## A Conversation for Pentominoes - a Puzzle

### 3-d Pentominoes

loneTree Started conversation Mar 21, 2003

A very good article. I disagree that the 20x3 is easiest to solve - as there is only one solution ( the 2 solutions mentioned in the article are partial mirror images of each other - so solving one gives you the other) versus finding any one of the thousands of the other sorts.

But aside from that, the author misses one dimension of the puzzle - if the squares are blocks, then the same shapes make 3-d figures and can be formed into a 3x4x5, 10x3x2 or 6x5x2 block.

In the third dimension, more arrangements of 5 squares also become possible - as far as I know, no-one has made a study of what challenges a puzzle of that nature might pose.

M-B Games made a game 30 years ago based on the flat puzzle with a little red tray of 10x6(I think), with many copies of each piece, and an instruction booklet with many exercises showing the starting places of some pieces, and progressively showing fewer pieces for each exercise. The great thing with the booklet was that it did give solutions.

l†

### 3-d Pentominoes

Baron Grim Posted Mar 25, 2003

The two solutions to the 3X20 are NOT isomorphisms. They are indeed unique solutions. If my memory serves me the X and U pieces are together in both solutions but most of the other pieces are in very different arrangments. You can not make one of the solutions and just rearrange the halves to get the other either. But I do agree that maybe the 3x20 is not indeed the easiest. It is for me, but that's only with hind sight. Since writing this article and making a set I gave to my local pub, one of the clientel has become somewhat obsessed with this puzzle and has found about thirty solutions. (On one day he claims to have found four solutions in an hour, but I suspect two were isomorphisms.) But he couldn't get the 3X20. I sat down at it and did it in about 30 minutes. I hadn't done it in over a decade, but I was able to find a solution. Of course I did vaguely remember the positions of the X and U for both solutions so that helped and I also knew that there were only two solutions and from that knowledge I figured out that the V *had* to be on and end (otherwise it would leave a verticle line in the middle which could be used to "isomorph" a half and find more than two solutions... Ok... some of this was faith.) My goal now is not finding solutions, but trying to find out any logic to finding solutions. Trust me, a much more difficult puzzle.

As far as the 3-D versions, I didn't exactly miss them. They really are a different puzzle. Most refer to them as pentacubes. There isn't a "nice" solution using all 29 pentacubes (12 of which are the original 2D pentominoes). There's not a solid block you can make using all 29 that I know of. But there are some very interesting shapes and solutions that include leaving internal "holes". Pentacubes could easily have an article of its own. Keep in mind there's also tetracubes, hexacubes, heptacubes, octacubes... you get the idea. I was trying to keep this article fairly short. Really an introduction. But if you want to learn more, I do recommend the link in the article to the Poly Pages as well as Solomon Golomb's book, "Polyominoes". You can also find a couple of puzzle makers online that sell some very nice 2D and 3D versions.

I gotta warn you though... these puzzles can be very addicting. They seem nice and cute and all, but suddenly you find yourself obsessed. They take over your brain if your not careful.

CountZero

### 3-d Pentominoes

loneTree Posted Mar 26, 2003

My recollection is that as the x piece spans the whole width, you can therefore flip half the puzzle over, and get a different solution. But I should just go rediscover the solution, that will set my mind at rest.

I don't doubt that someone could find many solutions in a short time - like chess, once you start seeing the patterns, solutions materialize quickly.

Yes, I often contemplate the program that finds solutions to these kind of puzzles - and I wonder how the total number of solutions is arrived at? Obviously, either an exhaustive search, or a mathematical proof. The latter would be more intriguing.

And I do like your article - it is a good introduction - I was suggesting that a few more hooks to other subjects might make it more interesting.

A similar, smaller set of 3-d cube shapes considers only 3 or 4 cubes, arranged in non-linear fashion - so the '|' is not allowed, and the 4 square is neither. That leaves one 3 cube shape, and six 4 cube forms. ('cause it is 3-d, two of the 6 shapes are mirror images.) Total 27 cubes, and makes a large number of 3x3x3 cubes.

Perhaps I should pen an article covering those...

lone†ree

### 3-d Pentominoes

Baron Grim Posted Mar 26, 2003

If you decide to work on a polycube article, be sure to check the poly pages.

As far as how they wrote the pentomino solving programs, I'm fairly certain that they used a brute force method. There's bound to be some s here who would know more about that.

Now as far as the 3X20, I'll give you another hint. The U and the X are on one end, so flipping these would just be a simple isomorphism. There are no symetric boundaries within the body of the solutions that would allow a partial isomorphic, but validly unique, solution.

If you'd like to try the 6X10 without buying or building a set, you can find a link to a javascript version from the link in the article.

Have fun.

### 3-d Pentominoes

Pimms Posted Mar 28, 2003

The Soma Cube is surely the polycube puzzle to start with. Six tetracubes and one piece of three cubes.

Search for Thorleif's SOMA page on the WWW (http://www.fam-bundgaard.dk/SOMA/SOMA.HTM) to find out much on the most popular polycube puzzle.

Are there any Entries on polycube puzzles? A quick search didn't find any.

Pimms

### 3-d Pentominoes

Baron Grim Posted Mar 28, 2003

I don't believe there are any other articles on polyforms at all besides the one on tetris, which doesn't exactly count as a puzzle. (excellent article though).

Thanks for the link to the SOMA puzzle. I'm almost certain that this is the puzzle I remember having as a child. I remember it formed a cube and was made from blue plastic, but I couldn't remember if it was all tetracubes or not. I do remember playing for hours with it forming different shapes.

### 3-d Pentominoes

Array Posted Mar 18, 2006

I have been doing pentominoes since I read the book by Arthur C Clarke as a teenager in the seventies, and can confirm that it can be both addictive and frustrating.

The friend in the pub may indeed have found four unique solutions in one day, as there are a few pieces which when placed side by side form a shape which can be rotated or flipped as one, giving a new solution (T and Y and X in a line can be flipped to give three distinc solutions). Also as "L" and "P" can fill the same space as "X" and "C" , some solutions immediately give rise to three others.

Perhaps because I don't have the skill, but I hope because I have the patience, I have made it part of my life's work to find as many solutions without computer aid myself. so far I have found 903.

I am considering using computer aide to record the solutions, as it is very frustrating finding a "new" one and checking it in the book, only to find it was solved (especially if the solution is closer to number 903 than number 1!). However the downside of that would be entering the existing 903 solutions from the books to the store!

### 3-d Pentominoes

Baron Grim Posted Mar 21, 2006

Well, if you don't want to see all the solutions avoid some of the links I gave in the entry.

(speaking of which, I better check to see if they are still working.)

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