## Party

cinelli
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### Party

My wife and I recently attended a party at which there were four other married couples. Various handshakes took place. No one shook hands with himself (or herself) or with his (or her) spouse, and no one shook hands with the same person more than once.

After all the handshakes were over, I asked each person, including my wife, how many hands he or she had shaken. To my surprise each gave a different answer.

How many hands did my wife shake?

Cinelli

UncleEbenezer
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### Re: Party

Let me guess. She shook four hands, and so did you?

She cannot have shaken 8 hands, because then there's noone to have answered 0. Similarly she cannot have shaken 0 hands.

Let's label the couples: A&a, B&b, Cinelli&MrsC, D&d, E&e.

One person shook 8 hands. That's everyone present except for that person and spouse. That also tells us everyone but the spouse shook at least one hand, so the spouse must be the one with no handshakes. Let's say that was E and e: the argument is unaffected.

So couples A-D shook 1-7 hands. Or 0-6 excluding the handshake with E. The same argument tells us another couple shook 6 and 0 hands excluding the Es, or 1 and 7 altogether. Let's say the Ds.

Iterate and we see the Bs shook 2 and 6 hands, and the As 3 and 5. You and Mrs C shook four hands each. Indeed, the same four hands, being the member of each couple who shook more than four.

jfgw
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### Re: Party

She cannot have shaken 8 hands, because then there's noone to have answered 0.

I do not follow this argument as, while 8 shakes imply that the spouse shook 0 times, 0 shakes does not imply that the spouse shook 8 times. Both Cinelli and one other person may have shaken 0 times.

Julian F. G. W.

UncleEbenezer
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### Re: Party

Er, what?

There are eight people excluding C and Mrs C. If Mrs C had shaken eight hands, that's all of them. But we know that one of the nine people C asked had shaken no hands at all, and had therefore not shaken Mrs C's hand.

The number of handshakes Cinelli himself made is immaterial at this point, though of course we can deduce it. He is excluded from the nine distinct numbers, and must of course duplicate one of them.

Of course there's a deeply suspect premise here, that nine people all remembered and answered correctly when Cinelli asked them.

Gengulphus
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### Re: Party

jfgw wrote:
She cannot have shaken 8 hands, because then there's noone to have answered 0.

I do not follow this argument as, while 8 shakes imply that the spouse shook 0 times, 0 shakes does not imply that the spouse shook 8 times. Both Cinelli and one other person may have shaken 0 times.

There are nine people other than Cinelli himself at the party. He asked all of them and got nine different answers. Unless one exploits a doubtless-unintended loophole in the puzzle (*), the maximum number of hands anyone can have shaken is 8, since they don't shake hands with either themselves or their spouse. So the nine different answers Cinelli got must have been the numbers 0-8, once each.

So someone - call them X - shook 8 hands, which implies (still barring the loophole) that they shook hands with everyone but themselves and their spouse. And someone else - call them Y - shook no hands, which implies that Y didn't shake hands with X, so Y must be one of the two people X didn't shake hands with. Obviously Y isn't the same person as X, so the only possibility is that they are X's spouse. Therefore X and Y must be one of the married couples - and they cannot be Cinelli and Cinelli's spouse, since Cinelli isn't one of the people asked, and X and Y both are.

So the argument isn't really either that 8 shakes implies that the spouse shook 0 times or that 0 shakes implies that the spouse shook 8 times, but that both X and Y exist, both of them must be people Cinelli asked (so neither is Cinelli) and they must be married to each other (so neither is Cinelli's spouse). One can follow that through to conclude that both Cinelli and Cinelli's spouse shook 4 times, and it then does follow both that 8 shakes imply that the spouse shook 0 times and 0 shakes implies that the spouse shook 8 times - but that's a consequence of the conclusion, not a reason for it, at least when following the easiest line of argument.

(*) If one does exploit it, the maximum number of hands someone could have shaken is probably 16, and the puzzle is unsolvable. This possibility could have been avoided by Cinelli phrasing the question he asked people slightly differently...

Gengulphus

UncleEbenezer
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### Re: Party

If we allow your loophole about double-handed shakes, shouldn't we also allow another loophole about non-exclusivity of four other couples (after all, unmarried people go to parties too)? Or even yet another about non-monogamous relationships?

I prefer to keep my loopholes to when a problem is so poorly expressed as to be genuinely ambiguous, or at least hard to parse.

jfgw
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### Re: Party

jfgw wrote:
She cannot have shaken 8 hands, because then there's noone to have answered 0.

I do not follow this argument as, while 8 shakes imply that the spouse shook 0 times, 0 shakes does not imply that the spouse shook 8 times. Both Cinelli and one other person may have shaken 0 times.

Blindingly obvious now I look at it again. If A shook hands with 8 people, a will be the only one who shook hands with none as everyone-else will have shaken hands with A.

Etc.

Julian F. G. W.

cinelli
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### Re: Party

Well, I hope you enjoyed this puzzle. There were no tricks or loopholes. UncleEbenezer answered very quickly and correctly although his solution doesn't inspire confidence as it begins. “Let me guess ...” Also he says that no one answered 0, which is wrong. I asked 9 people and got 9 different answers so those answers must have been 0, 1, 2, …, 8 (as Gengulphus points out). So the 8-person is married to the 0-person, 7 to 1, 6 to 2, and 5 to 3. That leaves two 4-people – me and Mrs Cinelli.

Cinelli

UncleEbenezer
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### Re: Party

cinelli wrote: Also he says that no one answered 0, which is wrong.
Cinelli

Erm, can anyone else parse that in what I wrote? My reply mentions explicitly the person who answered 0:
That also tells us everyone but the spouse shook at least one hand, so the spouse must be the one with no handshakes.

Gengulphus
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### Re: Party

UncleEbenezer wrote:
cinelli wrote: Also he says that no one answered 0, which is wrong.
Cinelli

Erm, can anyone else parse that in what I wrote? My reply mentions explicitly the person who answered 0:
That also tells us everyone but the spouse shook at least one hand, so the spouse must be the one with no handshakes.

Looks to me like it's probably your sentence "She cannot have shaken 8 hands, because then there's noone to have answered 0." It's correct, and a pretty standard usage to present an argument by contradiction: "X cannot be true, because then Y is true", is implicitly saying "X cannot be true, because if X were true then Y would be true and that contradicts the known fact that Y is not true". But I can imagine the part I've emboldened being misread as being equivalent to "because no one answered 0"...

Not saying there's anything very wrong about what you said, just that I can see how it was misread. Though one thing I might suggest is that when one wants to talk about things that aren't true, it's better to use subjunctive wording than indicative to make it harder to mistake it for a statement of fact. In this case, I think a similar misreading would have been quite a lot harder for "She cannot have shaken 8 hands, because then noone would have answered 0."...

But that's the sort of verbal polishing that one does for a published book, and one really wants someone (or preferably more than one someone) to independently look for errors, excessively-verbose wording, ambiguities, easily-misunderstood wording, etc, if one wants to do a really good job. That's what gets done by editors and proofreaders for published books, but is clearly completely over-the-top for TLF posts! So one just has to accept that what one posts will quite often have some poorly-polished bits...

Gengulphus