# A Hidato, but with no given numbers?

I can be a bit evil sometimes. Today, I am going to give you a Hidato with no numbers! It gets worse: This puzzle is on a Mobius Strip Y However, I will give hints as to what the numbers are, so this isn't going to be plain impossible. Also, to make this a bit easier, this is only a 3x3 Hidato. Here is the puzzle:

A B C
D E F
G H I

Hints:

1. $$B,C,G,E$$ are primes
2. $$A,H,I$$ are square numbers
3. $$C$$ does not connect with $$I$$
4. $$A+1\lt C\lt H\lt G\lt E$$
5. $$A$$ does not connect with $$D$$
6. There is only 1 connection that is made using a diagonal

The goal of Hidato is to fill the grid with a series of consecutive numbers adjacent to each other orthogonally or diagonally. The entire grid is required to be filled in.

Here's an example. Say we are on an 8x8 grid at R1C1. If we walk up, we go to R8C1. However, this is not a torus, so going straight from R1C1 to R8C8 is illegal, however, since this is Hidato, we can do $$R1C1\to R\color{red}2C8$$ due to our ability to move diagonally.

First off,

The four primes must be 2, 3, 5, 7 and the 3 squares must be 1, 4, 9.

A + 1 is at least 2 (since A can't be less than 1)
from (4), C, G, and E are all primes that are greater than 2 (A + 1 is at least 2)
thus C, G, and E are 3, 5, and 7 in that order, and that leave B as 2.

 -------------
|   | 2 | 3 |
-------------
|   | 7 |   |
-------------
| 5 |   |   |
-------------

Next,

The two numbers that are neither prime nor square are 6 and 8, and those must correspond to D and F (in some order).
If F is 6, we have a diagonal connection between 5 and 6, and there is no way to complete the grid without another diagonal, so F must be 8, and D is 6.

 -------------
|   | 2 | 3 |
-------------
| 6 | 7 | 8 |
-------------
| 5 |   |   |
-------------

Finally,

It doesn't take too much more figuring to place the 3 square numbers.
C is greater than A + 1, so A is obviously 1.
C and I aren't connected, so the only other place 4 can go is at H.
That leaves I for 9.

 -------------
| 1 | 2 | 3 |
-------------
| 6 | 7 | 8 |
-------------
| 5 | 4 | 9 |
-------------