Solution:
Let two consecutive odd numbers be 2k- 1 and 2k+ 1(k is a positive integer).
Then (2k+ 1)? -(2k- 1)? =(4k? +4k+ 1)-(4k? -4k+ 1)=8k
That is, the so-called mysterious number must be a multiple of 8.
(1) 28 is not a multiple of 8, so it is not a mysterious number. 20 16=8x252 is a multiple of 8, so it is a mysterious number.
(2) It has been proved before, see the previous process.
(3) Let two consecutive even numbers be 2k, 2(k+ 1), then [2(k+ 1)]? -(2k)? =8k+ 1, which is not a multiple of 8, so it is not a mysterious number.
[The word "even" in the title should be "odd"]