Solving Japanese Mosaic

Puzzle Analysis https://grandgames.net/mosaic/id347302

1. There is a cell with the number 0 highlighted in blue. This means that all adjacent empty cells can be marked as white (highlighted in blue).

2. Let's consider a pair of adjacent cells containing the numbers 2 and 4 (highlighted in red). Both of these cells border the edge of the field. Consequently, each of these cells has six cells analyzed when calculating the number in that cell. Out of these six cells, four are common. Since one cell contains the number 2 and the other contains 4, two cells (highlighted in blue) adjacent only to the cell with the number 2 must be white, and those adjacent only to the cell with the number 4 (highlighted in green) must be black.

NB: At this stage, a pair of cells with numbers 2–4 is considered, but identical conclusions are drawn for pairs 1–3 and 3–5.

3. Let's consider a pair of adjacent cells highlighted in red and apply reasoning similar to point 2.

4. Let's consider a pair of adjacent cells containing the numbers 7 and 4 (highlighted in red). Each of these cells has nine cells analyzed when calculating the number in that cell. Six of these cells are common. Thus, three cells (highlighted in blue) adjacent only to the cell with the number 4 must be white, and those adjacent only to the cell with the number 7 (highlighted in green) must be black.

NB: At this stage, a pair of cells with numbers 7–4 is considered, but identical conclusions are drawn for pairs 8–5 and 6–3.

5. Let's consider a pair of adjacent cells highlighted in red and apply reasoning similar to point 4.

6. Cells with numbers highlighted in brown have appeared on the field. This means that all adjacent empty cells can be marked as black (highlighted in green).

7. As long as there are numbers highlighted in blue or brown on the field, it is necessary to color the adjacent empty cells as shown in points 1 and 6.

8. Let's consider a pair of adjacent cells highlighted in red. For the cell containing the number 4, three adjacent cells are colored black, so one of its six adjacent empty cells must be black, and the rest must be white. For the cell containing the number 6, also three adjacent cells are colored black, meaning three of its adjacent cells must be black, and three must be white. The considered cells have four common empty cells. Obviously, the only way to satisfy all conditions is to color black two cells adjacent only to the cell with the number 6 (highlighted in green), and color white two cells adjacent only to the cell with the number 4 (highlighted in blue).

9. Numbers highlighted in blue and brown appear on the field again. Empty cells should be colored similarly to point 7.