Joseph Bolas wrote:It's been almost a week since this puzzle was first posted and the only person to get this right was Howard, so if Howard would like to explain his method of solving the puzzle and what the answer was please
Joesph's invited me to explain my method, and to give the answer.
I'll go part way, and show how I approached it. Then others might like to follow on using my method, and have another go at solving the puzzle.
Most of the statements made by the seven people are quite complicated, and can certainly be difficult to unravel if they're examined in detail. My way to approach these liars and truth-tellers puzzles is to realise that if you have a chain of any length, the statement:
X1 says that X2 says that X3 says that ... says that Xn says that Y is a fact.
simply boils down to
If Y is true, then there is an even number of liars in the chain, and if Y is false, there is an odd number of liars in the chain.
It says nothing more, and nothing less, until you know a little more about who are liars and who are truth-tellers.
This puzzle contains many such chains of length 3.
My start point was the simplest chain - the first thing said by Archer. Archer is saying that Davies is a liar. This means that of Archer and Davies, one is a liar and one a truth-teller, but we don't know which.
We do now know, however, that the fact given at the end of Brown's statement is false, and so using the reasoning I gave above, either Brown, Hart and Clark are all liars, or else just one of them is a liar and the other two are truth-tellers, because an odd number of them are liars.
We can now continue in this way, gathering more information, and eliminating a lot, before reaching a solution.
There is an alternative approach, which I abhor, and there is a lot of correspondence in various puzzle forums as to the legitimacy of such an approach. That is to use that fact that it is known that there is a unique solution to the problem in order to find that solution. Personally, I regard the task of solving any puzzle to include the requirement to prove, rather than assume, that the solution is unique.
However, to use that alternative approach here, we can think about what would be the situation if Hart was a liar. We would know that Archer didn't kill him. But how could the puzzle then lead to a unique solution? No other statements say anything at all about identifying who is, or who isn't, the murderer. The only possibility of a unique solution, if Hart was a liar, then is for there to be just three liars, Hart, Archer, and the murderer. So this alternative approach could start with us having eliminated all possibilities except:
(a) Hart is a truth teller and Archer is the murderer
(b) Hart, Archer and the murderer are the only three liars.
I'll leave it there, in case others want to have another go. I'm off to the Lake District again Monday to Wednesday next week, and will also be away on Thursday, but I'll try to look in on Wednesday evening to see how many have solved it now.
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so I will have to find it, but I don't think that this is right.