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Mr. and Mrs. Smith invited four couples to their home. Some guests were friends of Mr. Smith, and some others were friends of Mrs. Smith. When the guests arrived, people who knew each other beforehand shook hands, those who did not know each other just greeted each other. After all this took place, the observant Mr. Smith said "How interesting. If you disregard me, there are no two people present who shook hands the same number of times".

How many times did Mrs. Smith shake hands?

My question is regarding the use of "some" and "guests"

Some is defined as being at least one —used to indicate that a logical proposition is asserted only of a subclass or certain members of the class denoted by the term which it modifies.

But would "guests" imply more than one?

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  • $\begingroup$ No; "some elephants are blue" means the same as "some elephant is blue". $\endgroup$ – mjqxxxx Oct 28 '13 at 23:00
  • $\begingroup$ I suppose Mr. and Mr. Smith knew each other beforehand, so they shook hands with each other? Similarly, any two people who came as a couple shook hands with each other? $\endgroup$ – bof Oct 28 '13 at 23:13
  • $\begingroup$ The story is obscure and confusing, but I think the intended mathematical problem is the following; please correct if I guessed wrong. We have a graph with $10$ vertices $v_0,v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8,v_9$; the vertices $v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8,v_9$ all have different degrees; and the graph contains a perfect matching. What is the degree of the vertex matched with $v_0$. Is that it? (I get $5$.) $\endgroup$ – bof Oct 28 '13 at 23:25
  • $\begingroup$ That is correct. $\endgroup$ – John Thomas Oct 29 '13 at 0:52
  • $\begingroup$ And if you assume couples did not shake hands you get 4. $\endgroup$ – John Thomas Oct 29 '13 at 0:53
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In normal English usage the assertion Some guests were friends of Mr. Smith would in fact imply that more than one of the guests was friends with him. In this context, however, one has to assume that some is intended to be treated as a verbal equivalent of $\exists$ and should therefore be understood as a sloppy paraphrase of at least one.

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