# Puzzle! - Tetrads Filling a Square

Thread in 'Research & Development' started by schnuerle, 14 Oct 2008.

1. ### schnuerleUnregistered

I'm trying to work out a puzzle I came up with related to Tetris. I want to come up with a way to use the 7 Tetris blocks ( http://en.wikipedia.org/wiki/Tetris#Gameplay ) to fill up a square surface without any gaps.

Here are the rules:

1. Tetrads - Can only use the 7 Tetris shapes, and there can be no gaps or holes.
2. Square - The surface has to be a square.
3. Size - The square can be any size between 7x7 through 20x20. 10x10 would be good though.
4. Neighbors - No block can touch another block of the same type. IE, you can't fill the space with 2x2 O tiles. Diagonal corner touching is OK though.
5. Blocks - If the finished block is copied and put on any of the 4 sides of the first block, rule #4 still applies. EG, you can't have a 2x2 O shape at the bottom right and bottom left of the block, since if the whole block is copied to the right, the 2 shapes would be touching.

Obviously, #5 is the kicker! The created block has to be able to be used to fill up a larger grid of blocks and still hold to rule #4.

Here is an example of an 8x8 grid, but it breaks rules #4 and #5 since the same type of tetrads are touching (even though they are different colors) http://blog.craftzine.com/TetrisBlanket.jpg

Example of a 6x6 grid, also breaking rules 4 and 5: http://daddytypes.com/archive/oguro_tet ... blocks.jpg

8x5 grid, breaks rules 4 and 5, and 2 since it's a rectangle: http://www.inhabitat.com/images/tetris_new_big.jpg

Here is one that is closer. It is a 10x10 grid that actually tessellates to fill up a larger space (set it to your desktop wallpaper to see). It is close to fulfilling rules 4 and 5. But it breaks #1 since the square itself has gaps or cut off tetrad shapes. http://art1.server05.sheezyart.com/medium/75/754773.jpg

And here's an example of a larger number of blocks covering a large space without gaps, but obvious breaking rule #2. But it fulfills rule #4! http://i61.photobucket.com/albums/h44/d ... lpaper.jpg

Also check out Tetris Tiles for ideas: http://www.tetris-tiles.com/

Alright, I know this is a tall order. But I'm hoping that some experts here on this forum would know about some solutions to this already, or be able to come up with a solution or prove that it is impossible!

2. ### Kitaru

I was once asked to solve a puzzle with similar rules:

1) Tetrominos: Must use an equal number of each of the 7 tetrominos.
2) Square: The surface must be square.
3) Size: The dimensions must be as small as possible while still fulfilling all other rules.

After doing some work, I found the smallest possible square using an equal number of each tetromino is 28x28. I constructed 5 4x4 squares: one with 4 I, one with 4 O, one with 4 T, one with 2 L and 2 Z, and one with 2 J and 2 S. The final square calls for 7 I squares, 7 O squares, 7 T squares, 14 L+Z squares, and 14 J+S squares.

Here are some diagrams I made in the process of solving the puzzle: http://fumen.zui.jp/?m105@7eUKfBEgBkVB6 ... Y8e4GrbAAA

3. ### Kitaru

Earlier I thought one of the wordings of the rules said "Must use at least one of all 7 pieces," but upon re-reading found it said "Can only use the 7 Tetris shapes..."

http://fumen.zui.jp/?m105@7eHZaBSlBzaBG ... 7?eBAAAAAA is a solution for you puzzle, I think. Incorporating at least one of every piece is proving to be a bit more complicated.

4. ### schnuerleUnregistered

Kitaru,

Hey, that's terrific. I'm impressed you came up with a solution so fast.

I might need to add another rule though. While you solution does meet the 5 rules, it's a bit too visually repetitive for what I am looking for.

This might sound odd, but I want to use the final solution in a quilt pattern. I tailored the rules to allow the production of a standard quilt, ie. with multiple blocks of fabric stitched together. I want the end result to appear random, look good as well, and to me that means using a solution that is not too repetitive. If your 8x8 solution is used it would be tad dull across an entire quilt. And actually, what you have found is a 4x8 solution that you repeated 2 times in the 8x8 space.

Rule 6: Repetition - Don't repeat groupings of tetrominos within the solution. Use all 7 tetrominos if possible.

Is that helpful? I know this is going to be tougher.

Also, what tool are you using to create your step by step Tetris models? It's really neat.

5. ### Kitaru

http://www.tetrisconcept.com/wiki/index.php?title=Fumen has info to the tool and links to the page where you can use it. However, it appears Zeta's English version is down.

Anyway, here's another solution that should work for a bit. I'll try to come up with something with more variety later. I'm also trying to figure out how I'll be able to accommodate the T piece.
http://fumen.zui.jp/?m105@7eQZaBSlBzaBG ... B0IBmDBAAA

7. ### DumbledorsArmy

Heh. I found that same pic a little while ago, Photoshoped it and now use it for my background on my YouTube page. I know that you don't want any spaces or gaps, but it works great for wallpaper on a desktop. Tiles quite nicely and I like the neoness of it!!

Hey schnuerle: How big of a grid is your quilt going to be. In my experience if you take a group of pieces and tile them, they don't look random enough in my opinion. It still looks too 'planned'.

9. ### schnuerleUnregistered

DumbledorsArmy: I'm not sure how big of a grid it's going to be yet. It'll depend on the size of the solution, and how small/large I make the blocks. But you are right, it will probably look a bit 'repeaty' and not random enough. But I'm okay with that, since most quilts look like that anyway. Unless I can find a very large solution, like 20x20

DIGITAL: Hey, that second one works! So I might try to go with that 8x8 one, unless someone else can come up with a larger solution. 10x10 or larger would be nice... Great find though.

How did you find it, did you browse a collection somewhere? Do you know who make it, so when it's done I can give credit? Thanks!

11. ### schnuerleUnregistered

DIGITAL: That 10x10 is pretty cool, thanks.

Keep digging if you can. If I can find multiple solutions of the same size (ie, 8x8 or 10x10 or larger), I can use them all in the quilt to avoid the repetition issues that DumbledorsArmy was talking about.

12. ### DIGITALUnregistered

I'll have a go at some more 10x10 solutions later. Or would you rather have something like a 20x20? The same concepts apply really and it shouldn't be any more difficult.

14. ### schnuerleUnregistered

DIGITAL: Well, a big 20x20 would work well. I'd just tile 4 squares of 20x20 to make the quilt. Pretty sweet if it's possible.

Kitaru: Cool, another 8x8, thanks. I might be able to piece that block together with other 8x8s to make a pattern that repeats less.

15. ### Zaphod77Resident Misinformer

if yu have a 4 10x10 solutions that fail the tile rule, they may still tile into one 20x20 that passes the tile rule.

SO it's best to go for the 20x20 solution made up of 4 10x10 solutions, adn relax rule 5 on the 10x10 ones.

16. ### DIGITALUnregistered

Though doing it in this matter would assume that the edges of the 10x10 solutions won't conflict when pieced together. IMO, it's far easier to just make a 20x20 from scratch. My strategy so far is to work from the outside to the inside, making sure the edges don't conflict first.

18. ### schnuerleUnregistered

DIGITAL: Man, that is insane! You got a 20x20. How are you doing this, on paper, then transferring it to the web program? Awesome.

zaphod77: I think you are right about going for 4 10x10s. And that rule #5 becomes less important in that case, since the reason for the rule is to avoid visual repetition.

Ok, so I'm happy for all the help and glad to see what's possible. So maybe now I should think of the best possible requirements to make a nice quilt.

So I think that, knowing about how quilts are made and what it takes, there are 2 options:

1. Make a big square quilt with the 20x20 solution. Done, thanks to DIGITAL.

2. Make a rectangular quilt (for a bed) that is made up of multiple 10x10 or 8x8 blocks, with a final block setup of 3x2, 4x2, 3x4, or 3x5, etc.

The multiple block solution (2) is easier when quilting, since you create the smaller blocks first, then stitch them together. Making one big block is harder due to hand tool and sewing machine restrictions.

So we can consider it done, or if you all are having fun, creating 2 more 8x8 solutions and 3 more 10x10 solutions would be plenty to make the rectangular quilt. What do you all think?

When completed, I'm going to do a big blog post about it, thanking each of you, the contributors, and crediting each block made. It will take a while, since it will be a combination of me and my mother-in-law doing the work, and we live in different cities. But I think it will be awesome when done!

19. ### DIGITALUnregistered

Nah, I did it straight on there. It's nothing too complicated to be honest. Once you have the outer edges finished, working inward is very quick and easy. The only time you will run into problems is towards the very end where you may have some same piece conflicts, though nothing that can't be fixed with a little trial and error in rearranging minor portions.

20. ### schnuerleUnregistered

I just realized this is actually 2 10x20 solutions. Interesting.