Gaps - Square Numbers

Rules of Gaps:

1. Fill some cells with stars, so that each row and column contains exactly two stars.
2. The stars cannot touch each other, not even diagonally.
3. Numbers outside the grid indicate the number of empty cells between the stars in the respective row or column.

• Looks like a variation of Star Battle, try adding to the title: Star Battle: Gaps Commented Apr 25, 2021 at 10:24
• Penpa link for interested solvers: git.io/JO7Cq Commented Apr 25, 2021 at 13:20
• @Anonymus25-ReinstateMonica actually, I believe that Gaps grid puzzle is a stand-alone genre yet indeed share some similarity with Star Battle. (I recall Gaps grid puzzle is already existed for years... cmiiw) Commented Apr 25, 2021 at 17:31

The completed grid:

Step-by-step solution:

As a first step, note that for all the columns and rows with a gap of size at least 6, we know that some of the middle cells need to be empty:

This yields a number of other cells which also need to be empty, since they cannot form a gap of the required size:

Now comes the first deduction which is a bit trickier:

Consider the two rows with the 2 and the 5. In the case that the 5-row does not have a star in the first or last column of the grid, then it will block off cells in the 2-row above it in a specific pattern: the first few (0, 1, 2 or 3) cells are not blocked off, then 3 cells are blocked off, then 3 cells are not blocked off again, and finally a few (0, 1, 2 or 3) cells are not blocked off again. For example, if the 5-row has stars at positions 3 and 9, then the pattern of blocked off cells looks as follows (focus on the row with the 2):

Note that in such a situation, it is impossible to complete the 2 row in a valid manner. The only way of avoiding a situation like this, is by filling the 5 row as follows:

This leaves only one way of completing the 2 row. By furthermore marking some cells which cannot contain stars anymore, we get the following situation:

Next, we look at

the consecutive columns with the 1 and 6. These two columns can only be filled in a single manner: The 6-colums will have a star somewhere in the top four cells, which makes it impossible for the stars of the 1-column to go in the top four cells. This 1-column should therefore have a star in row 10, which in turn only leaves one possibility for the 6-column:

Now the 9-row can be completed in only one way, which in turn forces the position of one of the stars in the 4-column:

Now we are almost done!

The remaining 6-column (column 7) may not block off both possibilities for completing the 3-column next to it. This eliminates one of the possibilities of this column:

This leaves only one way of filling the 1-row, which leaves one possibility for completing the 4-column:

Now row 2 and column 11 can be completed, which quickly resolves the rest of the puzzle:

• On-point explanations! Very well done! :D Commented Apr 25, 2021 at 17:31
• Heh, that first major deduction step looks like someone with glasses awkwardly looking to their left
– HTM
Commented Apr 25, 2021 at 18:19