# Empty suguru (tectonic) puzzle

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:

1. A block of n squares contains all numbers 1 to n
2. 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).

• For the bonus, there's also a trivial 1x3 answer. Dec 13, 2022 at 2:31

Yes, I can solve it.

+---+---+---+---+---+
| 1 | 4   1 | 3   1 |
+   +---+   +---+   +
| 2   5 | 2 | 4 | 2 |
+   +---+   +   +---+
| 4 | 1 | 3 | 1   5 |
+   +   +   +---+   +
| 3 | 2 | 5 | 4 | 2 |
+---+   +---+   +   +
| 1 | 4   3 | 1 | 3 |
+   +---+---+   +---+
| 3   5   2 | 5   2 |
+---+---+   +   +---+
| 2   1 | 4 | 3 | 1 |
+---+---+---+---+---+

Starting with the 1 in the lower right corner (obvious), then

the 5 in the 4-square L, since anything else blocks the whole thing. Similarly, the 5 in the second row for the same reason with the long L.

The remainder of the 4s can be filled out once you realise that the top of the N piece must be the same as the lower Y piece, and therefore must both be 4. The next 4s are in the L (only one place) and the upper Y (there's only one if the four-square L can have a 4 in it, which also gives the 4 for that). The 4 and 1 from the Z have only one possibility, so can be filled out.

the one by two can only have one solution if the Z is to work, leaving only one place for a 2 in the lower Y. The 3s only have one square available for each region, and so go next.

The remainder becomes 'fill in the only places left for each number'.

For the bonus, we know that every 2×2 square in a Suguru must contain 4 distinct digits as a consequence of the rules. Therefore, the only boards with largest digit 3 or less must be 1×n. Furthermore,

the only board with largest digit 1 is the 1×1, as no two such regions can be adjacent.
The only boards with largest digit 2 are chains of dominos containing exactly one 1×1 - this can be shown by coloring the cells of a 1xn with alternating colors, noting that there must be an excess of one color for the solution to be unique, and then that such an excess can only ever be one.
There are no uniquely solvable boards with largest digit exactly 3. Each 3-cell region can have its 3 placed at either end, and then have the remainder treated as a domino.

As for boards with largest digit 4,

no 2×n boards exist. Every 2×2 square in the grid must contain a 4, and each of these 4's must be in a distinct 4-cell region. If the 2×2's are packed maximally, at most 2 cells are uncovered, and you need at least 6 to have a valid board (for the 1, 2, and 3-cell regions).
The smallest board with an even side that isn't ruled out by the above argument is 6×7, so let's look at odd-sided boards. 3×3 has too few cells, but 3×5 works, tying your example:

Boards with largest digit 5 or higher must contain at least 15 cells to have one region of each size, so our search is done.

• Nice work! Wish I had an extra V to hand out. There are indeed some interesting size constraints. Eg. My search algorithm finds plenty valid puzzles at 7x5, but none at 6x6. Dec 14, 2022 at 21:26
• A 6x6 puzzle must have a block of size 6: with a maximum size of 5 you would need at least 10 blocks (one each of size 1-5, and 5 more for the remaining 21 squares), and no 6x6 can have more than 9 blocks (otherwise two 1s would be adjacent). There is a related discussion at math.stackexchange Mar 3 at 13:51
• Thank you for pointing me to the other forum, predating my question by about 2 years. Looks like we independently went same journey. Mar 3 at 17:25