9:25 AM

@AlexA. Mods could also have purged the history. Just saying.

(Turns out, purging the history does not purge the last version... that's a bit stupid... I think you could also hard-delete the message right away though.)

1 messages deleted

doesn't help either.

so remember everyone: if you post sensitive information, first edit it out, then delete it...

Doorknob

11:28 AM

or, ya know, don't post sensitive information :P

aditsu

11:47 AM

< sensitive information >

Beta Decay

Please note the new bounty:

18

Q: Golfing strings in Fourier

Challenge Given a string as input, golf down the Fourier program which outputs that string. In Fourier there is no easy way to output a string: you have to go through each character code and output that as a character. Fourier The language is based upon an accumulator, a global variable which...

code-challenge metagolf

aditsu

you do the Fourier transform and encode the frequencies and amplitudes?

feersum

How can I start a program from the commandline in linux without blocking until it exits?

aditsu

append an ampersand

it will still be linked to the shell

feersum

cool, thanks

aditsu

11:53 AM

you can also check out ctrl+z, bg, fg and jobs

New Main Posts

1

Q: Dames, do some math!

The order of operations, PEMDAS, is a basic rule in mathematics telling us which order operations should be performed: "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction" The problem is, PEMDAS is not very versatile! What if you wanted to do it in another order? ...

code-golf

Martin Büttner

I've gotta say, the Voronoi wave also makes for a really good portrait-orientation desktop wallpaper

trichoplax

12:30 PM

@Status Sorry for the delay in getting back to you. All 4 images work fine for me, using Python 3.4.3 on Windows 8.1, with pillow version 3.0.0. They are JPEG images so the distortion makes the key image barely recognisable, and the restored image horribly corrupted, but they don't give any error messages, and converting them all to PNG first gives the intended results without corruption.

I'm trying to guess what might be different for you but I have no errors to work with at this end.

Anonymous

Hallo

trichoplax

Did you just say hello?

Anonymous

In Megoese, yes

trichoplax

Hallo :)

Anonymous

Megoese is like English, but there's an X% chance that a randum letter in a sentence will be swapped out

Anonymous

12:43 PM

X=rand()

Anonymous

Double random, what could it mean

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