Catching Geckoes

Suppose I discover that my cat has a taste for the geckoes that live in my back yard. In one week, he deposits the following on my front step:   six gray geckoes, twelve geckoes that had dropped their tails in an effort to escape capture, and fifteen geckoes that he'd chewed on a little.

Only one of the geckoes was gray, chewed on, and tailless; two were gray and tailless but not chewed on; two were gray and chewed on but not tailless. If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of "gray", "tailless", and "chewed on", how many were tailless and chewed on but not gray?

Adapted from Oswego City School District Regents Exam Prep Center

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