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# Exercise in logical thinking

There are 3 suspects, A, B, and C, for a robbery that presumably happened in a shop. We know that the following facts are true:

(1) Each of A, B, C was in the shop on the day of the robbery, and no one else was there on that day.

(2) If A was guilty, then he had exactly one accomplice.

(3) If B is innocent, then so is C.

(4) If exactly two are guilty, then A is one of them.

(5) If C is innocent, then so is B.

Was there a robbery in the first place?

#### Solution Preview

Claim: There cannot possibly have been a robbery in that shop on the day of the supposed robbery.

Proof of Claim:

By Fact (1), we know that A, B, and C were the only people in the shop that day (i.e., a robbery could not have been perpetrated by anyone other than A, B, and/or C), so if there was a robbery in the shop that day, either (i) one of them committed the robbery alone, or (ii) two or more of them committed it together.

Suppose A is guilty of the robbery. By Fact (2), either B or C (but not both) was A's accomplice. Thus if there was a robbery and A is guilty, then either (i) the robbers are A and B (but not C) or (ii) the robbers are A ...

#### Solution Summary

A complete, detailed solution is presented. The different combinations of possible robbers are explored, and the given facts are applied to obtain logical implications of those combinations and to determine whether a robbery actually occurred.

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