Monday, November 28, 2005

How Many Hands Did She Shake?

Eary stage VC's must make investments decisions with very little information. No wonder that barely half our investments turn a profit.

Here's a question I have often asked aspiring VC's to see if they persist in the face of seemingly incomplete information...

My wife and I went to a dinner party with four other couples. At the beginning of the party, some people shook hands. (Obviously, no one shook his or her own hand or spouse's hand, and no one shook hands with the same person twice.) During the party I surveyed all the other people as to how many hands each one shook. I got different answers from everyone. What did my wife say?

If the aspiring VC gives up on this solvable problem, I don't see how he or she will face down the much murkier puzzle of predicting a startup's success.

Update: The solution is presented visually in a nice way here.

44 comments:

  1. Anonymous7:08 AM

    "Why are you asking that bizzare question?"

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  2. Anonymous7:29 AM

    your wife said "That is impossible".

    This is because:
    5 couples = 10 people.
    All Different answers = "9,8,7,6,5,4,3,2,1,0". Sum=45

    So
    1. 9 means someone should his wife's hand.
    2. Also total # of handshakes should be even

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    Replies
    1. You have tallied 10 replies but I am one of the 10 people. I didn't ask myself. So I got back NINE different replies.

      Delete
  3. I have too much fantasy for such a question. My guess: The women kissed everybody. The guys shook hands with the other guys en kissed the women. Your wife shook hands with nobody. You shook hand with the other guys (4) and kissed all the women (4). Unless you didn't know the other people and I would also like to know how much wine you had!

    a lot of wine & unknow couples: hands at the start/kisses at the end

    Too much wine & unknow couples: hands and the start/hugs andd kisses with bot women and men at the end & slap in the face by wife

    So you see: I need more information!

    :-)

    Boris

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  4. ATTENTION: SPOILER

    Some VCs I know use the open source approach :-) Emulating them and mashing it up with Google I went to http://stuff.mit.edu/afs/sipb/user/golem/papers/898/handshake.als

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  5. Anonymous7:44 AM

    Daniel D., the spoiler analysis seems incomplete -- what if Hilary and Jocelyn were p0 and p6?

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  6. Nice problem, but given your ~50% success rate w/ investments, shouldn't you at least give us two chances to get one right? Not sure what a solvable problem has to do with predicting an uncertain future, but it was fun to go from "impossible" to an answer, once you get the trick. (I needed the almost incomprehensible "spoiler" to get the answer).

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  7. David...an interesting question, but moreso in terms of what you are looking for from an "aspiring VC." Wouldn't fluidity of thought and the ability to roll with the punches be integral parts of succeeding in early stage, too?

    So if you asked this of a prospective associate at your firm...and they proceeded to answer with a definitive number [e.g., she shook four hands...the two men and two women of the couples she doesn't know directly], would that be cause to dismiss him outright?

    I would think you would want to hear his ability to identify the variables in play [number of people, men vs. women, level of intimacy with each person, parameters of your wife's normal social demeanor] and then offer a range of outcomes?

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    Replies
    1. You are right. This is not a great interview question.

      Delete
  8. To Anonymous (7:44 am): Jocelyn in the spoiler is David here -- (s)he is asking the question -- (s)he is not one of the 9 persons asked. This singularity, the fact David's spouse is asked for a number but her own spouse (David) is not, is what make her "special" and why there is only one possible answer.

    David might claim there are other things that make his spouse special, but I'll leave that to him.

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  9. Jason,
    There is no range of answers--the only possible answer is indeed 4. But you are right, that in an interview, persistent critical thinking about the problem somewhat compensates for failure to solve.

    Anonymous (7:29) who thinks it's impossible--I said I "surveyed all the other people there," not myself. Just for kicks, I challenge you now to answer how many hands I shook as well.

    Daniel, thanks for the MIT link--I hadn't seen that analysis.

    David

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  10. You shook the same number of hands (i.e. 4). In fact, you shook the very same hands as your wife did!

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  11. Anonymous1:36 PM

    Hmm this is an interesting one.

    If I we were to stick strictly to the facts as provided, wouldn't the answer be 8?

    There are "four other couples," meaning 5 couples in total. No one shakes his or her own hand nor the hand of his or her spouse.

    Assuming that each individual shook hands with every other person except his/her own spouse, this would mean 8 handshakes.

    Four other couples means 8 hands to shake.

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  12. Anon,

    No, 8 is impossible. I never said that everyone shook everyone's hand except his/her own and spouse's. Each of the other 9 people responded with a different number (i.e. 0,1,2...8). Someone who isn't my wife's spouse shook zero hands, so my wife did not shake that person's hand, which means that 8 would be impossible.

    David

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  13. David,

    I appreciate the follow up. I guess having gone through those kinds of interviews years ago, and conducting them in recent years, I'm always interested more in HOW they think and their grace under pressure versus a specific answer to a logical conundrum.

    Jason

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  14. Anonymous3:22 PM

    Jason,

    I think that is exactly David's point. The candidate will illustrate his ability to handle a complex problem through his thinking process.

    Although, David, ironically, if a candidate asks for a hint, that might be interpreted as pure-VC behavior. After all, we succeed when we can identify quickly the person who can best help us evaluate and solve a problem. Who better to ask than the person who posed the question.

    Dennis

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  15. One little itty bitty implicit assumption that we're all taking for granted. We're assuming that all the surveyed guests are accurate in recollecting their handshaking activity. :-)

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  16. Anonymous8:07 PM

    Hmmm. I feel dumb after reading this.

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  17. Anonymous7:39 AM

    Read...User Illusion...by Norretranders.

    I personally don't believe you can predict anything. While I assume there is a mathematical logic to solving the handshaking problem, that while maybe good in helping you justify your investments to your overlords, it will not be of great use in deciding if any one investment is going to succeed.

    Maybe you have gotten used to the average, believing that is a prediction, but it is not...just an average experience, or in your case...better than average.

    Don't try to judge whether it is going to be a successs or not cuz you will not know until the end, but whether the probabilities are high for such an outcome. From that context, you do some computations, you assess the business proposition, you do all the things common sense would prescribe...then if you still like the team, their ability/ the idea/tech, the timing and u see the exit....u go for it.

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  18. So I took a crack at this and I think I got the solution. I put my solution on my blog so spoiler alert if you don't want to see it.

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  19. I had to solve this for a info analysis/problem solving 2nd year course and a friend sent me the link to your site just to torture me I'm sure. It's published in Effective Problem Solving,Second Edition by Marvin Levine (Prentice-Hall). Since I'm still looking for some practical use for the course I'm glad to see it will serve me well should I ever find myself an aspiring VC.

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  20. Anonymous2:06 PM

    I wonder if you extend this same thinking to those who come to you for funding. I would hope that you are looking to fund solutions to seemingly unsolvable problems. (Those problems that, once solved, provide a nifty profit, of course.)

    Would it not, therefore, be more important for a skilled VC to identify a problem and then encourage inventors to provide solutions?

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  21. I guess the Visto guys are still struggling to answer your question : )

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  22. No, there's enough information here.

    We know that there are eight people and that the speaker got a different answer from each of the other seven people. Since the maximum number of hands a person could possibly shake (which is a given) is 6 (everyone else except his/her spouse), we then know that the numbers of handshakes assigned to those seven people are 6, 5, 4, 3, 2, 1, and 0.

    Let's call the couples A1, A2, B1, B2, C1, C2, D1 and D2. (We need not decide which one the speaker is yet.) Without loss of generality, we say that A1 shook six hands (everyone else's except for A2). We need to assign 0 handshakes to *someone*, and the only possibility is A2, so A1=6 and A2=0.

    Again WLOG, we can assert that B1=5 (by shaking A1's hand, along with both C's and D's.) Therefore, only B2 could possibly have only shook one hand (A1's), we also know that B2=1.

    Again WLOG, we assert that C1, having shook the hands of A1 and B1, also shook the hands of D1 and D2 -- and this C1=4. However, this means that there's only one person remaining how shook two hands, and that's C2. So, C1=4 and C2=2.

    Finally, D1 and D2 have each necessarily shaken three hands. And, consequently, if the speaker had gotten a different answer from everyone asked, the speakere must be either D1 or D2, and their spouse is the other. Therefore, the speaker's spouse shook three hands.

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  23. Anonymous1:45 AM

    If you genreralize this to n couples with 0 to 2n as the number of handshakes surveyed, the answer would be n assuming your wife is part of the people you surveyed.

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  24. chris vls11:47 AM

    Great puzzle. Had to go to the whiteboard for it.

    And great area to plumb . . . when do we realize that we can divine more from the available data than what first meets the eye?

    The challenge, imho, with artificial problems is that you get specialized behavior from your candidate when you ask them. In other words, to answer this problem the candidate has to switch into Games magazine logic problem mode. And thus s/he makes it solvable by translating the key sentence "I got different answers" from the colloquial meaning, "no real trend emerged from the data", to a mathematically precise "each person gave a precise, unique, and accurate answer." If I know it's a game, I know I can make the (otherwise unrealistic) translation.

    But this is much easier than what we have to do in the business world. We see lots of ambiguous signs. Some of them turn out to be clear indicators of a deeper trend or structure. Some don't. The real trick is figuring out which is which.

    I'm not sure an artificial problem really tests that skill.

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  25. chris vls11:52 AM

    to continue & conclude. . . This explains why there are people who do very well with logic puzzles but don't have similarly outstanding insight into larger trends, people interactions, etc. . . .

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  26. I just interviewed with Bessemer yesterday and I wasn't asked the handshake question! I was so expecting to blow away the Larchmont partners with my instantaneous riddle-solving skills. Actually, I would have just revealed what an observant and honest blog researcher I am...

    -Ben

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  27. 8-0 eş, 7-1 eş, 6-2 eş, 5-3 eş. Bu aşamaya gelindiğinde artık 8 kişinin el sıkışma sayısı belirlenmiş durumda: 0, 1, 2, 3, 5, 6, 7, 8. Geriye 4 kalıyor. Yani 4-4, yani Bay Smith’in el sıkışma sayısı, yani Bay Smith’in de, Bay Smith’in eşinin de el sıkışma sayısı.

    First of all, I ask the question to everyone, and get a different answer. In other words, I ask 9 people (including my wife) and I should be getting 9 DIFFERENT answers. Since nobody is shaking their own hand or their spouses, teh largest number is 8. Then the answers are 0, 1, 2, 3, 4, 5, 6, 7, and 8. Which means, ONE person shook no hands and ONE person shook 8. WHose hand did the person, who shook 8 hands not shake? His (her) spouse's! In other words, the person who shook 8 hands and the person who shook 0 hands are a couple! Who else, and what answers remain? Take 7. Who can the person who answered 7 NOT have shaken hands with? The person who shook only 1 hand, because the hand of Ms. 1 was already shaken by Mr. 8. So, no more handshakes for Ms.1, who has to be Mr. 7's spouse. The pattern that emerges is that 8-0, 7-1, 6-2, 5-3 are spouses. At this point, we've identified these 4 couples. We're left with 4, who's the spouse of the other 4. I never asked myself the question, so that must be ME. So my wife shook 4 hands.

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  28. First of all, I ask the question to everyone, and get a different answer. In other words, I ask 9 people (including my wife) and I should be getting 9 DIFFERENT answers. Since nobody is shaking their own hand or their spouses, teh largest number is 8. Then the answers are 0, 1, 2, 3, 4, 5, 6, 7, and 8. Which means, ONE person shook no hands and ONE person shook 8. WHose hand did the person, who shook 8 hands not shake? His (her) spouse's! In other words, the person who shook 8 hands and the person who shook 0 hands are a couple! Who else, and what answers remain? Take 7. Who can the person who answered 7 NOT have shaken hands with? The person who shook only 1 hand, because the hand of Ms. 1 was already shaken by Mr. 8. So, no more handshakes for Ms.1, who has to be Mr. 7's spouse. The pattern that emerges is that 8-0, 7-1, 6-2, 5-3 are spouses. At this point, we've identified these 4 couples. We're left with 4, who's the spouse of the other 4. I never asked myself the question, so that must be ME. So my wife shook 4 hands.

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  29. Mike Hoyer9:43 AM

    Great Puzzle. The Answer requires that you re-evaluate what group you and your wife are to satisfy the condition that "I got different answers from everyone." Thus you and your wife both shook the hands of 4 people. :) I used pen and paper, is that allowed? :)

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  30. Interesting handshake problem. I enjoyed the question and the blog. Check out my blog if you have time
    http://niquel757.blogspot.com
    The blog includes good links, paintings, pictures and a mix of interesting and funny posts.

    Regards
    Javier

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  31. Edwin Guzman4:12 AM

    You never shook hands with the 4 couples you went to the party with right? How many people were already at the party?

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  32. Anonymous9:14 AM

    Your Wife said: 'Does it matter?'

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  33. Read the question. Answered reflexively. What did the wife say? She said, "My gawd. You're boring."

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  34. Anonymous3:28 AM

    you had the nearest seat to the ring and you are the only one privy to what your wife said. :-) nice post

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  35. DavidS6:05 AM

    The solution while clever and smart ignores the basic fact that you cannot always rely on people to tell the truth. What if your wife had lied? What if somebody had been worse for wine and answered 42?

    Most logic problems deal in absolutes, but the world is one of probabilities, likelyhoods and levels of confidence. To me that is more important an observation than the ability to solve the problem as stated.

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  36. Mathematically speaking, both you and your wife shook hands with 4 people.

    That is because the "different answers from everyone" does not include you. Cem sertoglu is right!

    This was one of the first questions given in my graph theory class. Cool to see it here. =)

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  37. Anonymous10:50 AM

    The answer is 4 shakes, and you also shook 4 hands.

    Not including you, there were 9 people, and the numbers they shook hands were 8,7,6...0
    It cannot be from 9,8,7... because that implies having a couple shaking hands with each other.

    The person shaking hands with 8 people shook hand with all except with person not shaking with anyone p(0). So they are couple. 8-0
    p(7) didn't shake hands with p(1) because that person already shook hands with p(8), it's only shake. p(7) and p(1) is also a couple.
    p(6) didn't shake hands with p(2), and so on... p(6) is couple with p(2)
    p(5) is couple with p(3)

    That leaves your wife and you with 4 each.

    HR

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  38. Anonymous3:19 PM

    Very nice and interesting blog! and very nice puzzle too!
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    Il viaggiasiti or my website Farm house in Umbria!

    ReplyDelete
  39. Anonymous10:58 PM

    also, you shook the same 4 hands that your wife did.

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  40. Symmetry. (Your wife can just as easily have asked this question, giving the same problem.) The answer is thus 4. K

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