Tag: grid puzzle

Tilings and Reconstructions

~ munizao ~ Leave a comment

There are nine proper 4-WFD-ominoes:

Recall that a proper WFD-omino is one where every spanning tree of W, F or D connections between the cells includes at least one of each. Recall that W, F, and D connections correspond to the moves of the Wazir, Ferz, and Dabbaba in historical chess variants. And recall that these moves are, respectively, a one space orthogonal move, a one space diagonal move, and a two space orthogonal jump respectively. (It’s fine if you don’t recall these things. I’ll recall them for you. Or you can read this post, and then recall them.) The full set is the right size to tile a 6×6 square. And indeed, it can!

And yet, you might find that graphic just a little unsatisfying. The solver that I use for polyomino tilings displays results with all of the pieces in the same color. This is fine for standard polyominoes, but for disconnected (or differently connected) polyominoes, it’s hard to see where all of the loose monominoes fit. But that just turns reconstructing the tiling into a logic puzzle. Indeed, there are entry points where a domino is constrained to be in a particular proper 4-WFD-omino, which removes certain monominoes from consideration and forces other dominoes to belong with their monominoes, and like a sudoku, the chain of logic eventually forces a complete solution. Try it!

But a 6×6 puzzle may feel a little small if you’re used to Sudoku or other puzzles in the Nikoli mold. So let’s try out the proper 4-WFA-ominoes. (Recall that the A (Alfil) is a two space diagonal jump.) There are 15 of these:

They have area 60, so they can tile a 8×8 square with corners removed, just like the 12 pentominoes.

Reconstructing the tiling here is a substantially harder puzzle than the previous one. The monominoes belonging with particular dominoes can be farther flung, which makes it harder to constrain which bits must be connected. I had to try out a large number of tilings before I found one where I could make some headway in the reconstruction puzzle. But rest assured that I did indeed solve this one.

Where to go from here? There are 21 proper 4-WFN-ominoes, (recall that N is for knight) but they have odd checkerboard parity, which interferes with trying to find a nice thing for them to tile. Another possibility would be to combine the proper 4-WFDs and 4-WFAs in a single puzzle. Either way would make it even more difficult to constrain which monominoes go with the dominoes. There may be solvable reconstruction puzzles in the space of tilings, but the strategy of picking tilings at random to find solvable ones will probably break down. I suspect that there are more good puzzles in this vein waiting to be discovered, but they may require more creativity in coming up with new piece sets that will work, or in puzzle design or discovery techniques.

Pentomino Painting Robots

~ munizao ~ Leave a comment

In the diagram below, each row (reading from left to right) and column (reading from top to bottom) gives instructions for painting one of the 12 pentominoes:

Sometimes an idea languishes in one of my notebooks for a few years before I can come up with the right iteration of it. My original idea here was to use a 4×4 grid. That would give me 8 pentominoes, (perhaps 10 using diagonals) but elegance surely requires all 12 to be present.

A combination of circumstances led me back to this problem. Some friends of mine have a tradition of playing RoboRally on New Year’s Day every year. This is a board game where you use cards with arrows on them to instruct your robot to move around a grid of squares. Also, in returning to the magic 45-omino problem, I was considering grids that could be used in sparse magic squares.

It might be possible to make an interesting grid puzzle, along the lines of sudoku, using this kind of grid as a basis. Most of the grid would start empty, except for a few squares in which arrows would be given at the start. Then the solver would fill in the rest of the grid by logical deduction so that the horizontal and vertical lines contain instructions for paining all of the pentominoes as above. Since the grid would have significantly fewer squares than a sudoku, this puzzle might be quicker to solve, but that doesn’t mean that it would necessarily be less interesting.

HEY, A BOGUS 9

~ munizao ~ 4 Comments

Dave Harper’s Polyomino Patterns page has some good stuff, looking at patterns of connections between squares in polyominoes, and processes of “integration” and “differentiation” on polyominoes. He enumerates all the possible patterns of connections of the cells in a 2×3 rectangular hexomino that make a connected whole. (There are ten.) These could also be considered as polysticks that touch all six vertices in a 2×3 lattice. The polysticks on a 2×3 lattice are precisely those that can be represented on a 7-segment LED, hence my presentation of them below:

It might be nice to have some puzzle using these. So here is one! Fill in segments on the figure below so that each of the ten patterns above is represented on a 7-segment LED shaped subsection of the figure.

Reflections and rotations of the patterns are considered equivalent. There are 13 7-segment LED shaped subsections of the figure, so three of them either can have other patterns, or can be duplicates.

Are there any other puzzle grids that would make for a puzzle using these patterns that is as good or better than this one?

Wanderings on a Six-Sided Die

~ munizao ~ 5 Comments

Here’s a little doodle on a grid based on a standard six-sided die:

I started by deciding that the pip positions should all connect North to south and East to West. It followed logically that I could have a puzzle where the solver could choose one of two possibilities for each empty cell: connecting North to West and South to East, or North to East and South to West. Because there are 33 non-pip squares, there would therefore be 233=8,589,934,592 ways to fill the grid. The lines on the outside of the grid show how the squares would connect when folded into a cube.

As an exercise, I found a way to make a single circuit, which is shown above. While that turned out to be about the difficulty of puzzle I can handle in something I am solving by hand, I’m sure there are more interesting specimens to be found.

Unfortunately, because the number of pips is odd, it’s impossible to have two circuits where each go through all of the pips. The circuits would have to cross each other an even number of times. But we could have one of the circuits cross itself once, and then have both circuits go through the remaining 20 pips. (Let’s call that problem number, oh what are we on, #12. By the way, the problem numbers are so that I can keep track of solvers of numbered problems and give them the fame they deserve. Nobody has solved any yet. You can be the first!)

Another possibility would be to use three circuits, each crossing itself once, and each visiting 12 pips in addition to the self-crossing. That’s #13.

I like big, wide ranging circuits here, so a constraint I like is to have circuits that visit all 6 sides. So bonus points on the preceding problem for having all three circuits do that. And that suggests #14: Maximize the number of circuits in a solution where all circuits present visit all 6 sides.

I thought early on that I could take this in the direction of a knot theoretical puzzle, but then one would have to keep track of which thread went under which in the crossings, which seemed like an unnecessary complication. I also think it would be interesting to make a multi-state maze (See Robert Abbott’s site for some good examples) using this template, but I haven’t yet had any good ideas for how that would work. If you have a good idea for a variation on this puzzle, I’d love to hear it. (This is of course true for all of my puzzles.)

For your solving convenience, I have an empty grid image here, and the Inkscape SVG file I used to produce it and the image above is here.