Meet the Fibnif!
The Fibnif, or fbNF for short, is a piece that moves either forwards and backwards as a Knight, or one square diagonally. Think of it as a narrow Knight.When I say "forwards and backwards as a Knight", this means that a Fibnif can go from e4 to d6 f6 d2 or f2, but *not* from e4 to c3 c5 g3 nor g5 -- those moves are somewhat more sideways than forwards or backwards, you see.
Here is a complete example of its power: from e4 it can go to d6, f6, d2, f2, d3, d5, f5, or f3.
This piece is no stronger (nor any weaker) than a "normal" Knight, which makes it one of the weakest pieces I've ever seen that has the potential to deliver a bare-King mate. Although you can construct a mating position, you cannot force mate; and you can prove this without calculating variations (an interesting logic puzzle).
You will notice that the Fibnif has different advantages and disadvantages than the Knight. On the negative side, it is not so quick at running from side to side of the board; but to look on the bright side, it always has the choice of either staying on the same color square or changing colors.
One reason that the fbNF should be as strong as a Knight despite its narrowness is that from f3 it attacks two different squares in the center (e4 and e5); and so its natural first developing move is a good one. Pieces like the WA and WD and fbNW cannot do this.
And the Wide Knight
The Sniff, or sNF, would be capable of moving from e4 to c5, c3, g5, g3, d3, d5, f5, or f3.I assume it would be noticeably weaker than a Knight because its power is less "forward", and because it would be more difficult to develop.
Other Variations
The fbNW would move from e4 to d6, f6, d2, f2, e5, f4, e3, or d4. I had thought the fbNW might be weaker than the fbNF, but am not so sure now.The fbNA or fbND are colorblind in a strange way; starting the game on the first rank, they can never visit the second, 4th, 6th, or 8th ranks!
fbNfbWfbD would move from e4 to d6, f6, f2, d2, e5, e3, e6, or e2; a very interesting piece.