Suguru (also knows as Number Blocks or Tectonic), is a popular type of grid-deduction puzzle:
In a rectangular grid of squares, the squares are grouped into blocks. As with a sudoku, some squares already have a number. The goal is to fill the remaining squares so that:
- A block of n squares contains all numbers 1 to n
- Adjacent squares (horizontally, vertically or diagonally) can not contain the same number
There's plenty of Suguru/Tectonic puzzles on paper and online (example)
So far the introduction...
I have always wondered if there exists Suguru puzzles without any initial numbers, that yet have exactly one solution.
It's relatively easy to show that such an empty Suguru puzzle would need blocks of all sizes (eg. if the biggest block has 5 squares, then you also need blocks consisting of 1,2,3 and 4 squares).
After a lot of manual attempts (which ended in puzzles with either no solution or multiple solutions), I eventually wrote a computer program to solve suguru and exhaustively search for empty ones. It seems that there aren't that many...
Here is an example of an empty suguru puzzle for you to warm up:
+---+---+---+---+---+
| | |
+---+---+---+---+---+
| | |
+ +---+---+ +---+
| | | |
+---+---+---+---+---+
Solution
+---+---+---+---+---+ | 1 2 3 | 2 1 | +---+---+---+---+---+ | 4 | 5 1 4 3 | + +---+---+ +---+ | 1 2 3 | 2 | 1 | +---+---+---+---+---+
(Notice how each block has all different numbers, counting up from 1, and how neighbouring grids never have the same number)
And here's one that is a lot more challenging:
+---+---+---+---+---+
| | | |
+ +---+ +---+ +
| | | | |
+ +---+ + +---+
| | | | |
+ + + +---+ +
| | | | | |
+---+ +---+ + +
| | | | |
+ +---+---+ +---+
| | |
+---+---+ + +---+
| | | | |
+---+---+---+---+---+
Can you solve the larger puzzle?
Bonus question: Do empty Suguru puzzles exist that are smaller than 3 by 5 and that have exactly one solution? (apart from the trivial 1 by 1, of course).