$\mathbb N$ queen

Question

A 8-queen problem is to find a function s.t.

1. $\ f : \{1, 2, 3, 4,5, 6, 7, 8\} \rightarrow \{1, 2, 3, 4,5, 6, 7, 8\}$
2. $\ f$ bijective
3. $\ f-\mathrm{Id}, \ f+\mathrm{Id}$ injective

If I modify the first limitation to $f : \mathbb N \rightarrow \mathbb N$, the function seems to exist(can be proved by induction?).

I was wondering if the set of these functions is

1. infinite (same by induction)?
2. countable or uncountable ?

Answer 1

This is answered in Invulnerable Queens on an Infinite Chessboard by D. S. CLARK, O. SHISHA. Theorem 2 of the paper says that the set of these functions has cardinality that of the continuum.

For existence of such queen configurations, Figure 1 of the paper shows a particularly simple one. It is reminiscent of the "rep-tiles".