I am relatively new to Sudoku and I tried recently to construct my first one. I intend to send it to a friend whose initials are "BH", which is why I confined myself to the following form (0: no clue, x: clue):

000 000 000

xxx 00x 00x

x00 x0x 00x

x00 x0x 00x

xxx 00x xxx

x00 x0x 00x

x00 x0x 00x

xxx 00x 00x

000 000 000

I don't know if it is visible, but this pattern yields the letters "BH".

After several failed attempts I ended up with the following:

000 000 000

842 003 005

900 104 007

100 209 008

298 006 753

300 708 001

400 805 009

529 001 004

000 000 000

It has two solutions and I intend to fix this ambiguity by requiring verbally that somewhere in the solution the "lucky number" 13 has to appear (that is, a 1 followed in the same row by a 3 in the column to the right).

It is not very difficult to solve, but, I believe, also not especially easy. X-wing can give you a head start, but it is not needed.

Although for the purpose of sending this to my friend I am quite happy with the result, I became curious, so here are my questions:

1.) Is it possible to construct Sudokus with unique solutions with the pattern above? I strongly expect the answer to be yes, but since I did everything by hand eventually I gave up because construction by hand is a bit tedious after some time.

2.) Is it possible to construct the Sudoku in such a way that X-wing is needed for the 4 corners and the 6th column? I conjecture a "yes" but I am less sure here.

3.) Is it possible to make an irreducible Sudoku with the pattern above? (by "irreducible" I mean that you cannot remove any of the clues and still get a unique result).

4.) What is the "most difficult" Sudoku that can be constructed with the pattern above? Now, this question will not have a unique answer, but what I mean is, how complicated can one rearrange the clues without destroying the BH pattern? Apart from the X-wing technique that might be needed for the corners there is surely a lot that can be done with forcing chains in the 7th and 8th column.

Please let me know if you have answers to these questions, I appreciate any input.