@Meduzia

Solving Japanese Mosaic

Puzzle Analysis https://en.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.

10. The puzzle is solved!

Как решать головоломку "Стены"

Разбор головоломки https://en.grandgames.net/walls/id389670

1. Рассмотрим клетку с числом 5 в левом верхнем углу (выделена красным). К ней примыкает два блока клеток, в которых можно провести линии (выделены синим). Однако для того, чтобы количество проходящих через эту клетку линий было равно 5, три клетки из вертикального блока должны обязательно содержать вертикальные линии (выделены зеленым).

2. Рассуждения, аналогичные пункту 1, применяются еще к 4 клеткам с числами (выделены красным). Блоки клеток, которые необходимо заполнить соответствующими линиями, выделены зеленым.

3. Рассмотрим клетку, выделенную синим. Если она будет содержать вертикальную линию, то количество линий, проходящих через выделенные красным клетки, будет не соответствовать числам в этих клетках. Следовательно, искомая клетка должна содержать горизонтальную линию.

4. Рассмотрим 4 клетки, выделенные красным. Для всех них остался единственный вариант заполнения примыкающих клеток, который позволит провести требуемое число линий через эти клетки.

5. После выполнения пункта 4 появились еще 2 клетки, для которых можно применить аналогичные рассуждения.

6. Появились еще 2 клетки с полностью определенными соседями.

7. Через клетку с числом 7, выделенную красным, уже проходит необходимое количество линий. Следовательно, клетка, выделенная синим, должна содержать горизонтальную линию.

8. Для 3 клеток с числами появились однозначно определенные блоки (аналогично пунктам 1 и 4).

9. Теперь аналогичные рассуждения можно применить еще к 2 клеткам.

10. К двум клеткам, выделенным синим, применяются рассуждения, аналогичные пункту 7. Таким образом, эти клетки должны содержать горизонтальные линии.

11. Для 4 оставшихся клеток с числами остался единственно возможный вариант заполнения примыкающих клеток.

12. Головоломка решена!

Solving the "Wind Rose" Puzzle

Puzzle https://en.grandgames.net/windrose/id213131

1. Let's consider the cell marked with the number 5, highlighted in red. The cells highlighted in yellow are those that can contain lines coming out of this cell. As seen, these are blocks of five, two, and one cell. Since five out of these eight cells must necessarily contain lines coming out of the considered cell, the first two cells of the longest block will contain such lines regardless of whether the blocks of two and one cell are filled.

2. Consider the two cells highlighted in green. All empty cells of the field must be filled, but after the previous step, there is only one cell from which a line can be drawn through the two highlighted cells. This is the cell with the number 2 (highlighted in red in the diagram). Similarly, the cell highlighted in blue is only accessible from the neighboring cell with the number 5.

3. Consider the two cells containing the number 5. Similar to step 1, for one of them, three blocks that can contain corresponding lines contain 5, 1, and 1 cell, meaning the first three cells of the five-cell block must be filled (highlighted in green). For the second considered cell, potential filled blocks contain 3, 2, and 1 cell, meaning the first two cells of the three-cell block and the first cell of the two-cell block must always be filled.

4. The block of three cells highlighted in green is now only accessible from the cell with the number 3 in the far left column (highlighted in red). Since lines cannot be drawn from this cell anymore, the two cells in the upper left and lower left corners (highlighted in blue) can only be accessible from the nearest cells with numbers in the first and last rows.

5. The cell highlighted in green is only accessible from the cell with the number 5. No more lines can be drawn from this cell, so the cell highlighted in blue is only accessible from the cell with the number 4.

6. Similar to the previous step, the cell highlighted in green is only accessible from the cell with the number 5 (highlighted in red). Another line needs to be drawn from this cell, obviously, it can only be the cell highlighted in yellow.

7. Two more lines need to be drawn from the cell with the number 5 highlighted in red. Since there are only two available cells left (highlighted in green), these lines are unequivocally determined.

8. The cell with the number 4 has two available cells left, and two more lines need to be drawn from it, meaning both these cells (highlighted in green) must be filled. Additionally, it is obvious that the cell highlighted in blue is only accessible from the neighboring cell with the number 1.

9. The cell highlighted in green is only accessible from the cell with the number 2 highlighted in red. There are still three single empty cells left, each of which has only one neighbor from which a line can be drawn.

10. The puzzle is solved!