If each person can only tell 1 lie, the Duchess can not be lying when she says both “I didn’t kill the Duke”, and “I’m not guilty”. So, her lie must be, “The gardener killed my husband”. That means, the gardener is NOT the killer. The Gardener says, “The Duchess is lying when she says I did it”. We know that is true, and that the Duchess was lying when she said that. So, it must also be true when the gardener says, “I am innocent”. The Gardener’s lie, then, is “The Butler is the murderer”. Now we know the Butler is not the killer either. The Butler says, “I am not the murderer”. That’s true. Now, we can not yet tell if he’s lying about the playing cards in the kitchen with the Chambermaid, or or he’s lying when he says the Duke was killed by the cook. Next, the Chambermaid tells us, “The Gardener did it”. We’ve already cleared the Gardener, so we know that is a lie. Which means, her other two statements must be true. She was telling the truth when she said, “I didn’t do it” and when she said “I wasn’t even here last night”. That means, the Butler was lying when he said playing cards in the kitchen with the Chambermaid. So, he must be telling the truth when he says the Duke was killed by the Cook. What does the Cook say about the situation? “It wasn’t me” sounds like the lie. So, “The Butler knows who did it” must be true. Sure enough, the Butler did say it was the cook. And, the cook must also be telling the truth about having worked for the Duke for 27 years.