If the vowel-consonant theory is correct (and subsequent additions to the list have not yet contradicted it), it follows that we have a complete knowledge of 1, 2 and 3-letter koans:
1-letter: All black
2-letter: VV white, others black
3-letter: VVV, VVC white, others black
However, for 4 letters and beyond, a koan can be white without having two consecutive vowels.