I love Simon Tatham's Puzzles, especially for killing the odd few minutes. One of the available puzzles is Galaxies, based on Tentai Show. I can usually solve these in anywhere from 30 seconds to a few minutes, but I stumbled upon one that has had me scratching my head for (off and on) the better part of an hour or more. Since this one seems unusually tricky, I thought I'd share it...


the puzzle

  • Add line segments along any grid edge to form non-overlapping regions.
  • Each region must contain exactly one circle.
  • Each region must have two-fold rotational symmetry about its circle.
  • The entire grid must be filled with regions.

Play online or with an offline client using Game ID 9x9:jexfctzndtfbmvzpzgdddwf.


I haven't yet given up on this, so I'm not going to look at solutions until I've either cracked it or thrown in the towel. Please be patient.

To be accepted, an answer must show how to solve the puzzle, preferably without bifurcation, and preferably using only the tools in the app. (That is, you can mark that a particular square is associated with a particular circle, but you can't mark a set of possible circles. Beware, however, that I'm not sure it's possible to solve under such limitations.)


2 Answers 2


That was a nice one. Made easier for me by the fact that you said it was hard!

Start with the obvious reflections:

step 1

The third square on the top row can only be associated with one of two centers, so I choose the less obvious first:

step 2

Fill out a few things from there:

step 3

Fourth item in row 3 can actually only be in one place, which limits things a lot:

step 4

And done!


There're a couple of pretty crazy shapes there that make it important not to make any assumptions. Tatham has nice back-tracking functions. In general, don't make any guesses. But if you have to, make sure you know what you guessed so you can go back and eliminate that option if it doesn't work out. You probably know this already.

  • 1
    $\begingroup$ Yes, as far as I can tell, there is no way to solve without bifurcating on either (1, 3), as you did, or (7, 3). Both have only two candidates for "ownership". (BTW, I did finally solve this myself before looking at your answer; one reason it took so long to reply!) $\endgroup$
    – Matthew
    Feb 26 at 17:07
  • 1
    $\begingroup$ I also found that I had to bifurcate here. Followed the logic around the entire grid until I just decided it had to be the solution because I didn't hit and contradictions. An unsatisfying end for something that took me about two hours! :) $\endgroup$
    – Goldstein
    Mar 2 at 18:03

Here is one immediate deduction. Not sure how to continue.

The tile (6,7) has to belong to the circle 1.5 rows above it. enter image description here

This cuts off access to (3,7) for all other circles, so that this circle sits in a 2x 4 area. enter image description here

This allows for some trivial reflections, but not for a full solution without bifurcation.


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