# Suguru puzzles with no given clues

A Suguru puzzle consists of a bunch of polynominoes fit together in a rectangular shape. The goal of the puzzle is to fill all the cells in all the polynominoes with the numbers $$1$$ to the amount of cells the polynomino has. A pentomino requires you to fill in $$1$$, $$2$$, $$3$$, $$4$$, and $$5$$, for example. A number is not allowed to touch the same number, not diagonally either!

Recently I was wondering if it is possible to construct a Suguru puzzle without any given clues (no given numbers) with exactly one solution. I was able to find some examples so I started wondering for which sizes this would work. So my question is:

For which $$a$$, $$b$$ is it possible to construct an $$a\times b$$ Suguru puzzle with no given clues and exactly one solution?

I was able to prove some cases, but not all cases.

• I was able to prove that if $$a$$ and $$b$$ are both odd, than there exists a possible arrangement with no given clues and exactly one solution unless both $$a$$ and $$b$$ are equal to $$3$$, in which case it is impossible to construct one.

• If one side is even and the other one is odd, I was able to proof that there exists an arrangement if the even side is at least $$6$$ and the odd side at least $$3$$. If the odd side is equal to $$1$$ than no arrangement with the other side being even exists.

• If the even side is equal to either $$2$$ or $$4$$, I am not sure whether it will ever be possible to construct a puzzle with no given clues and exactly one solution. So far I have not been able to find any working examples, so I believe there aren't any at all.

• With the help of the 8x8 configuration which was provided by Richard Tobin, I was also able to prove that an arrangement exists if $$a$$ and $$b$$ are both even and at least 8, however, I still haven't figured out what happens if $$a$$ and $$b$$ are both even, but not both at least 8.

This brings all the unsolved cases down to three categories. $$2\times n$$ for all $$n$$, $$4\times n$$ for all $$n$$ and $$6\times n$$ for all even $$n$$

Does anyone know how I could prove all cases, or just some of the cases I have not yet already solved?

• Please define a Suguru puzzle. Nov 29 '20 at 17:39
• A Suguru puzzle consists of a bunch of polynominoes fit together in a rectangular shape. The goal of the puzzle is to fill all the cells in all the polynominoes with the numbers 1 to the amount of cells the polynomino has. A pentomino requires you to fill in 1, 2, 3, 4 and 5 for example. A number is not allowed to touch the same number, not diagonally either! I hope this helps! Nov 29 '20 at 17:57
• You might be more likely to get an answer on puzzling.stackexchange.com Nov 30 '20 at 3:36

Here is an 8x8 suguru with no clues and one solution (so your conjecture that there are none with both sides even is wrong):

+---+---+---+---+---+---+---+---+
|           |           |       |
|---+---+---+---+---+   +---+---|
|               |   |       |   |
|---+---+   +---+   +---+---+   |
|   |   |   |       |       |   |
|   +   +---+   +   +   +---+   |
|   |       |       |   |       |
|---+   +   +---+---+   +   +---|
|   |       |   |       |   |   |
|   +---+---+   +---+---+---+   |
|   |           |   |           |
|   +---+   +---+   +---+---+   |
|   |   |   |       |       |   |
|---+---+---+---+   +---+   +---|
|               |       |       |
+---+---+---+---+---+---+---+---+

• Can I ask how you were able to find this one? May 15 at 13:03
• Using your 8x8 configuration I was able to prove that if a and b are both even and at least 8, it is also always possible to make a suguru with no given clues. So thank you for this! May 15 at 15:59
• I wrote a program to generate puzzles and see if they were uniquely solvable. I have not been able to find a 6x6 - if there is one, it must contain a polyomino of size 6. Sep 13 at 23:53