Give Me a Hand

A professor and their spouse are at a party. At the party there are four more couples (overall five couples).

During the party couples shake hands with the following rules:
a. One does not shake hands with oneself.
b. One does not shake his/her spouse’s hand.

At the end of the party the professor asked all the other guests at the party (including their spouse) how many different people they shook hands with. Each person tells him a different answer (meaning, if one person said “five,” no one else said “five”).

With how many people did the professor shake hands?

Communicated by Michael Geike (who heard it in college in 1999).

105 Comments

  1. Steven Miller on March 2, 2017 at 7:34 pm

    feel free to share but so slammed I have almost no time to do more now



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  3. Steven Miller on October 24, 2016 at 3:42 pm

    yes. –s



  4. kristin on October 24, 2016 at 3:24 pm

    Was there also supposed to be the direction of no one shook hands with the same person more than once??



  5. Steven Miller on August 1, 2014 at 2:22 am

    It can be the same.



  6. Nathan Ash on June 5, 2014 at 4:07 pm

    Can the number of hands that the professor shakes be the same as another guest’s, or does it need to be different?
    We (my class) would love a solution emailed to us….thanks!



  7. Steven Miller on June 29, 2013 at 8:12 pm

    glad you’re enjoying it

    there’s the AMS grad blog: http://mathgradblog.williams.edu/

    another is the math less traveled: http://mathlesstraveled.com/

    Tried emailing you but your address didn’t work.

    maybe why do math: http://www.whydomath.org/index.html



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  9. Indie on June 10, 2013 at 3:09 pm

    I want a riddle to make model on it can anyone help?????????????



  10. Johnny on May 21, 2013 at 1:57 pm

    Heyyy coach Jackson!!! We love you!!!! #AlgebraIsCool



  11. Steven Miller on December 8, 2012 at 1:46 am

    I’ll send a hint



  12. Curious on December 7, 2012 at 6:37 pm

    I would like an email on what the solution is please.



  13. Steven Miller on October 31, 2012 at 1:50 am

    Rage: nope — your second answer is right (sjm1 AT williams.edu)



  14. Rage on October 30, 2012 at 4:38 pm

    5..



  15. Steven Miller on October 1, 2012 at 6:30 pm

    anonymous: please include name/email: correct (sjm1 AT williams.edu)



  16. Steven Miller on September 15, 2012 at 1:14 am

    Nope — sjm1 AT williams.edu



  17. jjb on September 14, 2012 at 11:40 pm

    Has to be 5.



  18. Steven Miller on September 4, 2012 at 1:33 am

    sergio — correct; email me at sjm1 AT williams.edu



  19. Steven Miller on August 30, 2012 at 4:31 am

    Ravi: correct (sjm1 AT williams.edu)



  20. Steven Miller on August 13, 2012 at 7:50 pm

    quiz genious: correct, not posting as it’s the soln (sjm1 AT williams.edu to chat)



  21. Steven Miller on August 13, 2012 at 1:42 am

    quiz genious: correct, but why? (sjm1 AT williams.edu)



  22. Steven Miller on August 3, 2012 at 3:24 am

    [email protected]: correct answer but can’t tell if logic is write as didn’t provide details — email me at sjm1 AT williams.edu to chat further /s



  23. Steven Miller on July 14, 2012 at 3:46 pm

    nope — I’ll email you a hint. //s



  24. Sabina on July 14, 2012 at 9:50 am

    I started with drawing 5 couples
    If the first person shakes one hand, the second two etc and you put a ring on every person that shakes a hand and on the person that get his hand shook

    Then my answer is that the professor and his spouse both only shook hands with two people.



  25. Steven Miller on July 12, 2012 at 12:43 am

    ok



  26. felipe on July 11, 2012 at 2:56 am

    please sent me a hint to my email dont get it



  27. Steven Miller on June 27, 2012 at 4:31 am

    I’ll send a hint //s



  28. Tania on June 26, 2012 at 1:39 pm

    i’m not getting how to solve the problem:(



  29. Steven Miller on June 12, 2012 at 12:55 am

    there is a unique answer



  30. Mike on June 11, 2012 at 11:55 pm

    There’s nothing unique about the professor from any other person there. The answer could be several answers.



  31. Steven Miller on June 6, 2012 at 4:23 am

    Nope. Try doing small cases first (2 couples, 3 couples) and try and find a pattern.



  32. Anonymous on June 5, 2012 at 9:08 pm

    32. he and the other guests all have 2 hands. he shakes both hands from all people with one hand and again with another



  33. Steven Miller on May 7, 2012 at 4:28 am

    he gets her answer as well as the other 8, but doesn’t know which answer is hers.



  34. Mohamed Ilyas A on May 7, 2012 at 3:40 am

    Did the professor asked his spouse that how many different people she shook hands with?
    Or only the 4 other couples?



  35. Steven Miller on May 4, 2012 at 1:10 am

    no — there are only 10 people ./s



  36. Steven Miller on May 4, 2012 at 1:10 am

    no (sjm1 AT williams.edu)



  37. Eihunt on May 3, 2012 at 9:54 pm

    He shook hands with 8 people. right?



  38. Eihunt on May 3, 2012 at 9:50 pm

    Did he shake hands with 16 people?



  39. Steven Miller on April 30, 2012 at 1:23 pm

    Neveron: are you asking or are you telling me for your response? The number you sent *is* correct. sjm1 AT williams.edu



  40. Steven Miller on April 30, 2012 at 12:53 am

    No, a bit higher.



  41. Vicci on April 29, 2012 at 5:09 pm

    Is it 2? /:



  42. Steven Miller on April 26, 2012 at 6:27 pm

    It is not 10. He is not single.



  43. Anonymous on April 26, 2012 at 6:24 pm

    Ten. He is single



  44. Steven Miller on April 26, 2012 at 6:01 pm

    Nope, not 10. Also, he’s not single /s



  45. Anonymous on April 26, 2012 at 5:54 pm

    the answer os ten. FIVE couples, ten people, ten hand shake. He is single.



  46. Steven Miller on April 24, 2012 at 9:55 pm

    no — try doing special cases (just 2 couples, just 3 couples, …)



  47. Jann on April 24, 2012 at 6:48 pm

    the answer is 2



  48. Steven Miller on April 24, 2012 at 2:54 am

    anonymous: email me at sjm1 AT williams.edu



  49. Anonymous on April 24, 2012 at 1:36 am

    person 1 shook 1 person
    person 2 shook 2 people
    person 3 :3, person 4:4 and so on until person 8 shook 8 people



  50. Steven Miller on March 14, 2012 at 1:20 am

    nope. try special cases: 2 couples, then 3 then 4 //s



  51. Jakwanul Safin on March 14, 2012 at 1:03 am

    he shook hands with 0 people.



  52. Steven Miller on March 8, 2012 at 2:03 am

    This is a bit hard to follow. Try having just 2 couples, and go through all the possibilities of who shook 0 hands. Then do the same with 3 couples, then 4….



  53. Anonymous on March 7, 2012 at 11:29 pm

    Ok so let me get this straight, with two couples there are 3 answers-0,1,2, So if one of them says 0 then its safe to not count them so there are only 3 people that interchanged handshakes,if the proffesors wife says 0 then the proffesor shook two hands but if the other couples spouse says 0 then the professor only shook one hand….

    With three couples the same thing happens, if the professors wife says 0 then he shook 4 hands but if someone else said 0 then he only shook 3 hands.

    With four couples he either shakes 6 hands or he shakes five hands.

    and so on and so fourth……



  54. Steven Miller on February 25, 2012 at 1:48 am

    no, a bit higher — try doing just 2 couples, then just 3, then just 4. look for a pattern. //s



  55. Drew Graham on February 25, 2012 at 12:53 am

    Is 1 the correct answer



  56. Steven Miller on February 16, 2012 at 3:32 am

    nope — try smaller cases (just 2 couples, then just 3, then just 4). //s



  57. Mo on February 15, 2012 at 5:56 am

    Is it 2?



  58. Steven Miller on February 14, 2012 at 4:29 pm

    There’s a unique answer. The number of hands he shakes doesn’t depend on the order. It only depends on the restrictions of the problem (that one shaked exactly 0, one exactly 1, …).



  59. Melchor Castro on February 14, 2012 at 4:19 pm

    if there is five couples it depends in the order the profesor starts shaking hands if he goes
    1st he would shake 8 hands
    2nd he would shake 7 hands
    3rd he would shake 6 hands
    4th he would shake 5 hands
    If he is the last coulpe to shake hands he would only have 4 hands he has not shake



  60. Steven Miller on February 13, 2012 at 1:46 am

    no one left until the problem was solved.//s



  61. Sara on February 12, 2012 at 11:48 pm

    did anybody leave the party?



  62. Steven Miller on February 8, 2012 at 1:37 pm

    nope — try doing just 4 people, then just 6, then just 8….



  63. leah on February 8, 2012 at 11:57 am

    if im not mistaken, the answer is zero



  64. Steven Miller on February 8, 2012 at 1:52 am

    send your soln to me ([email protected]) and I’ll take a look at it. //s



  65. ed on February 7, 2012 at 7:58 pm

    I think I have it, but is the final answer just one answer or is it a depended of how others responds. In my solution I proved that the above is possibe however, the answer may vary.



  66. Steven Miller on January 24, 2012 at 3:12 am

    no, lower. try doing just 2 couples, then just 3 couples….



  67. Jean on January 23, 2012 at 10:50 pm

    Is it 9?



  68. Steven Miller on January 18, 2012 at 7:46 pm

    No, not 3. try special cases — try just 2 couples, then 3, then 4….



  69. Deshani on January 18, 2012 at 12:03 pm

    is the answer 3?



  70. Steven Miller on December 30, 2011 at 9:25 pm

    It’s not zero. Email me at [email protected] if you want a hint.



  71. Naj on December 30, 2011 at 9:23 pm

    0



  72. Steven Miller on December 7, 2011 at 1:42 am

    lower than 8. email me ([email protected]) for a hint



  73. Brandon on December 7, 2011 at 12:06 am

    8



  74. Steven Miller on November 17, 2011 at 3:03 pm

    I’m not sure, as you haven’t provided enough details. Do the case of 2 couples (4 people) very carefully. The person knows that among the three remaining people (which includes his spouse), one shook 0 hands, one shook 1 hand, and one shook 2 hands. You will see that the person ended up shaking 2 hands (as did his spouse). If you do three couples, you’ll find he shook 3 hands, NOT 4. One way to attack this is to ask: whom shook the maximum number of hands? You’ll find it isn’t the spouse, and then the person who shook 0 hands is the spouse of the one who shook the maximum.



  75. Debbie on November 17, 2011 at 2:37 pm

    I don’t see how it can’t be that he shook hands with 8 people. I first did 2 couples (4 people) results in 2 handshakes, 3 couples 4 handshakes, 4 couples 6 handshakes. The pattern I see is that the number of handshakes is always 2 less than the total number of people. This would make sense given that the professor doesn’t shake with himself or his spouse thus 10 people 8 handshakes. What am I missing? Is your definition of a couple different than 2 people?



  76. Steven Miller on November 11, 2011 at 2:18 am

    try doing smaller cases first — imagine there are just 4 people total, then 6, ….



  77. evan on November 10, 2011 at 9:31 pm

    5??? I really dont know.



  78. Steven Miller on November 7, 2011 at 2:29 pm

    yes, the prof can (and in fact MUST) say the same as someone else



  79. Hint on November 7, 2011 at 9:21 am

    I have a question about this:

    Can Professor say same number than someone else said or can his spouse?
    Cause with 2 couples there is only 2 possible amount of shakes, cause u can shake only with 0, 1 or 2 peoples and if someone shakes “0”, no-one cant shake 2 anymore. So with this function there is always 2 less ansrews that peoples.



  80. Steven Miller on November 5, 2011 at 1:55 am

    I think it must be zero, because everyone else would take the answers 1, 2, 3, 4, 5, 6, 7, and 8. The trouble is, I don’t know if anyone else said zero. I’m a 6th grade teacher posting this for my students, so if I don’t figure it out, I’m sure one of them will!



  81. Heather on November 4, 2011 at 9:59 pm

    I think it must be zero, because everyone else would take the answers 1, 2, 3, 4, 5, 6, 7, and 8. The trouble is, I don’t know if anyone else said zero. I’m a 6th grade teacher posting this for my students, so if I don’t figure it out, I’m sure one of them will!



  82. Steven Miller on November 1, 2011 at 4:50 am

    he doesn’t have to — he cannot shake his own or his spouse’s hand



  83. Shreya on November 1, 2011 at 3:50 am

    Does he have to shake hand to all except self and spouse?



  84. Steven Miller on October 30, 2011 at 1:13 am

    No, it isn’t 7. try doing just 2 couples, then 3, and see if you can detect a pattern.



  85. Steven Miller on October 30, 2011 at 1:12 am

    I’m having internet trouble tonight and can’t send email. Email me at [email protected] and let me know which riddle.



  86. sarbani on October 29, 2011 at 3:03 pm

    is the answer 7



  87. sarbani on October 29, 2011 at 3:02 pm

    i want to know the answer



  88. Steven Miller on October 19, 2011 at 5:13 am

    If you shake someone’s hand, your shake count goes up by 1, as does theirs; it does not matter how many people either of you have shaken.



  89. William Fox on October 19, 2011 at 5:05 am

    Im confused if he shakes a hand does it accumulate with the numbers he has shaken? So, he shakes hands with some who has shook five hands does that count as six? If i had to guess 4 as he doesnt shake the hands of spouses?



  90. Steven Miller on October 17, 2011 at 1:13 am

    Nope. start with two couples, then do three couples….



  91. Linh on October 16, 2011 at 11:53 pm

    6?



  92. Steven Miller on August 23, 2011 at 6:04 pm

    Try special cases. Imagine there are just 4 people total. Then try 6 people….



  93. Brown on August 23, 2011 at 3:17 pm

    ths riddle is tricky send me a clue



  94. Steven Miller on August 18, 2011 at 2:00 am

    correct again, not posting as it is the soln



  95. Steven Miller on July 28, 2011 at 2:50 am

    Richard: not posting your solution as it is correct; well done!



  96. Steven Miller on July 2, 2011 at 5:50 am

    it’s possible. it’s not 9 answers for 10 people.



  97. Benny on July 2, 2011 at 5:48 am

    Is it possible? There would be 10 people if you exclude yourself and your spouse you leave 8 potential people to shake hands with. Assuming you only shake hands with each person once you have the answers to the question 0-8 which is only 9 answers for 10 people so one person would either have to shake hands with the same person twice to enable 10 different answers or two people have to give the same answer. Unless you allow polygamy



  98. Steven Miller on June 23, 2011 at 2:08 am

    yes — don’t shake the same hands twice ..s



  99. mark on June 22, 2011 at 8:07 pm

    Can I assume that they don’t shake the same hand twice?



  100. Steven Miller on June 16, 2011 at 1:36 pm

    I’ll email you as I don’t want someone to see the hints by mistake.



  101. jihadhamwi on June 16, 2011 at 1:30 pm

    Hi, i send you the message about helping me with some riddles for example this one, fruitful problem, bridge over troubled students, and a peeny for your thoughts.
    Please repley.



  102. Steven Miller on May 23, 2011 at 3:20 am

    Just email me at [email protected] for hints / solns



  103. Steven Miller on May 23, 2011 at 3:13 am

    Nope. The professor did not shake hands with 8. If you want a hint, let me know.



  104. The Queen's Class on May 20, 2011 at 5:02 pm

    oh…and we need some answers before June 3rd…if possible!!!!



  105. The Queen's Class on May 20, 2011 at 4:14 pm

    We know you don’t print answers, but we need to know if we are right….

    The professor shook hands with everyone but himself and his spouse which would mean he shook hands with eight people!

    Yes????



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