Thalia R. Hoyi L., Eric P. and Christopher P. all submitted the same answer, 2. Although it is the correct answer, no one has an acceptable solution. Please see the display case outside the main office for their solutions.
Here is one way to reason this out:
Let’s focus on Hillary first. Hillary either shook four hands, or she did not. If she shook four hands, then it would not be possible for anyone to report shaking 0 hands. Therefore Hillary did not shake four hands. If Hillary did not shake four hands, then one of the remaining four people did. Let’s call this person A, and let S(A) represent the “spouse of A”. Let’s also define the second couple as B and S(B). If A shook four hands, then they must be with Bill, Hillary, B, and S(B) since A could not shake hands with his spouse. This also means S(A) shook no hands. We now need to figure out who shook 1, 2, and 3 hands. If Hillary shook only one hand (that would be A), then neither B nor S(B) could shake 3 hands since the only person remaining to shake hands with them is Bill. So Hillary did not shake hands just once. Without lost of generality, let’s assume B shook hands once, then the only possible placement for 2 and 3 hand shakes would be for S(B) to shake hands with Hillary and Bill and for Hillary to have 2 hand shakes (that would be with S(B) and A).
