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November Puzzle Periodical – The Heist

By Jared Z., Nicole H., and Benjamin E., NSA, Mathematicians

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Submitted by Jared Z., Nicole H., and Benjamin E., NSA, Mathematicians

The chief detective hurried down to the police station after hearing big news: there was a heist at Pi National Bank! The police had brought in seven known gang members seen leaving the scene of the crime. They belonged to the nefarious True/False Gang, so named because each member is either required to always tell the truth or required to always lie, although everyone is capable of engaging in wrongdoing. The chief also knew from his past cases that any crime committed by the gang always included one truth teller.

When the chief showed up, he asked the gang members the following questions:

  • 1) Are you guilty?
  • 2) How many of the seven of you are guilty?
  • 3) How many of the seven of you tell the truth?

Here were their responses:

  • Person 1: Yes; 1; 1
  • Person 2: Yes; 3; 3
  • Person 3: No; 2; 2
  • Person 4: No; 4; 1
  • Person 5: No; 3; 3
  • Person 6: No; 3; 3
  • Person 7: Yes; 2; 2

After looking these answers over, the chief prepared to arrest those responsible.

Which of these seven did the chief arrest?

Click to see the answer!

Solution

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.