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Make an optimal machine that accepts any letter combination that begins with a consonant and alternates vowel consonant from then on. ("Y" is considered both vowel and consonant)
My solution has 5 nodes.
(EDIT: I made it too complicated, and a 3-node solution is possible. I'm leaving this answer as-is, because it works and is interesting.)
Note that the Consonant and Vowel labels in the picture exclude Y, e.g. Vowel means AEIOU only.
It does not accept an empty string.
It accepts a string of Y's.
It accepts a string starting with any even number of Y's only if it is followed by a consonant, but not if it is followed by a vowel.
It accepts a string starting with any odd number of Y's only if it is followed by a vowel, but not if it is followed by a consonant.
The left two nodes of the square deal with any Y prefixes. The right two nodes are used once a non-Y has been seen. I needed the extra starting node in order to disallow an empty string, since the question says the string must begin with a consonant.