黑白点数独
2
1
4
5
7
4
9
8
2
1
7
4
2
7
3
8
4
6
3
1
4
9
8
1
2
3
7
3
5
6
7
4
2
7
6
4
9
1
白点 = 连续数字；黑点 = 一个数字是另一个的两倍
什么是黑白点数独？
Kropki Sudoku在相邻cells pairs之间放置dots。White dot表示两个digits恰好相差1（consecutive）。Black dot表示一个digit恰好是另一个的两倍（例如，2和4，或3和6）。没有dot出现的地方，neither condition applies。一些Kropki puzzles使用'negative constraint'，其中absence of a dot guarantees neither condition holds，从而提供更多information。
一览
| 约束类型 | 单元格关系 |
| 典型给定数 | 18–24 |
| 难度评级 | ★★★☆☆ 3/5 |
| 平均解题时间 — 简单 | 8 分钟 |
| 平均解题时间 — 中等 | 18 分钟 |
| 平均解题时间 — 困难 | 35 分钟 |
| 平均解题时间 — 专家 | 62 分钟 |
如何解黑白点数独
| 技巧 | 它的作用 | 级别 |
|---|---|---|
| White Dot Pairs | White dots mark consecutive digit pairs: (1,2), (2,3), …, (8,9). Cross-reference with row and column constraints. | 高级 |
| Black Dot Pairs | Black dots mark double-relationship pairs: (1,2), (2,4), (3,6), (4,8). Only these four combinations are valid. | 高级 |
| Negative Constraint | No dot between two cells means they are neither consecutive nor in a 1:2 ratio — a powerful elimination on every unmarked adjacent pair. | 高级 |
| Dot Chain Analysis | A sequence of connected dots forms a digit-value chain. Enumerate valid digit sequences for the entire chain simultaneously. | 高级 |
| 1-2-4 and 2-4-8 Chains | A black dot adjacent to a white dot often forces the common 1→2→4 or 2→4→8 chain. | 高级 |
平均解题时间
简单
8 分钟
中等
18 分钟
困难
35 分钟
专家
62 分钟
常见问题
什么是Kropki数独？
Kropki Sudoku is a variant where dots appear between adjacent cells. A black dot means one of the two cells holds exactly double the value of the other. A white dot means the two cells hold consecutive digits (differing by 1). Standard Sudoku rules apply, and crucially: if there is no dot between two adjacent cells, they are guaranteed NOT to be consecutive and NOT to have a double relationship.
What does a black dot mean?
A black dot between two cells means one digit is exactly twice the other. For example: 1&2, 2&4, 3&6, 4&8. Note that 1&2 satisfies both black (1×2=2) and white (|2-1|=1) conditions — in Kropki Sudoku, a black dot is used when only the doubling relationship holds, not the consecutive one, unless specified otherwise.
What does a white dot mean?
A white dot between two cells means the digits differ by exactly 1: they are consecutive. Valid pairs are 1&2, 2&3, 3&4, 4&5, 5&6, 6&7, 7&8, 8&9. The order doesn't matter — either cell can hold the higher or lower digit.
How does the no-dot constraint work?
In standard Kropki Sudoku, every possible black dot and white dot relationship is marked. This means that if no dot appears between two adjacent cells, you can be certain those cells are neither consecutive nor in a 1:2 ratio. This negative information is just as powerful as the positive dot clues.
Is Kropki Sudoku harder than regular Sudoku?
Easy Kropki puzzles are accessible and make excellent teaching tools for the variant. Hard and expert puzzles are significantly more challenging than classic Sudoku because you must simultaneously track dot relationships and the negative no-dot constraint across every pair of adjacent cells.
How long does Kropki Sudoku take to solve?
Easy puzzles take 8–18 minutes. Medium puzzles run 18–35 minutes. Hard puzzles average 35–65 minutes and expert puzzles often exceed 90 minutes. Mastering the no-dot constraint is the key to faster solve times at all levels.