Spontaneous Saturday 4/11/15: Sym-a-pix, the puzzle that I underestimated

If you recall on June 14, 2014, when I wrote my first Conceptis article (http://wp.me/p4GI1b-u), I stated the following:

Looking back, that was a shallow judgement on my part, and I regret it wholeheartedly. It dawned on me last week when I decided, “Hey, why don’t I do every puzzle instead of just ignoring those three in particular?” I mean, Dot-a-pix is still child’s play and Maze-a-pix is average at best, but when I did last week’s Sym-a-pix puzzle, I found out the hard way that there is more to Sym-a-pix that meets the eye.

Screen Shot 2015-04-11 at 12.48.51 AM

You see, when I first looked at a board like this, I thought, “Well, just draw the smallest possible box around all the colored circles, simple as that.” For a while, I was going along with that logic and thought, “Eh, this is getting boring.” That was where the shallow judgement came from. Then, that one fateful week, I tried to apply my shallow-minded way of thinking to that week’s weekly puzzle (not the one shown above) and, surely enough, found out to my dismay that I had 64 wrong walls. Going through the many errors that I had committed, I was taught the true nature of Sym-a-pix. That is the focal point of discussion on this Spontaneous Saturday.

 

For simplicity, the example will not be the one above, but instead a more basic one.

Screen Shot 2015-04-11 at 1.09.43 AM

The point of Sym-a-pix is not to draw the possible box around all the circles, but to create walls on the grid such that:

  • All enclosed areas have one and only one circle within them.
  • All enclosed areas have rotational symmetry, meaning you could theoretically rotate them 180º and still see the same shape.

Things to note when solving Sym-a-pix puzzles:

  1. Start with corners and edges. If a circle is taking up one of the corner squares, it is actually okay to apply my “shallow judgement algorithm” (SJA, as I’ll call it), and surround the circle with the smallest possible box (1×1 for circles in the center of a square, 2×1 for circles on an edge, and 2×2 for circles on a corner). If a circle is taking up one of the edge squares, create walls that are both parallel and as close as possible to the edge in question. Applying this step to the example should result in the following:
    Screen Shot 2015-04-11 at 1.24.09 AM
  2. Divide adjacent circles. In this case, adjacency refers to the concept that the smallest possible boundary boxes of two circles will overlap at at least one edge. This edge or set of edges should be highlighted, as it is the means by which the “adjacent” circles will be divided. Applying this step to the example should result in the following:
    Screen Shot 2015-04-11 at 3.36.06 AM
  3. Consider the symmetry. This is the sole most important rule of Sym-a-pix, and where the “sym” part of the name comes from. As mentioned before, the boundary surrounding a circle should be rotationally symmetrical, meaning that it can be turned 180º without its appearance changing. Similarly, if you take any borders that surround a circle and rotate them 180º, using the circle as an axis, the resulting image should depict borders that are to be highlighted if they are not already. This is a complex process, so here are some elementary derivatives:
    Circle in center: For any boundary that is a part of the SJA box (the box generated by my Shallow Judgement Algorithm, mentioned above) of the circle, the line on the other side must also be highlighted. Applied to the example:
    Screen Shot 2015-04-11 at 3.37.15 AM
    Circle on edge: This is slightly (but only slightly) more complicated. The dimension of length 1 works the same as any dimension of the SJA of a circle in the center, but the dimension of length 2 is a different case. If a border lies on that dimension, not only must its “reflection,” so to speak, be on the other side in terms of its own orientation, but in terms of its perpendicular orientation as well. This goes back to the previous definition of the rule; whether it’s on the left or the right (relative to the circle), if you rotate the boundary 180º, it should still be on whichever of the left or the right it was on before. It’s really difficult to explain this any further, so I’ll just fill in the example:
    Screen Shot 2015-04-11 at 3.50.55 AM
    Circle on corner: Similar to circle on edge, only both dimensions are of length 2. In the example (not much change here):
    Screen Shot 2015-04-11 at 3.51.51 AM
  4. Consider which tiles belong to which circles. The easiest way to think about this step is by looking for the “dead ends” of the puzzle—for instance, row 3 column 9 and row 10 column 1.
    Screen Shot 2015-04-11 at 3.53.28 AM
    Anything not covered by any of the first two rules or the first check of rule 3 will probably fall under this rule. On some occasions, this one included, you can keep doing rule 4 checks until you’ve reached an impasse, like I will for this example…never mind. It turns out simple dead end analysis is really effective for low-difficulty puzzles like this one (mind you, this one is low difficulty relative to Sym-a-pix, not puzzles in general). However, sometimes, you will have to refer back to rule 3, and from there back to rule 4 again, in order to complete the puzzle.

The sample, if completed successfully, should look like http://i.gyazo.com/671ef2c147982cd4fff22c4fcbf62f06.png

 

Nowi Wins Once again, I have been, in a manner of speaking, slapped in the face and told to check myself before I wreck myself. (It may not be “once again” in terms of Vouiv-review, but overall I’ve made dumb errors like this too many times to count.) In closing, I take back what I said before about Sym-a-pix; it requires plenty more thinking than I gave it credit for, even more so than I can explain (at the moment), although this does not mean it’s the most difficult form of puzzle out there (personally, I find Pic-a-pix more difficult). That said, I hope this post has been of use somehow, and y’all have fun solving puzzles!

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