# Grid Puzzle - Paint

A student was found making Graffiti in the teacher's office.
He was punished and was asked to repaint the wall of the room.
The teacher gave him a grid to show which part of the wall should be painted white,
and which one should be blue.

Solve this grid as a Nurikabe puzzle.

.....
..5..
.....
.....


There is only one solution.

Prove it.

My solution:

The top-left 2x2 block must have at least one white cell.
The top-right 2x2 block must have at least one white cell.
The third row must have at least 2 white cells to break up the bottom right and bottom left 2x2 blocks.

This leaves one possibility

...... 5 .. . ......

where the . dots are blue cells.

The four white cells are all placed symmetrically in the same positions. There is one in each 2x2 corner block. They can't be placed anywhere else, otherwise more then 5 white cells are needed to connect them all.

• Yea, something like that. I found it too easy, maybe it would be better if harder kind of this puzzle would be posted with some twist. – Jan Ivan Nov 14 '19 at 12:39
• @JanIvan I agree. I have never come across this particular type of puzzle. After looking at the linked page, I solved it mentally within a few seconds. – Weather Vane Nov 14 '19 at 12:40

If any corner is white, there aren't enough white cells to avoid 2x2 blues in at least one other corner. Likewise for the cells adjacent to the corners. The last cell in each 2x2 corner block must then be white.

So we have

BB-BB
BwwwB
Bw-wB
BB-BB

This uses up all 5 white cells, so the remaining "." cells must be blue.

Solution:

BBBBB
BwwwB
BwBwB
BBBBB

In this case W is not allowed to be a 2x2 square. To prevent this, you need to draw B in between. And these Bs need to be connected to the 5.

 W W W W W
W B B B W
W B W B W
W W W W W

• Use ">! your spoiler here" for EVERY line of the spoiler. – Scratch---Cat Nov 14 '19 at 12:48
• ...without any empty lines. – Weather Vane Nov 14 '19 at 12:50

All the X's would need at least 3 white cells to get there, and it is not possible to block all 2x2 blue squares on the right with the one remaining white cell.

 .....
..5..
X....
XX...
Using symmetry, we get the same for the other side, and this gives:
 .....
..5..
.W.W.
.....
To make the white cells connected, and remove the left and right possible 2x2's, we get:
 .....
.W5W.
.W.W.
.....