Exercise: recognizing Hawaiian plaintext
Let's assume that the incorrectly decrypted messages look like random sequences of letters.  That assumption might not be exactly correct, but for most modern ciphers it should be pretty close.  In any case, the mention of "random noise" in the assignment suggests that this is indeed the assumption that we're supposed to make.

According to the given rules of Hawaiian morphology, no word may contain two consonants in a row or end in a consonant.  This implies that the correct plaintext can never contain two consonants in a row.

In an incorrect plaintext, the probability of each letter being a consonant is 7 out of 12.  From this, you should be able to calculate (or at least approximate) the probability of two consonants appearing adjacent to each other in the incorrect plaintext as a function of its length.

Given this probability, you can then calculate the probability of all the $2^{50}-1$ incorrect plaintexts having at least one pair of adjacent consonants.  Then pick some reasonably high target probability (say, 0.9999) and calculate the minimum plaintext length needed to meet it.  (Of course, you really should give the full formula for the plaintext length needed to reject all the incorrect plaintexts with a given probability, so that the person reading your answer can choose a higher or lower target probability if they want.)