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Nov 29, 2017 at 16:21 comment added Gareth McCaughan There's nothing very fancy. To fill in a single-space gap you can look for things that produce squares when added to both (known) neighbours, and in the cases here there was only one possibility. For a two-space gap you look for possible neighbours of the two known cells on either side, and then see which pairs add up to make a square. For longer gaps you need (or at least I needed) to draw a graph (in the vertices-and-edges sense) showing what can be next to what, and then look for paths through it.
Nov 29, 2017 at 16:19 comment added geeky me Any good approach for the answer?
Nov 29, 2017 at 3:18 comment added Jakob Lovern Isn't it obvious? The circle is the one used by the OP.
Nov 29, 2017 at 2:40 vote accept Bernardo Recamán Santos
Nov 29, 2017 at 2:39 comment added rudra From Where did you get the circle
Nov 29, 2017 at 2:27 history answered Gareth McCaughan CC BY-SA 3.0