Circles and Squares - A Grid Puzzle

Question

This relatively new grid puzzle genre is invented by Anonymus25.

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Rules of Circles and Squares:

1. Shade some cells so that all shaded cells form one orthogonally connected area and each orthogonally connected area of unshaded cells is in the shape of a square.
2. Cells with black circles must be shaded, and cells with white circles must not be shaded.
3. No 2x2 region may be entirely shaded.

Answer 1

To get started, r1c3 must be a 1x1 square. Any square bigger than that would either have a white circle next to it (making it a non-square) or the black circle inside it (forbidden by rules).

After connecting that square's known-shaded neighbours out, r2c2 will obviously be a 1x1 too.

Now, the shaded cells in r1c5 must be able to connect out, so r2c4 must be a 1x1. To prevent a shaded 2x2 on the top two rows then, at least one of r1c6 and r2c6 must be unshaded. However, it cannot be r1c6, because there is no possible unshaded square size for that to happen.

After that deduction, we can easily finish the top 3 rows.

Then, on row 4, one of the two rightmost cells must be unshaded to prevent a square. Whichever it is, the only way to make an unshaded square is to make a 4x4. Then we need to connect the shaded cells below it out, and avoid any 2x2 shaded areas at the bottom.

Next, the white circle at r4c4 must be a 1x1: if it weren't it would need to be a 4x4, which would cut off the connectivity of the shaded region. To avoid a shaded 2x2, r4c2 must be unshaded. Continuing by surrounding known squares with shaded cells and checking all 2x2 squares with 3 shaded cells in them, we get this far

At this point, only two deductions remain:

• To maintain connectivity of the shaded region, r6c4 must be a 1x1 square
• To prevent a shaded 2x2, at least one of the bottom two cells in column 3 must be unshaded. Whichever it is, the only way to make an unshaded square is a 3x3.

And we are done:

Answer 2

Completed grid

Dark grey is shaded, light grey is unshaded.

Solution explanation

I'll explain the first few steps and then provide a gif to illustrate the rest.

• Obviously, the first step is to mark the black-circle cell as shaded and all the white-circle cells as unshaded.

• There's a T-shape of unshaded cells in the top left; consider the single cell in the middle of that T. If it's unshaded, then we must have a big square of unshaded cells: at least 3x3, then at least 4x4 because of the next white circle down, then at least 5x5 because of the next one, contradiction because of the black circle. So that centre cell in the T is shaded, and then we can shade in lots of other cells in the top left because we know that shaded cells are all orthogonally connected and blocks of unshaded cells must form squares.

• Now we have this. Look at the row just below the top left completed block. If the second cell in that row is shaded, then the ones on either side of it must be unshaded (by the 2x2 rule), but then the square rule gives a contradiction. So the second cell is unshaded, and we have a 2x1 block of unshaded cells which must extend left or right to a square. If it extends right to a 2x2 square, then actually it must be a 4x4 square and we get a contradiction in the bottom left.

• Now we're at the stage where I can show the rest by just a gif: