I used warm water on a wet rag. I put some Dawn dish soap on it and poured rubbing alcohol on the rag too. I rubbed the Silly Putty; (which was hardened) out in a few minutes. It worked great! The dog didn't complain a bit either.
Source: I used information from this site; and modified the information, making a new "recipe" to see what would happen. It worked great with no harsh toxic chemicals.
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My son has put silly putty in his hair, I don't know what is best to break it down to get it out, any ideas?