## Solution to Puzzle #165: Truth & Lies – 7 of them!

This was a very interesting puzzle, and I got many correct solutions, and I was delighted to see many children come back with solutions this time. I got correct solutions from Mahi Saraf and a joint team of Abhiram, Anisha & Arushi. Also got correct answers from the regulars – Abhinav Jain, Suman Saraf, Vishal Poddar and Pratik Poddar, in addition to Ananda Bhaduri. Well done and thanks.

Answer is that the Chief will arrest 2, 3 and 4. Here is the explanation:

All truth-tellers will give the same response to question 2 as all other truth-tellers, so we can divide suspects up into groups based on how they responded to question 2. Then, we can consider the implications from each group potentially being the truth-telling group. There are 5 possibilities for the truth-telling group: (1), (4), (3, 7), (2, 5, 6), or none of them. Let’s consider each case:

Person 1: Yes; 1; 1
If this is the group of truth-tellers, then we have:
1 truthful confession (#1) + 4 liars claiming innocence (#3, #4, #5, #6) = 5 guilty, not 1 as claimed.

Person 4: No; 4; 1
If this is the group of truth-tellers, then we have:
0 truthful confessions + 3 liars claiming innocence (#3, #5, #6) = 3 guilty, not 4 as claimed.

Person 3: No; 2; 2

Person 7: Yes; 2; 2
If this is the group of truth-tellers, then we have:
1 truthful confession (#7) + 3 liars claiming innocence (#4, #5, #6) = 4 guilty, not 2 as claimed.

Person 2: Yes; 3; 3

Person 5: No; 3; 3

Person 6: No; 3; 3
1 truthful confession (#2) + 2 liars claiming innocence (#3, #4) = 3 guilty, as claimed, so this scenario is logically consistent.

If ALL the suspects are liars, then we have:
0 truthful confessions + 4 liars claiming innocence (#3, #4, #5, #6) = 4 guilty, which was claimed by Person #4, who was assumed to be a liar in this case.

The truth-tellers must be #2, #5, and #6; the chief should arrest #2, #3, and #4.

Hope you all enjoyed the puzzle!

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