"Fragment numbers" (order unchanged), for example, the "fragment numbers" of 186 are $1,8,6,18$, 36, a total of 5. Find the $n$-digit natural numbers that satisfy the following conditions respectively.
(1) It is a perfect square, and all its "fragment numbers" are perfect squares;
(2) It is a prime number, and all its "fragment numbers" are prime numbers.
Solution (1) All non-fog digits of $N$ are one-digit "fragment numbers" of $N$, which can only be $1,4,9$.
$N$ cannot contain the digit 0, otherwise there might be two-digit "fragment numbers" such as $10,40,90$ that are not perfect squares.
$N$ cannot contain the digit 1, otherwise there might be two-digit "fragment numbers" such as $11,14,19, 41,91$ that are not perfect squares.

Among the natural numbers composed of 4 and 9, $44,94,99$ are not perfect squares, so 49 is the only natural number that meets the criteria.
(2) According to the definition, the usable digits can only be $2,3,5,7$.
It is easy to know that among the two-digit numbers, $23,37,53,73$ are the ones that meet the criteria.
Among the three-digit numbers, if 2 is used, then 2 can only be in the hundreds place (otherwise there would be two-digit "fragment numbers" that are not prime), and 5 cannot be used (otherwise there would be "fragment numbers" such as $25,35,75$ that are composite). The sum of 2, 3, and 7 is 12, which is a multiple of 3, so there are no natural numbers that meet the criteria in this case.

If 5 is used, then 5 can only be in the hundreds place, and the sum of 5, 3, and 7 is 15, which is a multiple of 3. So there are no natural numbers that meet the criteria in this case either.

Only 3 and 7 can be used, but neither can be used consecutively, and 737 is a multiple of 11. After checking, 373 is the one that meets the criteria.
Therefore, there are a total of 5 natural numbers that meet the criteria, which are $23,37,53,73,373$.