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KingGeorge

[SOLVED] Here's a neat little combinatorics/graph theory question. Mr. Smith and his wife invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse and nobody shook hands with the same person twice. After that, Mr. Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody. How many times did Mrs. Smith shake someone’s hand? PS. This took me about 5 minutes to solve when I was presented with it. Can you beat me?

  • one year ago
  • one year ago

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  1. ParthKohli
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    8 people.

    • one year ago
  2. ParthKohli
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    I mean, if four other couples were invited, then 4*2 = 8

    • one year ago
  3. KingGeorge
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    it's a little more complicated than that. Remember that everyone shook hands a different number of times (with Mr. Smith as the exception).

    • one year ago
  4. ParthKohli
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    Yes, I excluded Mr. Smith.

    • one year ago
  5. KingGeorge
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    Btw, the answer isn't 8.

    • one year ago
  6. KingGeorge
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    Barely legible hint in white LaTeX below.\[ \color{white}{\text{what is the maximum/minimum number of times} \text{someone could shake hands?} }\]

    • one year ago
  7. Ishaan94
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    0? Hmm Total of 10 people and Mr. Smith shook hand with everyone right? 8 is the maximum number of handshakes? and you're asking Nine people. So, 0 for Mrs. Smith?

    • one year ago
  8. KingGeorge
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    It isn't 0 either. Perhaps draw a picture, and start drawing handshakes with some random person. Also, remember that he asked 9 people how many handshakes they did. What is the maximum number of handshakes a single person could have?

    • one year ago
  9. dumbcow
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    isn't it 0 and 8 as max/min you can't shake your own hand nor your spouse, which leaves 8 people

    • one year ago
  10. dumbcow
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    he received a different answer from everyone...0 thru 8 gives 9 different answers right

    • one year ago
  11. KingGeorge
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    That's exactly the conclusion to make. Good work so far.

    • one year ago
  12. KingGeorge
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    Perhaps another leading question. How many people shook hands 8 times? How many 0 times?

    • one year ago
  13. dumbcow
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    i believe its impossible for 2 people to have shook hands 0 or 8 times thus mr. smith shook hands somewhere between 1 and 7 following that logic, i will say 4 times is answer

    • one year ago
  14. KingGeorge
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    Correct. Basically, he didn't shake hands with himself or his wife. So that leaves a maximum of 8. One person shook hands 0 times, so max of 7. One person shook hands 1 time, but it had to have been with the person who shook hands 8 times. So we have a max of 6. Continue this way, and you get a max of 4. At this point, if you draw it out, 5 people have shaken hands 0, 1, 2, 3, 4 times, and 4 people have shaken hands 4, 5, 6, 7 times. To fix this, Mr. Smith has to shake hands with the last 4 people. So 4 shakes.

    • one year ago
  15. KingGeorge
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    Can you figure out how many times his wife shook hands?

    • one year ago
  16. dumbcow
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    well its not 0 for sure...kinda stuck, i think its possible she could be the one to have shaken 8 times ?

    • one year ago
  17. KingGeorge
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    Suppose she did shake 8 times. Who would have shaken 0 times?

    • one year ago
  18. dumbcow
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    oh duh, haha ok so not 8

    • one year ago
  19. dumbcow
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    wait i see the pattern for this to work each couple must complement each other so their sum will be 8 handshakes 0,8 1,7 2,6 3,5 4,4 Mrs. smith shook 4 times as well

    • one year ago
  20. KingGeorge
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    Correct again!

    • one year ago
  21. KingGeorge
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    I was actually going to ask "How many times did each couple shake hands next" :P Looks like you already solved that.

    • one year ago
  22. dumbcow
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    haha :P these are fun...takes me awhile to visualize the problem

    • one year ago
  23. KingGeorge
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    The way I see it for these problems, is to draw it, and to see what happens at extremes. I.e. seeing what would happen if Mr. Smith shook hands 8 times/0 times. Then try to extrapolate.

    • one year ago
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