Tag: tri-diamonds

Notation Notions: Operations on Ominoes

~ munizao ~ Leave a comment

Here’s a tiling of the 3+2-ominoes:

This use of a plus sign seems natural enough, but we might want to think a bit more about what it implies. We have established an operation on polyform sets, and a notation for that operation. This raises some questions: what other operations might we want to use? How should we notate them? And finally, can we design a notation system that readably describes a wide variety of polyform sets? (And should we?)

After addition, a notation for multiplication would be handy. We’ve recently looked at di-triamonds and tri-diamonds. We can call these 2·3-iamonds and 3·2-iamonds respectively. Notice that this multiplication, unlike the addition, doesn’t commute. But it does decompose into addition in the natural way; the 3·2-iamonds are the same as the 2+2+2-iamonds.

In a way, we were already using polyform multiplication to define n-forms in the first place. The pentominoes are essentially the 5·monominoes. In the interest of brevity, we can use symbols for common base monoforms:▲, ■, ⬣, and ◣ for the moniamond, monomino, monohex and monotan respectively. If we are consistent with the above examples, a n■ has subdivisions for the individual cells. That may seem a little weird, but it can be useful; a 2×1 rectangle could be either a domino or a tetratan, and we’d like to be able to know which. I won’t show these subdivisions in my graphics unless it aids with clarity.

We would also like to combine sets together into a larger one. This is multiset addition rather than set union, because we could want to work with multiple copies of the same polyform. I’ll use circled operator symbols for multiset operations, even though that’s a little nonstandard. They’re nicely readable, and the circle will be our mnemonic that we’re doing multiset things. The tetrominoes and pentominoes together would be 4⊕5■. We can read the ‘⊕’ as “and”, so 4⊕5■ is read as “the four and five -ominoes”. Making a set from multiple copies of the same set is the same as scalar multiset multiplication. So five copies of the tetrominoes is 5⊙4■. As before, this is non-commutative left multiplication; the dot is our mnemonic for that. And it decomposes as expected into multiset addition: 5⊙4■ = 4⊕4⊕4⊕4⊕4■. I can’t think of any reason I would ever want to do element-wise multiset multiplication with polyforms, but ⊗ is there if I ever need it.

Now that we have multiset operations and polyform connection operations, we can start to combine them. There are 22 4+1■. I hope to share more problems involving them soon, but one thing I noticed was that with some smaller pieces included I could get an area of 144, and make a square. With my notation system, I can call these 2+1⊕3+1⊕4+1■. Or I could write that as (2⊕3⊕4)+1■. Polyform addition distributes over multiset operations!

(Well, I could have made a square. I’m showing this shape instead because PolySolver wasn’t finding solutions for the square with separated monominoes. Thanks to Bryce Herdt for showing me a technique for getting PolySolver to find solutions with this property.)

Finally, I must address the final question from the start of this post. Is a notation system for polyform sets actually a reasonable thing to develop, given that I am a lone crank and nobody else is likely to use this stuff? And I think that I am finding, for my own explorations with polyforms, that the answer is yes. With algebraic notation, the concepts behind the notation can be expressed with words, and were for a long time. But symbols are easier to mentally manipulate, and formulas that could not fit into working memory as a paragraph can do so as a modest number of symbols. I am already finding it easier to think about polyform sets because I have symbolic notation for them. As I hinted in my fuzzy polyominoes post, I’m working on notation for related concepts, so more posts on polyform notation are sure to follow.

Component Colorings II: Diamonds and Triamonds

~ munizao ~ 2 Comments

Here’s a nice coincidence: the numbers of tri-diamonds and di-triamonds are both 9, which is the right amount to tile a regular hexagon of side length 3. And both sets can! Behold the di-triamonds:

3-coloring the triamonds here isn’t hard. The tiling seems to want to have a bunch of points where four triamonds meet, which disrupts chains of forced colors. The challenge is adding more challenges on top of 3-coloring. I suspect that there is no strict 3-coloring of the triamonds. One possibility is a sort of meta-coloring of the di-triamonds where no two di-triamonds with the same color pair may be adjacent. The above diagram doesn’t qualify because there are blue-red di-triamonds touching each other. Problem #60: Find such a meta-coloring.

The diamonds in the tri-diamonds are even easier to 3-color. Enough so that 3-coloring them so each tri-diamond has all three colors (the equivalent of the poorly thought out problem #58 with the tri-dominoes) was no challenge at all. Perhaps there is something to be done with symmetry. Notice that, ignoring color and the tri-diamond outlines, the diamonds in the figure below have an an axis of reflection symmetry. I wonder if, for some tiling, some form of symmetry on the diamonds is possible where colors are included.

The meta-coloring idea above suggests a way to salvage Problem #58. Instead of a three coloring of the dominoes in a tri-domino tiling, we could look for a 4-coloring of the dominoes where every tri-domino contains 3 of the 4 colors, and there is simultaneously a meta-4-coloring of the tri-dominoes where no two adjacent tri-dominoes are missing the same color.

The Rune Where It Happens

~ munizao ~ 4 Comments

A while back, I was finalizing my drawing files for laser-cut frames for my Cross Products puzzles, which I was intending to sell at Gathering for Gardner 13. I wasn’t happy with just wasting the material in the center of the frames, so I looked for a simple idea that would make use of it. The shape that was being cut out was a rectangle with a 3:4 aspect ratio. I could cut that into 12 squares, and then engrave something on each of them. Now what might there be 12 of?

It will probably not surprise anyone here that my mind went straight to the pentominoes. Tiles with pentominoes could be useful for choosing one at random. (Sure, I already owned a 12-sided die with the pentominoes on it, which I bought from Eric Harshbarger, but with tiles one could select without replacement, or even effectively shuffle an ordering of the entire set.)

The tiles I made for my G4G13 exchange gift didn’t have these fancy swirled colors.

The tiles reminded me of rune stones, with the pentominoes forming a cryptic alphabet. I thought it would be amusing to make sets of them my exchange gift for G4G13, along with a slip that instructed the user on how to use the arcane power of the pentominoes to divine the future. It would be the kind of playful deadpan jab at ungrounded mysticism that Gardner’s alter ego, Dr. Matrix, might have enjoyed making. But to really justify the effort, it couldn’t be just that. I’d need to include some activity using the pieces that would have genuine recreational math interest. Perhaps a puzzle.

What I found was a variation on the common superform framework that incorporated squares with pentomino runes. The basic common superform problem is to find a figure into which any of the polyforms in a set can be placed. Usually, the object is to minimize the area of such a figure, but in this case, the area will be set by each particular challenge using the pieces. We add a couple of restrictions:

  • Each set of five tiles that forms a pentomino must contain the corresponding rune.
  • Each rune must be contained in at least one set of tiles that forms that pentomino.

The above figure was included as an example on the instruction sheet. The challenges provided were to find a valid tile arrangement using nine of the tiles, to do so with all of the tiles but one, and to do so with the complete set. Remarkably, this is possible!

Show Solution

I’ve since looked for other sets of polyforms that are able to make valid rune configurations with a complete set. Here are the tri-diamonds:

And here are the tetrahexes:

Can you find others?

Incidentally, at G4G13, I gave a few pentomino readings to fellow conference goers, one of whom reacted with cheerful amusement, and another with stony skepticism. Honestly, I could not have hoped for anything better.