# Heyacrazy: Empty Space

This is a Heyacrazy puzzle.

Rules of Heyacrazy:

• Shade some cells of the grid.

• Shaded cells cannot be orthogonally adjacent; unshaded cells must be orthogonally connected.

• When the puzzle is solved, you must not be able to draw a line segment that passes through two borders, and does not pass through any shaded cells or grid vertices.

For an example puzzle and its solution, see this question.

Though I generally try to avoid requiring bifurcation or deep hypotheticals, that might be the easiest way to get the last few shaded cells. I would have liked to overclue it to make the ending clean, but when you solve the puzzle it should be clear why I couldn't do that.

• Can the line segment of rule three pass through a vertex? Eg if your set up is a chess board with A1 at the bottom left and you shaded G2 and H3, could you pass a line segment from H2 to G3 and count that as a ligit line segment that passes through one border? Commented Aug 25, 2019 at 22:56
• @DrXorile No, that's stated at the end of the rule: "...and does not pass through any shaded cells or grid vertices."
– Deusovi
Commented Aug 25, 2019 at 22:57
• Oh right. That paragraph was just too long for me! Apologies! Commented Aug 25, 2019 at 23:04
• I'd love to see one of these in 3d! :)
– JMP
Commented Aug 26, 2019 at 6:13

This cell next to the bottom-right corner cannot be empty, because we would not be able to block all these three lines without hemming in unshaded cells in the corner.

The corner square must be filled as well to block both of these lines.

Only one way to block this line.

One of the pink cells must be filled to block this line. This means the circled cell must be empty.

Only one way to block this line now.

Also, only one way to block this line.

One of the pink cells must be filled to block this line, so the circled cells must be empty.

Two of the pink cells need to be filled to block all these three lines. There is only one way to legally fill in two of the three.

Only one way to block this line.

Filling the pink square is the only way to block all these three lines without leaving unshaded cells stranded.

Some cells can be marked as empty because every way to block a nearby line would force them to be empty.

We must fill a diagonal of these pink cells in order to block these lines. Only one way to do this without isolating the unshaded cells in the corner.

Regardless of how we block this line, the circled cell must be empty.

The pink cell cannot be empty, because it would not be possible to block both of these lines without leaving unshaded cells on the left stranded.

Only one way to block these lines now.

Block this one as well. The entire left half is now solved.

No matter how this line is blocked, the circled cell must be empty.

The pink cell here cannot be shaded because that would lead to a contradiction.

Like so. No possible way to block both lines.

Same story with the pink cell here. Shading this quickly leads to dead end.

Like so. No way to block both lines without cutting off the unshaded cells on the right.

From here, this line can only be blocked one way.

This cell must be empty regardless of how this line is blocked.

The pink cells cannot both be shaded, because that would make it impossible to block the red line. Furthermore, leaving the left-hand pink cell empty would make it impossible to block both blue lines. So the left-hand pink cell must be filled.

From here, only one way to block this line. This forces some cells near the corner to be empty.

Somehow I was blind to this final step and thought there were two legal possibilities to finish off the grid. (In my defence, it was very late at night.) Thanks to Dr Xorile for pointing out the actual final step.

Final position.

• Must be the left one (G5) or there will be a straight line from A4 to H6 Commented Aug 25, 2019 at 23:52